A switching approach for perfect state transfer over a scalable and routing enabled network architecture with superconducting qubits
IIndian Institute of Technology KharagpurDepartment of Mathematics and Department of PhysicsA switching approach for perfect state transfer over a scalable androuting enabled network architecture with superconducting qubits
Master Thesis Project (PH57002)
Student:
Siddhant Singh (15PH20030)
Supervisors:
Bibhas AdhikariSonjoy Majumder
July 7, 2020 a r X i v : . [ qu a n t - ph ] J u l bstract We propose a hypercube switching architecture for the perfect state transfer (PST) where we prove that itis always possible to find an induced hypercube in any given hypercube of any dimension such that PST canbe performed between any two given vertices of the original hypercube. We then generalise this switchingscheme over arbitrary number of qubits where also this routing feature of PST between any two vertices ispossible. It is shown that this is optimal and scalable architecture for quantum computing with the feature ofrouting. This allows for a scalable and growing network of qubits. We demonstrate this switching scheme tobe experimentally realizable using superconducting transmon qubits with tunable couplings. We also proposea PST assisted quantum computing model where we show the computational advantage of using PST againstthe conventional resource expensive quantum swap gates. In addition, we present the numerical study ofsigned graphs under Corona product of graphs and show few examples where PST is established, in contrastto pre-existing results in the literature for disproof of PST under Corona product. We also report an errorin pre-existing research for qudit state transfer over Bosonic Hamiltonian where unitarity is violated.1 ontents
Perfect State Transfer for qudit (d-level systems) networks 47 ist of Figures s tothe receiver spin r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Part (a) shows the first four Cartesian product of K or Q with itself. These are the first fivehypercubes. The new edges formed due to the fourth Cartesian product (cid:3) ( G ) is shown inred for clarity. Part (b) shows the first two Cartesian product of two-link graph G with itself. 232.2 The corona product of G ◦ G is shown in (c), (d) with canonical and plurality markingfunctions on G i , i = 1 ,
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Examples of the Corona graphs (a) Seed graph K (b) Corona graph for G (1) , and (c) Coronagraph for G (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 2-clique signed graph. Simplest case for a signed graph that is net-regular and balanced withmarkings as shown. All edge weights are ±
1, with solid lines as +1 and dotted lines as -1. . . 283.2 First corona product ( m = 1) of signed 2-clique. . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Plot of maximum fidelity max[ F ( t )] vs m between nodes 1 & 3 and 2 & 4. . . . . . . . . . . . 293.4 Another non-trivial example forming a signed hexagon K . . . . . . . . . . . . . . . . . . . . 293.5 Another non-trivial example forming a signed octagon K . Near perfect state transfer is possible. 303.6 First non-tivial example that satisfies the theorems and is unbalanced. . . . . . . . . . . . . . 303.7 Another variation of signed net-regular K with more connectivity. . . . . . . . . . . . . . . . 313.8 Two conjugate signed versions of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.9 Another variation of signed net-regular K with more connectivity. . . . . . . . . . . . . . . . 323.10 Another variation of signed 2-clique K with more connectivity. . . . . . . . . . . . . . . . . . 323.11 Another variation of signed net-regular K with more connectivity. . . . . . . . . . . . . . . . 323.12 Typical flow of fidelity F ( t ) w.r.t. time t for a pair of nodes of example 9. . . . . . . . . . . . 333.13 Non-cyclic signed graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1 Quantum circuit for our CQC-hopping scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 456.1 The part (a) shows the isolated pair of coupled qubits with the ancillary coupler acting as theinter-mediator. The situation for every pair that has an edge between them is the same. Part(b) shows our qubit architecture for n = 6 qubits with the ancillary couplers involved. Thecontrol parameter for edge switching here is the external control on the capacitance for eachcoupler, denoted as yellow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.2 Schematic circuit diagram for a pair of connected tunable transmon qubits. Each connectedpair which forms an edge on the graph has this structure. . . . . . . . . . . . . . . . . . . . . 566.3 Variation of the dynamic tunable coupling ˜ g ij w.r.t. the detuning ∆ i for each qubit. Thereexists a cutoff value, in this case ∆ i = − .
426 GHz, corresponding to ω cij off = 5 .
426 GHz. Forall configurations, such a cut-off value can always be obtained. . . . . . . . . . . . . . . . . . 586.4 Variation of the Perfect State Transfer time 2 t w.r.t. qubit detuning ∆ i for all qubits. Thetypical PST time is around 1.5 ns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 hapter 1 Introduction to Quantum PerfectState Transfer (PST)
In quantum computation, it is often required to transfer an arbitrary quantum state from one site to another[1]. These two sites may belong to the same quantum processor or different processors. The latter oneis not trivial for many quantum information processing (QIP) realizations, such as, solid state quantumcomputing and superconducting quantum computing [2]. This is because the state transmission channel isnot a computational space for either processor and cannot involve manipulation. In large scale quantumcomputation, it is a very important task to be able to transfer a quantum state within a processor as wellas between two physically distant QIP processors with robust transmission lines. It is also important to findthe physical systems also which support this quantum information exchange between distant sites. For shortdistance communications (say between adjacent quantum processors), alternatives to interfacing differentkinds of physical systems are highly desirable and have been proposed, for example, for ion traps [3][4],superconducting circuits [5][6], etc. The task of state transfer is thought with the intention of reducing thecontrol required to communicate between distant qubits in a quantum computer [7].Quantum state transfer with 100% fidelity is known as perfect state transfer (PST) and this idea usinginteracting spin-1/2 particles was first proposed in [8] and established the connection between graph theoreticnetworks and actual quantum networks for quantum processors in the first excitation subspace of many-body qubit network [9]. For graph structures, usually the XY coupling Hamiltonian and the Heisenbergspin interaction are considered. Due to this connection, a quantum architecture can be designed purelyin graph theoretic fashion (determining the qubits’ mutual connectivity) and can be realized by physicalsystems. It was established that PST is possible in spin-1/2 systems and other bosonic networks withoutany additional action and manipulation from senders and receivers [10]. PST only requires access to twospins at each end of the spin network while all other spins in the network act like a channel for transferand are not computational spins. In general, this involves mixed states of the network qubits [11], however,showing PST for pure states in a graph suffices to prove the phenomenon. PST can be used in entanglementtransfer, quantum communication, signal amplification, quantum information recovery and implementationof universal quantum computation [12][10][13][14].PST in graphs is a rare phenomenon and only very few graphs and class of graphs are known to exhibitthe phenomenon of PST. For this reason, the idea of pretty good state transfer is also studied, where thefidelity is a little less than unity but offers a large number of graphs that support state transfer [15][16][17].The task is to find graph structures which support PST for as many pair of vertices as possible and possiblygrow under some operation (scalability of networks). It is important to find the class of graphs where itoccurs and equally important to find graphs where it does not occur [18][19]. Researchers aim to find classof graphs as well as as various products of graph to establish a growing scalable network supporting PST[20][21], and in general these graphs can be weighted [22]. More general graphs such as signed graphs [23]and oriented graphs [24] are also studied. In this way, we essentially define a quantum computig architecture.PST for qudits or higher dimensional spins over weighted graph is also classified for some networks [25][26].This was further developed for arbitrary states and large networks in [20] and [21].5ST scheme established in [20] and [21] allows PST over arbitrary long distances with the use of Cartesianproduct of one-link and two-link graphs which support PST under the XY as well as Heisenberg interactionof spins. It is known that one-link and two-link chain graphs exhibit PST between the end vertices [20].And this feature carries over to the antipodal vertices of the resulting Cartesian product of these graphswith themselves, which become the pair of vertices exhibiting PST in the same time. First shortcoming inthis model is that of the impossibility of routing [27][28]. The second being that only a pair of antipodalvertices of the graph support PST which becomes less useful as the graph scales to a larger network. Thiswould involve constructing a very large network just to enable PST between a pair of antipodal qubits of anetwork. Third being that this architecture scales the number of qubits with the factor of 2 which will bevery large gap for the larger dimensional hypercubes as the network scales up. This motivates for finding aquantum computing architecture which would allow us routing to any given vertex of the graph as well asenables arbitrary number of vertices while still preserving perfect fidelity and routing to any vertex startingfrom any other given vertex. A switching was proposed in [29] where in a complete graph K n , switching offone link establishes PST in non-adjacent qubits. This enables PST for more vertices but still does not enablerouting to different vertices and there is no scalability, the graph remains fixed. One attempt at switchingand routing is proposed in [30] which involves creating new edges and coupling for qubits, however, is stillnot scalable. Routing in special regular graphs was proposed in [31][32]. It leads us to the motivation thatonly quantum mechanical processes may not not be sufficient to fulfill our requirements, that is the perfectstate transfer between any two vertices of a graph of arbitrary number of vertices. Therefore, we propose ahybrid of classical combinatorial and quantum information theoretic method, such that, a perfect quantumstate transfer is possible between any two vertices of the graph.In this work, we propose a solution to both the problems of routing and scalability for quantum architec-ture where the network will be enabled with PST from all-to-all nodes for any arbitrary number of qubits,which fits with the idea of Noisy Intermediate-Scale Quantum (NISQ) processors [33]. And this task can beaccomplished in just two quantum operations only. Our architecture also features the addition of any arbi-trary number of qubits in an already constructed network according to our scheme while preserving both theproperties. Thus, this is a possible and optimal solution for a scalable architecture for quantum informationprocessing. Our results in this work hold both for XY-coupling as well as the Laplacian interaction Hamilto-nian. We also propose the idea of PST assisted quantum computation where PST can be used in contrast tolarge number of SWAP gates between any two distant qubits when the quantum circuit depth is very highand thereby reducing the complexity of a large quantum circuit. This involves PST over the computationalqubits of a quantum processor. We also present analytically and simulate numerically the experimental im-plementation of our architecture using superconducting circuits thereby showing the implementation of PSTin superconducting circuits for the very first time. Apart from these main results, we also report the PST inqudit systems and analysis of PST under Corona product of certain special graphs. Chapters 1 and 2 formthe part of the preliminary literature and chapters 3,4,5 and 6 are the original contributions from this thesis. The idea for perfect state transfer of arbitrary states is to establish the connection between the graph theoreticapproach and spin Hamiltonian. The system of spins can be translated into a corresponding graph wherethe dynamics can be explored by the structure of the graph governed by its adjacency marix and Laplacianwhich are in one-to-one correspondence with the connectivity of the spins in the physical picture. Arbitrarystate of spin in a lattice is simply a qubit state. The principle problem is that the Hilbert space of a graph G with n vertices is given by C n , and the Hilbert space of a spin-1/2 (or generally any many-body qubitnetwork) particle attached to each vertex of the same graph G is C n , which is exponentially larger. Graphsand their products will be discussed in detail in chapter 2. For spin-dynamics we have generally two kind ofinteraction Hamiltonians: The XY coupling adjacent interaction and the Heisenberg interaction model. Wewant to establish a connection between the dynamics of n number of spin-1/2 particles interacting accordingto these two kind of interactions on a graph G and the structure of G itself.The central idea of this equivalence is that complicated physics of a system of distinguishable spin-1/2particles interacting pairwise on a simple geometry given by an undirected simple graph G are equivalent tothe sometimes physics of a single free spinless particle hopping on a much complicated graph G (which is somedisjoint union of the graphs G k , which are related to G ) [9]. This can be understood as direct application ofa special graph product, called the wedge product of graphs which is discussed in section 2.3.1.Consider n = | V | distinguishable spin-1/2 subsystems, each at one vertex of the graph G ( V, E ), where6 ( G ) is the finite vertex set and E ( G ) is the edge set (a two-element collection of vertices) of the graph. Wesay distinguishable spins because we are able to label them with the labeling of the graph vertices. We fixa labeling i = 1 , , ..., n of the vertices. This labeling induces an ordering of the vertices which we write as v i > v j if i > j , where v i is the i th vertex of the graph G . The degree deg( v i ) of a vertex v i is equal to thenumber of edges which have i as an endpoint. The adjacency matrix A ( G ) for G is the { } -matrix of size | V ( G ) | × | V ( G ) | which has a 1 in the ( i, j ) entry if there is an edge connecting v i and v j . Define the Laplacian L ( G ) of the graph as D ( G ) − A ( G ) where D ( G ) is the degree matrix for G defined as D ( G ) = diag { deg( v i ) } .We also define the Hilbert H ( G ) space of the graph G to be the vector space over C generated by theorthonormal vectors | i (cid:105) , ∀ i ∈ V ( G ), with the canonical inner product (cid:104) i | j (cid:105) = δ ij . For details in graph theory,refer to chapter 2. Now we define the two mentioned interaction Hamiltonian for the pairwise interactionsbetween the spins. The first is the XY model in two spatial degrees of freedom, H XY = (cid:88) ( i,j ) ∈ E ( G ) J ij (cid:0) σ xi σ xj + σ yi σ yj (cid:1) = (cid:88) ( i,j ) ∈ E ( G ) J ij (cid:0) σ + i σ − j + σ − i σ + j (cid:1) , (1.1)= 12 (cid:88) (cid:104) i,j (cid:105) (cid:0) σ xi σ xj + σ yi σ yj (cid:1) (for J ij = 12 ∀ (cid:104) i, j (cid:105) ) (1.2)where (cid:104) i, j (cid:105) denotes an adjacent pair of vertices on the graph (which have an edge between them) and σ + , − i are the ladder operators for the i th spin (qubit) such that σ xi = σ + i + σ − i and σ yi = iσ + i − iσ − i , with σ x,yi asthe Pauli matrices for spin-1/2 system at the i th vertex. The second connection is via the three-dimensionalHeisenberg model, H Hei = − (cid:88) ( i,j ) ∈ E ( G ) J ij (cid:126)σ i · (cid:126)σ j + (cid:88) j B j σ zj (1.3)= − (cid:88) (cid:104) i,j (cid:105) ( (cid:126)σ i · (cid:126)σ j − I i I j ) (for J ij = 12 ∀ (cid:104) i, j (cid:105) ) (1.4)where, (cid:126)σ i is Pauli matrix vector (cid:126)σ i = ( σ xi , σ yi , σ zi ) for the i th spin and I i is the identity operator for the i thvertex. We take J ij = 1 / B j at the j th site such that (cid:80) j B j σ zj = I (which makesthe Hamiltonian coincide with the Laplacian of the graph)for the uniformly coupled system with edge weightunity for both the models.There is one peculiar conservation property of both of these Hamiltonians that they conserve the totalspin along the z -axis of the whole system. Formally, we define S z := (cid:80) i ∈ V σ zi , and it can be verified that (cid:2) S z , σ xi σ xj + σ yi σ yj (cid:3) = 0 ∀ i, j ∈ V (1.5)and also [ S z , (cid:126)σ i · (cid:126)σ j ] = 0 ∀ i, j ∈ V. (1.6)For Heisenberg Hamiltonian, it even commutes with total spin along x - or y -axis also, or generally alongany axis with suitably defined spin operators along that axis. These commutation relations are enough toestablish the idea of different excitation subspaces for the action of the Hamiltonian. For example, if thesystem had two spins excited and all other in ground state, then throughout the quantum evolution of thesystem under the spin Hamiltonian will conserve this total spin as two excitations. Therefore, the action of H XY and H Hei breaks the Hilbert space
H ∼ = C n into a direct sum H ∼ = n (cid:77) k =0 Γ k (1.7)where the vector subspaces Γ k constitute the elements as followsΓ = {| ... (cid:105)} , Γ = {| ... (cid:105) , | ... (cid:105) , ..., | ... (cid:105)} , Γ = {| ... (cid:105) , | ... (cid:105) , ..., | ... (cid:105)} , ...Γ n = {| ... (cid:105)} . (1.8)7his implies that starting with a ket in Γ k , the system will evolve in the linear combination of vectors inΓ k strictly. Projectors P k can be defined for each Γ k , then the total spin for each Γ k is Tr( S z P k ) = k (use standard basis) which is also justified by the physical argument for total spin in each subspace. Thedimension of Γ k is dim (Γ k ) = (cid:18) nk (cid:19) (1.9)We also note that the subspace Γ matches exactly with the vertex space of the graph G itself. Dynamics inthis subspace is simply the hopping between different vertices while at each point in time one of the verticesis excited. There is a deep relationship between the subspaces Γ k and the exterior vector spaces via thewedge product. The vector space Γ k is generated by the k th wedge product of G denoted as ∧ k ( H G ). Forthe first wedge product of G , we get Γ and action of corresponding adjacency matrix A ( G ) is identical tothe action of the Hamiltonian H XY . And the action of the graph Laplacian is identical to the action of theHeisenberg hamiltonian H Hei . In the first excitation space, we denote these Hamiltonians as H XY and H Hei respectively. For general higher order wedge product we formulate it in section 2.3.1.
Simplest and fundamental system for the construction of network for perfects transfer are the ferromagneticchains (or network) in Heisenberg model for spins. Consider the general graph shown in figure 1.1, where thevertices are spins and the edges connect spins which interact. Say there are n spins in the graph and theseare labeled 1 , , ..., n . The Hamiltonian is given by H Hei = − (cid:88) (cid:104) i,j (cid:105) J ij (cid:126)σ i · (cid:126)σ j − n (cid:88) i =1 B i σ zi (1.10)as before where (cid:126)σ i = ( σ xi , σ yi , σ zi ) are the Pauli spin matrices for the i th spin, B i > J ij > (cid:104) i, j (cid:105) represents pairs of adjacent spins which are coupled. H G describes an arbitrary ferromagnet with isotropic Heisenberg interactions. We now assume that the statesender A is located closest to the s th (sender) spin and the state receiver B is located closest to the r th(receiver) spin (these spins are shown in figure 1.1. All the other spins will be called channel spins as they areinvolved in transferring the state of the qubit (spin) identical to a quantum channel. In the original idea [8],it is also assumed that the sender and receiver spins are detachable from the chain. In order to transfer anunknown state to Bob, A replaces the existing sender spin with a spin encoding the state to be transferred.After waiting for a specific amount of time, the unknown state placed by A travels to the receiver spin withsome fidelity. B then picks up the receiver spin to obtain a state close to the the state Alice wanted totransfer. As individual access or individual modulation of the channel spins is never required in the process,they can be constituents of rigid 1D magnets (for instance).Perfect transfer of a state in a many-qubit system modeled as a combinatorial graph in which the edgesof the graph represents coupling of qubits, is defined by starting with a single qubit state ρ A qubit on somevertex A , with ρ in in the state of the rest of the qubits, and after evolution for some time t under a fixedHamiltonian H , the output state e − iHt ( ρ A qubit ⊗ ρ in ) e iHt = ρ B qubit ⊗ ρ out (1.11)is produced, thereby transmitting the input qubit to another desired vertex B of the graph. In general, ρ qubit is a density matrix, however, in this thesis we consider that it corresponds to a pure state (which is sufficientto demonstrate the idea of PST). The most simplified case for such realization is the one-dimensional chainof qubits.We assume that initially the system is initially cooled to its ground state | (cid:105) = | ... (cid:105) where | (cid:105) denotesthe spin down state (i.e., spin aligned along − z direction) of a spin. This is shown for a 1D chain in theupper part of figure 1.1. We set the ground state energy E = 0 (i.e., redefine H G as E + H G ). We alsointroduce the class of states | j (cid:105) = | ... .... (cid:105) (where j = 1 , , ..s, ..r, .., n , i.e., the vertex space of G ) inwhich the spin at the j th site has been flipped to th | (cid:105) state. To start the protocol, A places a spin in the8nknown state | ψ in (cid:105) = cos( θ/ | (cid:105) + e iφ sin( θ/ | (cid:105) at the s th site in the spin chain. We can describe thestate of the whole chain at this instant (time t = 0) as | Ψ(0) (cid:105) = cos( θ/ | (cid:105) + e iφ sin( θ/ | s (cid:105) (1.12) B wants to retrieve this state, or a state as close to it as possible, from the r th site of the graph. Then he A BA B (a) (b) NN-1 N-2
Figure 1.1: The part (a) of the figure shows the quantum communication protocol. Initially the spin chainis in its ground state in an external magnetic field. Alice and Bob are at opposite ends of the chain. Aliceplaces the quantum state she wants to communicate on the spin nearest to her. After a while, Bob receivesthis state with some fidelity on the spin nearest to him. Part (b) shows an arbitrary graph of spins throughwhich quantum communications may be accomplished using this protocol. The communication takes placefrom the sender spin s to the receiver spin r .has to wait for a specific time till the initial state | Ψ(0) (cid:105) evolves to a final state which is as close as possibleto cos( θ/ | (cid:105) + e iφ sin( θ/ | r (cid:105) . As [ H G , (cid:80) i σ i ] = 0, the state | s (cid:105) only evolves to states | j (cid:105) and the evolutionof the spin-graph (with (cid:126) = 1) is | Ψ( t ) (cid:105) = cos( θ/ | (cid:105) + e iφ sin( θ/ n (cid:88) j =1 (cid:104) j | e − iH Hei t | s (cid:105)| j (cid:105) (1.13)For the Perfect State Transfer (PST) to occur from the s th site to the r th site, we should have (cid:104) r | e − iH G t | s (cid:105) := f nr,s = e iη (where the phase e iη can be compensated and corrected later by B ) or simply | f nr,s | = 1 for somefinite t = t , is enough for pure states. The state of the r spin will, in general, be a mixed state, and can beobtained by tracing off the states of all other spins from | Ψ( t ) (cid:105) . This evolves with time as ρ out ( t ) = P ( t ) | Ψ out ( t ) (cid:105)(cid:104) Ψ out ( t ) | + (1 − P ( t )) | (cid:105)(cid:104) | (1.14)where | Ψ out ( t ) (cid:105) = 1 (cid:112) P ( t ) (cid:18) cos θ | (cid:105) + e iφ sin θ f nr,s ( t ) | (cid:105) (cid:19) (1.15)with P ( t ) = cos ( θ/ ( θ/ | f nr,s | is the normalization of the state at any time t and f nr,s = (cid:104) r | e − iH Hei t | s (cid:105) .Note that f Nr,s ( t ) is just the transition amplitude of an excitation (the | (cid:105) state) from the s th to the r th site ofa graph of n spins. It is also equal to the fidelity between these states for the case of pure states. It is decidedthat B will pick up the r th spin (and hence complete the communication protocol) at a predetermined time t = t . We show later in chapter 6 that the phenomenon of perfect state transfer is not only restricted tospin Hamiltonians but is a general property of many Hamiltonians which are similar in action with the H XY or H Hei models. We show that superconducting transmon qubit network Hamiltonian also allows perfectstate transfer along with the power of quantum computation.9 .4 Review on fidelity and distance measures
Fidelity is a measure of how apart two states are in the Hilbert space, or to say, simply the overlap of states(see chapter 9 in [34] for distance measures). Fidelity for two density matrices ρ and σ is defined as F ( ρ, σ ) = (cid:113) √ σρ √ σ ∈ [0 ,
1] (1.16)which takes the form (when σ = | ψ (cid:105)(cid:104) ψ | , that is, a pure state) of F ( ρ, σ ) = (cid:112) (cid:104) ψ | ρ | ψ (cid:105) . (1.17)We can treat σ = | ψ (cid:105)(cid:104) ψ | as the final desired (pure) state of the system and ρ as the noisy obtained state(after an evolution) and check the fidelity of the two as a check for closeness. Fidelity is related to the tracedistance D ( ρ, σ ) by the following mathematical relations1 − F ( ρ, σ ) ≤ D ( ρ, σ ) ≤ (cid:112) − F ( ρ, σ ) (1.18)where the trace distance is defined as D ( ρ, σ ) = Tr | ρ − σ | . They can be used interchangeably but we willstick with the measure of the fidelity for the purpose of perfect state transfer. First let us recall that for the general Hamiltonian (for the first excitation subspace) H = (cid:88) i
G, σ ). We assume that σ is balanced or anti-balanced signing of G . Then dueto lemma 1 in [23], we have that if a graph G has perfect state transfer, then so does the signed graphΣ = ( G, σ ). Suppose G has perfect state transfer from vertex A to B . If σ is a balanced or antibalancedsigning of G , then there is a diagonal ± A ( G σ ) = ± Θ − A ( G )Θ. Thus, we have (cid:104) A | e − iA ( G σ ) t | B |(cid:105) = (cid:104) A | Θ − e ± iA ( G ) t Θ | B (cid:105) = ±(cid:104) A | e ± iA ( G ) t | B (cid:105) . (1.45)Therefore, G σ has perfect state transfer from A to B .Directed or oriented graphs can be defined in similar manner. A directed graph (or digraph) is a graphthat is made up of a set of vertices connected by edges, where the edges have a direction associated withthem. A directed graph is an ordered pair G = ( V, A ) where • V denotes the vertex set, and • E is now a set of ordered pairs of distinct vertices, called directed edges.For oriented graphs, the direction of edges is modeled with a sign. Hence, the adjacency matrix is skewsymmetric. We take the adjacency matrix of an oriented graph to be the matrix A with rows and columnsindexed by the vertices of the graph, and A ij equal to 1 if the the edge { i, j } is oriented from i to j , equal to − { i, j } is oriented from j to i , and equal to zero if i and j are not adjacent. Consequently A ( G ) is skew symmetric [24]. If A ( G ) is a skew symmetric matrix, then iA ( G ) is Hermitian and so U ( t ) = exp( − it ( iA ( G ))) = exp( tA ( G )) (1.46)is the transition matrix of a continuous quantum walk (a quantum evolution with adjacency or Laplacianmatrix over the vertex space of a graph is simply a continuous quantum walk). We note that U ( t ) is realand orthogonal. Similarly, a general weighted graph can be obtained if weight w ( i, j ), some real number, isassigned to each edge ( i, j ) for all possible edges. Weighted chain for perfect state transfer in last chapterwas an example of weighted graph. 13 .7 Impossibility of routing and need for a custom architecture Routing of an initial state means the freedom to be able to choose which vertex we wish to perfectly transferthe state at the initial given vertex. The idea of routing is eventually related to the complex nature of theHamiltonian involved [27]. If for a given graph G , perfect state transfer is possible between A and B , andthe minimum time for which this is possible is t AB , then we have e − iH t AB | A (cid:105) = e iφ | B (cid:105) . (1.47)Furthermore, since the Hamiltonian is real, all the ϕ j are 0 or π (see equation (1.21)). Then for twice thetime, 2 t AB , we will have perfect revivals as e − iH t AB | A (cid:105) = e iφ | A (cid:105) . (1.48)which is periodic dynamics. Now, if perfect routing is possible, this means that we must have a time t AC < t AB such that e − iH t AC | A (cid:105) = e iφ (cid:48) | C (cid:105) (1.49)and similar revivals as e − iH t AC | A (cid:105) = e iφ (cid:48) | A (cid:105) . (1.50)These equations together give e − iH (2 t AC − t AB ) | B (cid:105) = e i φ (cid:48) − φ | A (cid:105) (1.51)But this is simply the perfect state transfer between B and A with time | t AC − t AB | < t AB , which isimpossible by the initial assumption that t AB was the shortest time for perfect state transfer between A and B . Therefore, the transfer from A to C cannot exist. As a result, if there is perfect state transfer to one sitefrom a given site, there cannot be a perfect state transfer from this given site to any other sites. However,this can be tackled as shown in [28] by considering the dynamics of complex Hamiltonians.This means that for a real Hamiltonian, we can have only pair of vertices where perfect state transfer ispossible. If the network is very large, then this becomes less useful because given a vertex only one othervertex can be reached out with the transfer. This calls for the possibility of an architecture which allows morefreedom for routing. In chapter 4, we propose such an architecture with additional conditions to this idea suchthat routing is possible to any arbitrary site in the network for real Hamiltonians. Such an architecture willbe very useful for the era of large scalable quantum processors where a state needs to be perfectly transferredto any given qubit in the processor with maximum fidelity. Moreover, in chapter 6 we also show that ourarchitecture works with the conventional quantum computing architecture which can be used to perfectlytransfer or swap arbitrary states between any two given qubits in just two steps. Our scheme greatly reducesthe circuit depth if the swap has to be performed between distant qubits using the universal quantum gates. Some weaker bounds for transfer rate for spin chains are known. Transfer rate has been discussed in [35][12].Consider assigning a second state at site A before the first state has been moved to site B , and impose thecondition that the first state should still arrive at B perfectly. After the existence of a transfer Hamiltonianis guaranteed, the necessary condition for the ability to insert a second quantum state into the spin networkto the same initial input qubit at some time t without disturbing the first quantum state is that (cid:104) A | e − iH t | A (cid:105) = 0 . (1.52)This condition is necessary and sufficient for chains. However, for general networks which are not chains,more conditions are required. More generally, consider inserting many different states at different times, butthe condition for the chain remains the same for all possible times. This will certainly not be the case forthe dynamically changing many-excitation states in an arbitrary network. Yet this is a necessary condition.Therefore, given p unique time intervals t i < t at which (cid:104) A | e − iH t i | A (cid:105) = 0, perfect state transfer can occurto a site | B (cid:105) at a distance of d in time t . With p time intervals, one can have p unique time intervals t i byimposing fixed intervals. For each transfer distance m = 1 , , ..., d − (cid:104) B | H m | A (cid:105) = 0 . (1.53)14his can be expressed as n (cid:88) j =1 e − iϕ j λ mj a j = 0 (1.54)with the same a j = |(cid:104) A | λ j (cid:105)| which can be re-expressed again after removing the degeneracies ( n reduces to n (cid:48) number of unique eigenvalues) from the system in the linear equation form as d − (cid:88) m =0 n (cid:48) (cid:88) j =1 e − iϕ j λ mj a j | m (cid:105)(cid:104) j | n (cid:48) (cid:88) j =1 a j | j (cid:105) (1.55)Each of the d − n (cid:48) (cid:88) j =1 (cid:104) j | n (cid:48) (cid:88) j =1 a j | j (cid:105) (1.56)We can now add conditions corresponding to (cid:104) A | e − iH t i | A (cid:105) = 0 and further divide these into real andimaginary components. The real part is p (cid:88) i =1 n (cid:48) (cid:88) j =1 cos( λ j t i ) | i (cid:105)(cid:104) j | n (cid:48) (cid:88) j =1 a j | j (cid:105) (1.57)and the imaginary component gives p (cid:88) i =1 n (cid:48) (cid:88) j =1 sin( λ j t i ) | i (cid:105)(cid:104) j | n (cid:48) (cid:88) j =1 a j | j (cid:105) (1.58)Given that all these times ti are less than t , the half period of the system, all of these rows must be linearlyindependent from each other. (Since we are assuming the Hamiltonian is real and performs perfect transfer,the system is periodic with a period 2 t .) Hence, if a suitable set of an is to possibly exist, it must be thecase that 2 p + d ≤ n (cid:48) ≤ n (1.59)Ideally, we want the maximum transfer distance, which would be n − p = 0. Theonly way to increase the perfect transfer rate is to reduce the transfer distance. However, you cannot alsolower the state transfer time (as you would expect by shortening the transfer distance). This is because theMargolus-Levitin theorem [35] imposes a minimum time for evolving between two orthogonal states, such asa | A (cid:105) as an input state and the | A (cid:105) required for the next input. Hence the transfer time is bounded frombelow by ( p + 1) π/ (4 (cid:80) j J j ). In some sense, the “standard” perfect state transfer chains saturate the boundfor a chain of n qubits, any state | j (cid:105) transfers a distance D = n + 1 − j , but there are j − t i such that (cid:104) j | e − iH t i | j (cid:105) = 0. Unfortunately, however, these times are not equally spaced, so they are notuseful for achieving a high rate of transfer. It is desirable to maximise the distance over which communication is possible for a fixed number of qubits.The simplest and optimal arrangement, in this case, is just a linear chain of n qubits, where A and B arethe qubits at opposite ends of the chain.Let us start with the XY chain of qubits, with uniform couplings J i,i +1 = 1 for all 1 ≤ i ≤ n −
1. TheHamiltonian reads H = 12 n − (cid:88) i =1 σ xi σ xi +1 + σ yi σ yi +1 (1.60)In this case, one can compute f AB ( t ) explicitly by diagonalizing the Hamiltonian or the correspondingadjacency matrix. The eigenstates and the corresponding eigenvalues are given by | ˜ k (cid:105) = (cid:114) n + 1 n (cid:88) j =1 sin (cid:18) πkjn + 1 (cid:19) (1.61)15nd λ k = E k = − kπn + 1 (1.62)with k = 1 , ..., n . Thus f AB ( t ) = 2 n + 1 n (cid:88) k =1 sin (cid:18) πkn + 1 (cid:19) sin (cid:18) πknn + 1 (cid:19) e − iE k t (1.63)Perfect state transfer from one end of the chain to another is possible for n = 2 and n = 3, where we findthat f AB ( t ) = − i sin( t ) and f AB ( t ) = − sin ( t/ √
2) respectively. Hence, for perfect state transfer, that is, tohave | f AB ( t ) | = 1, we have t = (cid:40) π/ , for n=2 ,π/ √ , for n=3 , (1.64)in the units of energy inverse. We have shown that perfect state transfer is possible for chains containing 2or 3 qubits. It can be now shown that it is not possible to get perfect state transfer for n ≥
4. This workwas originally done in [21]. A chain is symmetric about its centre. Hence the rationality for eigenvaluescondition equation (1.38) for perfect state transfer applies for all longer chains. If we pick specific values l = 2 , j = n − , l (cid:48) = 1, and j (cid:48) = n , then using the expression for eigenvalues for the chain, this conditionbecomes cos πn +1 cos πn +1 ∈ Q (1.65)to hold for the perfect state transfer. The concept of algebraic numbers can be used to find the value of n for which the above condition holds. An algebraic number x is a complex number that satisfies a polynomialequation of the form a x m + a x m − + ... + a m − x + a m = 0 , (1.66)with integral coefficients a i . Every algebraic number α satisfies a unique polynomial equation of least degree.The degree of this polynomial is called the degree of α . If α satisfies a polynomial of degree m , then it i calledan algebraic integer of degree α . An algebraic integer of degree m is also number of degree m . Rationalnumbers are algebraic numbers with degree 1, and numbers with degree ≥ n >
1, and gcd( k, n + 1) = 1 then cos[ πk/ ( n + 1)] is an algebraic integer of degree φ (2( n + 1)) /
2, where φ is the Euler phi function and we have that φ (2( n + 1)) / ≥ n ≥
6. See
Irrational numbers by Lehmer(Mathematical Association of America, 1956).It we assume that the expression of the formcos 2 θ cos θ = pq ∈ Q p, q ∈ Z (1.67)(with θ = π/ ( n + 1) is an algebraic number of degree ≥
3) is rational then using trigonometric identitycos 2 θ = 2 cos θ − θ − p q cos θ −
12 = 0 (1.68)which has rational coefficients. This means that cos θ is algebraic with degree ≤
2. Hence, this is a con-tradiction and cos 2 θ/ cos θ must be irrational. Therefore, this strictly proves that for n ≥
6, perfect statetransfer is impossible (between the end vertices of the chain) as deg( n ) ≥
3. Furthermore, for n = 4 and n = 5 similar calculations show that cos 2 θ cos θ = pq / ∈ Q . (1.69)We, therefore, in conclusion, have that it is impossible to perform perfect state transfer in unmodulatedchains of constant coupling for number of nodes n ≥ J i = (cid:112) i ( n − i ) and the chain magically supports perfect state transfer. But suchmodulations over a considerable length of chain is very hard to engineer experimentally.16 .10 Perfect State Transfer in long and weighted chains The workaround to enable perfect state transfer for chains with length n ≥
4, the idea of projecting a hypercube (see chapter 4 for detailed study on hypercubes) onto a spin chain was originally studied in [21].The hypercube Q k resulting from the k − fold Cartesian product (see section 2.3.2) of one-link graph has theproperty that it can seen as arrangement of its vertices as columns such that there are no edges between thevertices within any column and edges only join vertices in different columns. And furthermore, each vertexin column i must have the same number of incoming (from column i −
1) and outgoing (to column i + 1)edges as all other vertices in that column (it is simply due to the property of hypercubes that each vertexhas the same number of adjacent vertices).Let Q k be arranged in n c columns, call the graph as G . The size of each column is c i := | G i | = n c − C i − and label the vertices in each column as G ij with j = { , , ..., c i } . Start with a vertex A , then the i th columnis i − A . From each column there are edges going backward to the previouscolumn and edges going forward to the next column (except for the end columns). These are denoted as C fori := { ( G ij , k ) : j ∈ { , ..., c i } , k ∈ { , ..., f i }} C backi := { ( G ij , k ) : j ∈ { , ..., c i } , k ∈ { , ..., b i }} (1.70)where f i and b i denote the number of forward and backward edges, respectively, for the i th column. If allthe edges are to have ends, then | C fori | = | C backi | . Since there is only one vertex (qubit) in the first column( c = 1), each vertex in the second column has only a single edge going backward, implying b = 1. Startingfrom this boundary condition, and that f i and b i must be integers for all 1 ≤ i ≤ n c , we have the conditionthat c i f i = c i +1 b i +1 (1.71)which implies f i b i +1 = n c − ii . (1.72)The solution for this is to choose f i = n c − i and b i = i −
1. Therefore, we end up with a graph such thatfor every pair of numbers ( i, j ), G ij ic onnected with n c − i columns in G i +1 and each vertex in G i +1 isconnected with i − G i .We define the vectors that span the column space H c as, | col i (cid:105) := 1 √ c i c i (cid:88) j =1 | Gij (cid:105) . (1.73)Class of networks with this column representation have the special property that throughout the quantumevolution with the adjacency matrix the instantaneous state always remains in the column space H c . Thus,it can be seen as the problem of perfect state transfer from G to G n c , for instance. Also note that the antipodal vertices of a hypercube constructed from one-link and two-link graphs admit perfect state transfer(see next section). The matrix elements of the adjacency matrix of G , restricted to the column space aregiven as J i := (cid:104) col i | H G | col i + 1 (cid:105) = (cid:112) i ( i − n c ) . (1.74)The matrix form is J = J ... J J ... J J ...
00 0 J ... J n c − J n c − (1.75)This is because (cid:104) col i | H G | col i + 1 (cid:105) = 1 √ c i c i +1 c i (cid:88) j =1 c i +1 (cid:88) j (cid:48) =1 (cid:104) G ij | H G | G i +1 j (cid:48) (cid:105) (1.76)= 1 √ c i c i +1 c i ( n c −
1) = (cid:112) i ( n c − i ) . (1.77)17learly, this is identical to the matrix form of the XY -model chain with just specially engineered couplingstrengths { J i } sch that the Hamiltonian is H G ≡ H XY = 12 n c − (cid:88) j =1 J j (cid:0) σ xj σ xj +1 + σ yj σ yj +1 (cid:1) (1.78)where J j is given by equation (1.74). Such a chain must allow perfect state transfer over any length n c (where we redefine | A (cid:105) := | col 1 (cid:105) and | B (cid:105) := | col n c (cid:105) ) because the hypercube does (see next section). Similarweighted chain can be realised with the Heisenberg model with local magnetic fields as H G ≡ H Hei = 12 n c − (cid:88) j =1 J j (cid:126)σ j · (cid:126)σ j +1 + n c (cid:88) j =1 B j σ zj (1.79)with B j = ( J j − + J j ) − n c − (cid:80) n c − k =1 J k which has been specially chosen to cancel the diagonal elementsto bring the XY and Heisenberg model on equal grounds. This model is now perfectly equivalent to thetransfer dynamics of a weighted chain with n c vertices (qubits). With this idea of projecting a hypercube toa spin chain, we see that the chain is enabled for perfect state transfer with the difference being that it isnow modulated with special coupling strengths which gives rise to a weighted chain graph. Perfect state transfer over arbitrary distances is impossible for a simple unmodulated spin chain (limitedto n = 2 and n = 3 only!). Clearly it is desirable to find a class of graphs that allow state transfer overlarger distances. One approach to achieved this apart from modulation of spin chains is to construct largerarbitrary graphs using the graph products of small blocks of n = 2 or n = 3 which serve as the fundamentalbuilding blocks for such construction. One well explored construction is through the Cartesian product oflinear chains proposed in [21]. We examine the d -fold Cartesian product of one-link (two-vertex) and two-link(three-vertex) chain G . We denote this by G d := (cid:3) d G where the square denotes the Cartesian product of G with itself. See section 2.3.2 for details and construction of Cartesian product of graphs. Following thebinary and ternary representation for the vertex labeling as in chapter 4, consider two antipodal vertices A (0 , , ...,
0) and B (1 , , ...,
1) (labels of length d each) for for one-link. Similarly, for two-link hypercube wehave the antipodal points as A (0 , , ...,
0) and B (2 , , ...,
2) respectively. This can be proved that for anydimension d | f AB ( t ) | = 1 for t = t = π/ t = t = π/ √ t is the perfect state transfer time for transfer between antipodal vertices A and B for G d also.The first sign of perfect state transfer for hypercubes can be seen due to equation (1.38). For hypercubesfrom one-link and two-link seed graph G , the ratios of differences of all possible eigenvalues are rational,which permits perfect state transfer. Furthermore, it can be proved strongly by construction. As alreadyestablished, the Hamiltonian dynamics of XY interaction Hamiltonian is identical to the dynamics of theadjacency matrix in the first excitation subspace. This holds equally for the Cartesian product of G , byconstruction. Hence, H = A ( G d ) = d − (cid:88) j =0 I ⊗ j ⊗ A ( G ) ⊗ I ⊗ d − j − (1.80)and e − iHt = ( e − iA ( G ) t ) ⊗ d (1.81)Thus, if we evolve the system for time t , we get perfect state transfer along each dimension. Each termin the tensor product applies to a different element of the basis. We therefore achieve perfect state transferbetween A and B as well as between any qubit and its mirror vertex qubit. The fidelity of the state transferis simply the d th power of the fidelity for the original chain: F G d ( t ) = [ F G ] d ( t ) = (cid:40) sin d ( t ) , for 2 d verticessin d ( t/ √ , for 3 d vertices . (1.82)18his formalism also extends over to the Heisenberg couping Hamiltonian. This is because, in the case of atwo-qubit chain, the Hamiltonian in the single excitation subspace is represented by a matrix with identicaldiagonal elements, and hence is the same as the Hamiltonian of an XY model up to a constant energyshift, which just adds a global phase factor. Hence, the same hypercube transfer dynamics holds true forHeisenberg scheme. Thus, any quantum state can be perfectly transferred between the two antipodes of theone-link and two-link hypercubes of any dimensions in constant time. The above discussion motivates the idea of finding other graph products or class of graphs which mightsupport perfect transfer. Cartesian product is the simplest such product which has been explored. Otherproducts are not so physically relevant as the Cartesian product. Basically, this defines a growing architecturescheme for connectivity in a quantum processor. However, for large d , the difference between 2 d and 2 d +1 is very large and it makes little sense experimentally to add this many qubits in a system to allow forperfect state transfer. Moreover, for large d , the cost of adding so many edges (physically establishingprecisely the same coupling strength) in the system just to establish perfect state transfer between only pairof antipodal nodes, is too high. In this thesis work, we propose our scheme which is based on the hypercuberesult for long distance transfer which primarily resolves these two challenges to the hypercube architecture.Our scheme in chapter 4 enables perfect state transfer from all-to-all nodes for arbitrary number of qubits!Hennce, the problem of routing of states can be resolved. It also features that one qubit can be addedeach time individually in our architecture. This also complies with the current experimental challenges forthe realization of quantum computing where a small number of qubits can be added into the processor forscalability. 19 hapter 2 Graph Theory
In this chapter we briefly discuss some fundamental concepts related to graphs, and matrices associated withgraphs. We primarily focus on finite, simple graphs: those without loops or multiple edges. Further detailscan be found in [36].
A graph is an ordered pair G = ( V, E ), where • V denotes the set of vertices { v i } (also called nodes or points), and • E ⊆ {{ x, y }| ( x, y ) ∈ V × V, x (cid:54) = y } denotes the set of edges (also called links or lines), which areunordered pairs of vertices (i.e., an edge is associated with two distinct vertices)The adjacency matrix associated with a graph G is denoted by A ( G ) = [ a ij ],where a ij = (cid:40) , if ( i, j ) ∈ E , otherwiseLet G be a graph on n vertices, that is, | V | = n. Then obviously, A ( G ) is a symmetric matrix of order n × n. Let v i ∈ V. Then the degree of v i is defined as deg( v i ) = (cid:80) nj =1 a ij . The degree matrix D ( G ) of G is a diagonal matrix where D ( G ) ≡ [ d ii ] = deg( v i ) (2.1)The graph Laplacian matrix L ( G ) associated with the graph G is defined as L ( G ) = D ( G ) − A ( G ) (2.2)which is equivalent to saying L ( G ) ≡ [ l ij ] = deg( v i ) , if i = j − , if i (cid:54) = j, and ( i, j ) ∈ E , otherwise . The signless
Laplacian matrix corresponding to G is defined by L + ( G ) = D ( G ) + A ( G ) . (2.3) Following are some more general definitions of the graphs which may play a role in PST for specific class ofgraphs. 20
Signed graph:
A signed graph is an ordered tuple G = ( V, E, σ ) where V denotes the set of nodes, E ⊆ V × V , the edge set, and σ : E −→ { + , −} is called the signature function. This another degreeadded to the definition of a graph. An obvious way to construct a signed graph from a marked graphis be defining the sign of an edge of the marked graph as the product of signs of its adjacent vertices.Thus, the sign of an edge is the product of its signs of vertices it connects. • Marked graph:
We can assign a marking {±} to the nodes or vertices along with naming them. Thisadds more degree of freedom to the graph which can be captured by additional functions. A graph iscalled a marked graph if every node of the graph is marked by either a positive or negative sign. Thusa marked graph is a tuple G = ( V, E, µ ) where V is the node set, E the edge set and µ : V → + , − iscalled the marking function. There are various possible marking schemes. Let us look at the followingtwo conventional ways of marking the vertices. – Canonical marking scheme: Defined from a signed (see next definition) graph G = ( V, E, σ )by defining the marking of a node v ∈ V as µ ( v ) = (cid:89) e ∈ E v σ ( e ) (2.4)where E v is the set of signed edges adjacent at v . – Plurality marking scheme: We define plurality marking of a node v of a signed graph G =( V, E, σ ) as µ ( v ) = (cid:40) + , if max { d + ( v ) , d − ( v ) } = d + − , OtherwiseHence a node is negatively marked in plurality marking scheme only when d − ( v ) > d + ( v ). • Balanced graph:
A signed network is balanced if and only if all its cycles are balanced. A signedcycle is called balanced if the number of negative edges in it is even. In other words, a graph is balancedif all its cycles (vacuum-loops) or cliques are balanced. More rigorously it can be stated as follows. Let G be a graph with vertices v , .., v n and cliques C , .., C m . The (0,1) matrix A = ( a ij ) where a ij is 1 iffvertex v i belongs to clique C j is called a clique matrix of G . A (0,1)-matrix is balanced if it does notcontain the vertex-edge incidence matrix of an odd-cycle as a submatrix (that is, it contains no squaresubmatrix of odd order with exactly two 1s per row and per column). A graph is balanced if its cliquematrix is balanced. • Regularity:
A regular graph is a graph where each vertex has the same number of neighbors; i.e.every vertex has the same degree or valency. A regular directed graph must also satisfy the strongercondition that the indegree and outdegree of each vertex are equal to each other. A regular graph withvertices of degree k is called a k -regular graph or regular graph of degree k . Also, from the handshakinglemma, a regular graph of odd degree will contain an even number of vertices. A signed regular graphcan be defined in the sense that signed regularity (say, d + ( v i ) − d − ( v i ) = d , constant) for all vertices v i ∈ V ( G ).Therefore, a graph generally is a 4-tuple G ( V, E, σ, µ ). To take products of two graphs we first start bydefining a graph with signed edges or marked vertices and find the other using a scheme we wish to followin accordance with the graph operation involved. Then, the new edge signing in the new product graph canbe found using the same scheme(s).
Two graphs can be operated with a defined operation that gives another resultant graph. A graph product isa binary operation on graphs. Specifically, it is an operation that takes two graphs G and G and producesa graph H with the following properties: • The vertex set of H is the Cartesian product V ( G ) × V ( G ), where V ( G ) and V ( G ) are the vertexsets of G and G , respectively. 21 Two vertices ( u , u ) and ( v , v ) of H are connected by an edge if and only if the vertices u , u , v ,and v satisfy a condition that takes into account the edges of G and G . The graph products differin exactly which this condition is.Graph product is a very important operation for this work as it defines a new ’larger’ graph from the initialgraphs. This is helpful in describing a growing network which multiplies according to some defined graphproduct rule. In this thesis, we are specifically concerned with Wedge product, Cartesian product and Coronaproduct of graphs. This section follows the wedge product as proposed in [9] and describes the general action of couplingHamiltonians in different excitation spaces
Definition 1.
We define the wedge product ∧ k G of a graph G to be the graph with vertex set V ( ∧ k G ) := { ( v , v , ..., v k − ) | v j ∈ V ( G ) , v k − > v k − > ... > v } . We write vertices of ∧ k G as v ∧ v ∧ ... ∧ v k − . Weconnect two vertices v ∧ v ∧ ... ∧ v k − and w ∧ w ∧ ... ∧ w k − in ∧ k G with an edge if there is a permutation π ∈ S k ( S k is a permutation group on k distinguish entities) such that w j = v π ( j ) for all j = 0 , , ..., k − j = π ( l ) where ( v l , w π ( l ) ) ∈ E ( G ) is an edge in G . The Hilbert space H ∧ k G of the graph ∧ k G is isomorphic to ∧ k ( H G ). Exterior vector space ∧ k G is spanned by vectors | v ∧ v ∧ ... ∧ v k − (cid:105) whereno two vectors | v j (cid:105) , | v k (cid:105) , ∀ j (cid:54) = k , are the same.Wedge product for corresponding Hilbert space ∧ k G is defined as ∧ : H G × H G × ... × H G −→ (cid:78) k − l =0 H G with action on the vectors as | v ∧ v ∧ ... ∧ v k − (cid:105) := 1 k ! (cid:88) π ∈ S k (cid:15) ( π ) | v π (0) , v π (1) , ..., v π ( k − (cid:105) (2.5)where (cid:15) ( π ) is the sign of the permutation π . This defines the basis for ∧ k G as {| v ∧ v ∧ ... ∧ v k − (cid:105)} with v j ∈ V ( G ) and v k − > v k − > ... > v . Furthermore, Γ k (defined in section 1.2) and ∧ k have the samedimension and they are isomorphic vector spaces over C . The correspondence can be assigned by identifyingthe state | v , v , ..., v k − (cid:105) ∈ Γ k which has a 1 at positions or vertices v k − > v k − > ... > v and zeroselsewhere, with the basis vector | v ∧ v ∧ ... ∧ v k − (cid:105) ∈ ∧ k H G .We show how the structure of ∧ G is captured by the adjacency matrix for for ∧ G . And then show itsaction is identical to H XY in that excitation space. Let B ( H G ) be the space of all bound operators on H G .And let M ∈ B ( H G ) be a linear operator from H G to H G . Define the operation ∆ k : B ( H G ) −→ (cid:78) k − j =0 B ( H G )as ∆ k ( M ) := k − (cid:88) j =0 I ...j − ⊗ M j ⊗ I j +1 ...k − . (2.6)Dimension of (cid:78) k H G is greater than that of ∧ k H G . We define the projection Alt : (cid:78) k H G −→ ∧ k H G byAlt | φ , φ , ..., φ k − (cid:105) := 1 k ! (cid:88) π ∈ S k (cid:15) ( π ) π [ | φ , φ , ..., φ k − (cid:105) ] (2.7)= 1 k ! (cid:88) π ∈ S k (cid:15) ( π ) | φ π (0) , φ π (1) , ..., φ π ( k − (cid:105) (2.8)where the action of the symmetric group S k on the basis kets is evident and Alt † = Alt (because it is aprojection). The analogous adjacency matrix (signed version) for ∧ G is generally defined by C ( ∧ k G ) = Alt ∆ k [ A ( G )] Alt (2.9)which generally contained negative entries also. It can be calculated that (cid:104) v ∧ v ∧ ... ∧ v k − | C ( ∧ k G ) | v ∧ v ∧ ... ∧ v k − (cid:105) = ± v j = w j for all j except at exactly one place j = l where ( v j , w l ) ∈ E ( G ). Allother entries are exactly zero. The unsigned adjacency matrix A ( ∧ k G ) is the matrix obtained after replacingall instances of − C ( ∧ k G ). Using the spectral decomposition of A ( G ), thespectral properties of C ( ∧ k G ) can be obtained as in [9].22s evident from equation (1.1), the Hamiltonian H XY action on | φ (cid:105) = | v , v , ..., v k − (cid:105) ∈ Γ k is to movethe 1 at position i to j if and only if there is no 1 in the j place. In this way, we can say that the Hamiltonian H XY maps the state | φ (cid:105) to an equal superposition if all states which are identical to | φ (cid:105) at all indices exceptat one place. So, a 1 at a given place has been moved along an edge e ∈ E ( G ) as long as there is no 1 at theendpoint of e . This is identical to the action of A ( ∧ k G ) on Γ k . Similarly, an observation can be made forLaplacian matrix in Γ k .The action of H XY (and respectively H Hei ), when restricted to Γ k is the same as that of the adjacencymatrix (and respectively Laplacian) of ∧ k G . Definition 2.
The Cartesian product of two graphs G := { V ( G ) , E ( G ) } and H := { V ( H ) , E ( H ) } is a graph G × H whose vertex is a set V ( G ) × V ( H ) and two of its vertices ( g, h ) and ( g (cid:48) , h (cid:48) ) are adjacent iff one ofthe following conditions hold • g = g (cid:48) and { h, h (cid:48) } ∈ E ( H ) • h = h (cid:48) and { g, g (cid:48) } ∈ E ( G ).Furthermore, if | k (cid:105) is an eigenvector of A ( G ) with corresponding eigenvalue E k and | l (cid:105) is an eigenvectorof A ( H ) with corresponding eigenvalue E l , then | k (cid:105) ⊗ | l (cid:105) is an eigenvector of A ( G × H ) with correspondingeigenvalue E k + E l . Here, A ( · ) is the adjacency matrix. This happens due to the underlying construction A ( G × H ) = A ( G ) ⊗ I V ( H ) + I V ( G ) ⊗ A ( H ) . (2.10)This is exactly as forming the composite system out of two sub-systems in quantum theory. All the sameconstruction applies. (a) (b) Figure 2.1: Part (a) shows the first four Cartesian product of K or Q with itself. These are the first fivehypercubes. The new edges formed due to the fourth Cartesian product (cid:3) ( G ) is shown in red for clarity.Part (b) shows the first two Cartesian product of two-link graph G with itself.23 .3.3 Corona product of graphs We state a relatively new and special kind of product called the Corona product. Corona product of graphswas introduced by Frucht and Harary in 1970 [37][38][39]. Given two unsigned and unmarked graphs G and H , the corona product of G and H is a graph, we denote it by G ◦ H , which is constructed by taking n instances of H and each such H gets connected to each node of G , where n is the number of nodes of G .Starting with a connected simple graph G , we define corona graphs which are obtained by taking coronaproduct of G with itself iteratively. In this case, G is called the seed graph for the corona graphs. Using aseed is exactly the same approach as the chain building blocks were considered previously.Now we state the definition of the signed Corona product of two graphs. Definition 3.
Let G = ( V , E , σ , µ ) and G = ( V , E , σ , µ ) be signed graphs on n and k nodesrespectively. Then corona product G ◦ G of G , G is a signed graph by taking one copy of G and n copiesof G , and then forming a signed edge from i th node of G to every node of the i th copy of G for all i . Thesign of the new edge between i th node of G , say u and j th node in the i th copy of G , say v is given by µ ( u ) µ ( v ) where µ is a marking scheme defined by σ i , i = 1 , G ◦ G of signed graphs G and G is shown in figure 2.2. Note thatcanonical and plurality marking are same for the graph G . For G the marking of the nodes 1,3 are samefor canonical and plurality markings, whereas the canonical and plurality markings of node 2 are − and +respectively. Thus the choice of the marking function produce different corona product graphs.Figure 2.2: The corona product of G ◦ G is shown in (c), (d) with canonical and plurality marking functionson G i , i = 1 , G = G (0) be a simple connected graph [39]. Then the corona graphs G ( m ) corresponding to the seedgraph G are defined by G ( m ) = G ( m − ◦ G (2.11)where m ( ≥
1) is a natural number. For example, the corona graphs G (1) and G (2) corresponding to the seedgraph K are shown in figure 2.3.The following are some observations associated with corona graphs. • The number of nodes in G ( m ) is | V ( G ( m ) ) | = n ( n + 1) m (2.12) • If k is the number of edges in the seed graph G (0) then the number of edges in G ( m ) is | E ( G ( m ) ) | = k + ( k + n )(( n + 1) m − −
1) (2.13) • The number of nodes added in i th ( i ≤ m ) step during the formation of G ( m ) is n ( n + 1) i − .24igure 2.3: Examples of the Corona graphs (a) Seed graph K (b) Corona graph for G (1) , and (c) Coronagraph for G (2) We can do the similar construction of the signed graphs using a signed and marked seed. Let us define theadjacency and the Laplacian matrix of the signed graphs Corona product as well. We focus on the spectralproperties of G ◦ G . Let G = ( V , E , σ , µ ) and G = ( V , E , σ , µ ) be two signed graphs with n and k number of nodes, respectively. Suppose V = u , ..., u n and V = v , ..., v k . Let us denote the markingvectors corresponding to vertices in G and G as µ [ V ] = [ µ ( u ) µ ( u ) ... µ ( u n )] and µ [ V ] = [ µ ( v ) µ ( v ) ... µ ( v k )] (2.14)where µ j ( u ) = 1 if marking of u = +, otherwise µ j ( u ) = − j = 1 ,
2. Defining a matrixdiag( µ [ V ]) = diag( µ ( u ) , µ ( u ) , ..., µ ( u n )) (2.15)defined using the marking µ [ V ], and similarly for V .Then with a suitable labeling of the nodes the adjacency matrix of G ◦ G is given by A ( G ◦ G ) = (cid:20) A ( G ) µ [ V ] ⊗ diag( µ [ V ]) µ [ V ] T ⊗ diag( µ [ V ]) A ( G ) ⊗ I n (cid:21) (2.16)where A ( G i ) denotes the adjacency matrix associated with G i , i = 1 , ⊗ denotes the Kronecker productof matrices, I n is the identity matrix of order n .Similarly we have the definition for the Laplacian of the Corona product of two graphs G and G as L ( G ◦ G ) = (cid:20) L ( G ) + kI n − µ [ V ] ⊗ diag( µ [ V ]) − µ [ V ] T ⊗ diag( µ [ V ]) ( L ( G ) + I k ) ⊗ I n (cid:21) (2.17)with rest similar definitions as for the adjacency matrix of the product. We can use these definitions torecursively construct G ( m ) using G = G ( m − and G = G and so on. Theorems on construction of eigenvalues and eigenvalues for product of corona graphs
Constructing the higher order adjacency and Laplacian matrices using the previous section definitions is easywhen done recursively. However, we need the eigenvalues and eigenvectors for these matrices which can bevery tedious for even m ≥ n = 4. We thus require some algorithm to construct the eigenvalues andeigenvectors recursively. This in general is not known, but for certain special graphs, this is possible if thesespecial constrains on the graphs being multiplied are satisfied. We state two such extremely useful theorems(Theorem 2.3 and Theorem 2.6) extracted from [37] (without stating the proof here).25 heorem 1. Let G be any signed graph on n nodes and G be a net-regular signed graph on k nodeshaving net-regularity d . Let ( λ i , X i ) be an adjacency eigenpair of G , and ( η j , Y j ) be an eigenpair of G , i = 1 , ..., n , and j = 1 , ..., k . Let η k = d . Then an adjacency eigenpair of G ◦ G is given by ( λ ( i ) ± , Z ( i ) ± ), i = 1 , ..., n where λ ( i ) ± = d + λ i (cid:112) ( d − λ i ) + 4 k , and Z ( i ) ± = X iµ ( v ) λ ( i ) ± − d diag( µ [ V ]) X iµ ( v ) λ ( i ) ± − d diag( µ [ V ]) X i ... µ ( v k ) λ ( i ) ± − d diag( µ [ V ]) X i (2.18)In addition, if all the nodes in G are either positively or negatively marked, that is µ [ V ] = I k or − Ik then (cid:18) η j , (cid:20) Y j ⊗ e i (cid:21)(cid:19) (2.19)is an eigenpair of G ◦ G where j = 1 , ..., k −
1, and { e i : i = 1 , ..., n } the standard basis of R n . Theorem 2.
Let G = ( V , E , σ , µ ) be a signed graph on n nodes and G = ( V , E , σ , µ ) be a signedgraph on k nodes. Let V = v , ..., v k . Let ( λ i , X i ) be a signed Laplacian eigenpair of G , and ( η j , Y j ) aresigned Laplacian eigenpairs of G , i = 1 , ..., n, j = 1 , ..., k . Let d − = d − j denote the negative degree of everynode v j in G . Then a signed Laplacian eigenpair of G ◦ G is given by ( λ ( i ) ± , Z ( i ) ± ) where λ ( i ) ± = 2 d − + 1 + λ i + k + ± (cid:112) [(2 d − + 1) − ( λ i + k )] + 4 k Z ( i ) ± = X i − µ ( v ) λ ( i ) ± (2 d − +1) diag( µ [ V ]) X i − µ ( v ) λ ( i ) ± (2 d − +1) diag( µ [ V ]) X i ... − µ ( v k ) λ ( i ) ± (2 d − +1) diag( µ [ V ]) X i (2.20)where i = 1 , ..., n . Let η k = 2 d − . In addition if all the nodes in G are marked either positively or negativelymarked then an eigenpair of G ◦ G is (cid:18) η j + 1 , (cid:20) Y j ⊗ e i (cid:21)(cid:19) (2.21)is an eigenpair of G ◦ G where j = 1 , ..., k −
1, and { e i : i = 1 , ..., n } the standard basis of R n .Both these theorems together give all the ordered eigenpairs for higher order Corona products recursively.They can be used when these special conditions mentioned in the theorems are satisfied and approaching thecalculation for perfect state transfer is a lot easier. 26 hapter 3 Perfect State Transfer under Coronaproduct of signed graphs
This chapter corresponds to Part-I of the thesis project and was aimed at studying the perfect state transferfor signed graphs under the Corona product (as discussed in section 2.3.3). This consists of some importanttheorems and numerical results. Results from Part-II of the thesis are contained in chapters 4,5 and 6.
A general discussion for conditions for PST under Corona product of graphs is presented in [40]. A negativeresult indicating that no perfect state transfer is possible between any vertices of resulting Corona productof two given graphs, is obtained in [19]. We first present the the Theorem 4.1 in this paper.
Theorem 3.
Let G be a connected graph on n ≥ (cid:126)H = ( H , . . . , H n ) be an n -tuple of graphson m ≥ G ◦ (cid:126)H .The proof of this theorem can be found in [19]. Here, the second graph is a tuple of n graphs for eachvertex of G . This is more general scenario of Corona product where each vertex of G is associated witha different graph H j . Construction of this product is also given in section 3 of the same paper. This is avery strong result for Corona product. This result holds for the Laplacian evolution of the graph (that is,under Heisenberg coupling interaction of qubits) and is related to the impossibility of Laplacian perfect statetransfer for trees [18]. Therefore, now adjacency state transfer remains to be explored for Corona product.More freedom can be explored in state transfer by using signed graphs which may support PST. The previous section indicates that there is no PST in Corona. However, there is a concept of pretty goodstate transfer where the fidelity is below 100% but still close to it [15][17]. Work done in [19] presents twotheorems on pretty good state transfer in Coronas that we restate here without their proof. These theoremsare very strong results for state transfer under Corona product. Proofs can be found in the original work.
Theorem 4.
Let G be a graph on n vertices and (cid:126)H = ( H , . . . , H n ) be an n -tuple of graphs on m ≥ G has perfect state transfer between vertices u and v , and let 2 r be the greatest power oftwo dividing each element of the eigenvalue support of u . If 2 r +1 divides m + 1, then there is pretty goodstate transfer between vertices ( u,
0) and ( v,
0) in G ◦ (cid:126)H .These are the sufficient conditions for pretty good state transfer under Corona product of graphs. Theorem 5.
Let (cid:126)H = ( H , H ) be a pair of graphs on m ≥ K ◦ (cid:126)H has pretty good statetransfer between the vertices of K . 27his theorem establishes a particular class of Corona graphs where the pretty good transfer is possible.In the light of these three theorems and subsection 1.6.3 what remains to explore are the signed balancedgraphs under the Corona product with XY coupling. And also the unbalanced signed XY and Heisenbergcoupling based graphs under Corona. We use Theorem 1 and Theorem 2 to construct some specific graphsand numerically study perfect state transfer and pretty good state transfer under Corona product (for themarking schemes mentioned in section 2.3.3) for both XY and Heisenberg interactions. Some conclusive results based on the previous section study are presented here. All the results and conclusionsfor the current progress are based on numerical study and examples based on construction. We start withconstructing the Hamiltonian (and thereby the adjacency and the Laplacian matrices) of a possible graphand then calculate the fidelity of a perfect state transfer from a given node to another given node of thegraph. Perfect transfer is possible if a non-zero and finite time t exists at which fidelity is unity. Mostoften the value of this t is not very important for our work as is the check for a perfect transfer. If perfecttransfer is possible for some time t , it is enough evident to classify the possible graphs which support perfectquantum state transfer. We plot the fidelity as a variation of time as a parameter in the problem. Formirror-symmetric graphs between a chosen pair of nodes, the fidelity is periodic as a function of time for thegiven pair of nodes. If periodicity does not hold, then fidelity follows quite complicated variation w.r.t. timeevolution in large graphs. Numerics has been performed in Wolfram Mathematics 11.3. Example: 1
This is the case of a 2-clique signed graph as shown in figure 3.1. This is the simplest example of net regularbalanced signed graph that satisfies Theorem 1 and Theorem 2. Perfect transfer is possible from node 1 tonode 3 and vice versa and also node 2 to node 4 and vice versa (shown in green). Perfect transfer is forbiddenfor the rest combinations, which are all adjacent (shown in red). All vertices are negatively marked accordingto canonical marking scheme. Graph is mirror symmetric between 1 & 3 and 2 & 4 and hence the transfer isperiodic with a period 2 t = π , with t as the time for perfect transfer from one node to another. All theseproperties are summarised by finding that the fidelity is F ( t ) = sin ( t ) (3.1)for both the cases. For most cases, the fidelity is not at all simple to calculate in compact analytical form asabove and instead only numerical computation is possible.Figure 3.1: 2-clique signed graph. Simplest case for a signed graph that is net-regular and balanced withmarkings as shown. All edge weights are ±
1, with solid lines as +1 and dotted lines as -1.The first self corona product is shown in figure 3.2More self-corona products with the seed can be found out in similar fashion. The fidelity between 1 &3 and 2 & 4 is observed to decrease as m increases. This is plotted in figure 3.3. This implies that Coronaproduct for this graph does not support perfect state transfer and it satisfies the conclusions drawn in theprevious section. These corona products are balanced and any signed balanced graph has the same transferdynamics as its unsigned version where (in this case), corona product perfect state transfer is forbidden forLaplacian and adjacency constructions.Inferences from this example are as follows: 28igure 3.2: First corona product ( m = 1) of signed 2-clique.Figure 3.3: Plot of maximum fidelity max[ F ( t )] vs m between nodes 1 & 3 and 2 & 4. • Fidelity between any two nodes in any graph decreases as m increases (more graph bulk increases) formost (not all) graphs. • Symmetry is important factor. In many cases, symmetric ends support perfect transfer.
Example: 2
There is not much similarity between graph dynamics when we change a graph slightly. This can be seen inthis example. It does not support perfect state transfer between any given pair of nodes over all vertices.Fidelity below 90% is practically very bad in terms of experiments and hence this example is a bad archi-tecture. The maximum fidelity allowed by this seed graph is only 0.68. However, the first corona product ofthis graph allows the same fidelity between 1 & 3 which does not decay (another important observation).Figure 3.4: Another non-trivial example forming a signed hexagon K .29 xample: 3 In this example of signed K , the near perfect quantum state transfer is possible between diametricallyopposite ends. Also notice that the effective distance between these ends is more than the length of a n = 3chain yet a near perfect transfer is possible due to the signed graph nature and also the cyclicity imposed.The corona products of the graph offers decaying fidelity similar to the 2-clique graph and much lessfidelity on the rest of the pair of nodes.Figure 3.5: Another non-trivial example forming a signed octagon K . Near perfect state transfer is possible. Example: 4
This is the fist non-trivial and simplest example of a signed unbalanced graph that satisfies the net-regularityand net-negative degree to make use of the concerned theorems for the adjacency and Laplacian eigenvaluesand eigenvectors. It allows prfect state transfer for two symmetric adjacent nodes as shown in figure 3.6. Theimportance of this graph is that it is not proved to show forbidden perfect state transfer for corona products.However, the dynamics of the corona product follows the similar results as for many other examples that theperfect state transfer is not supported after the corona products and fidelity between any pair of given nodesdecays as the order increases. But it leaves the question of exploring other variations of similar graphs whichare signed and unbalanced.Figure 3.6: First non-tivial example that satisfies the theorems and is unbalanced.
Example: 5
This is another signed variation of the K with more connectivity, however, the graph is still net-regular totake the advantages of the theorems. The graph is not symmetric. In contrast to symmetry argument before,this still allows perfect transfer between 1 & 5 and near perfect transfer in other three pair of nodes markedwith yellow in 3.7. The graph is not complete as every node is not connected to every other node as can beseen trivially. This seed as a graph is very well constructed example for perfect state transfer. And this isa very peculiar example with the property that the fidelity for perfect transfer between 3 & 5 is preservedeven with corona products of higher order and it is also conserved for the other three near-perfect transfer30nds. This graph for these pair of nodes behaves as being invariant for fidelity for corona products. This isgenerally true for other nodes as well that the fidelity does not decay with the product order m . However,after the corona product, new nodes are observed to have less fidelity than any pair on the seed itself.Figure 3.7: Another variation of signed net-regular K with more connectivity. Example: 6
These are two signed and conjugate pair variations of K as shown in figure 3.8. These two examplesindicate that signs and connectivity of graphs change the perfect transfer with strong dependence. Perfecttransfer is allowed between 3 & 6 ends in both graphs. This also proves that these conjugate signing schemesare equivalent in perspective of perfect state transfer problem, however they are two very different graphs.Symmetry argument holds for this graph and perfect state transfer is periodic. For the corona product,fidelity between any two nodes chosen initially is conserved and also perfect transfer is possible between thesame nodes even with the higher order corona products much similar to example 5.Figure 3.8: Two conjugate signed versions of K . Example: 7
This example is a fully connected version of K . One negative edge allows perfect transfer between thesame edge, indicating the importance of negative edges. There is very rich dynamics of this graph to see thedifferent combinations of signed edges in different signed versions. This is symmetric between nodes 7 & 8. Example: 8
Another signed and complete version of the 2-clique graph. Allows perfect state transfer between the samepair of nodes as the initial graph. has exactly the same dynamics. The fidelity decreases with m . No perfect31igure 3.9: Another variation of signed net-regular K with more connectivity.state transfer possible in the higher corona products implying it is also a bad seed graph to start with. Thegraph should be chosen such that the fidelity is non-decreasing between any pair of nodes with increasingbulk on the graph.Figure 3.10: Another variation of signed 2-clique K with more connectivity. Example: 9
One signed unbalanced graph and another its completed connected version. Both these graphs allow onlybelow 0.5 fidelity and the corresponding corona products are even far below this value. Also, for thesenon-symmetric graphs, the flow of fidelity with time is very irregular and chaotic as shown in figure 3.12.Figure 3.11: Another variation of signed net-regular K with more connectivity. Example: 10
This is not a net-regular graph and hence we cannot take the advantages of the theorems for the eigenpairconstruction. However, the manual computations reveal that no perfect state transfer s possible betweenany nodes in seed as well as the corona products. It gives very low fidelity for any pair of nodes on the seedas well as the corona products. It can be looked in contrast to the linear chain of two-links where perfect32igure 3.12: Typical flow of fidelity F ( t ) w.r.t. time t for a pair of nodes of example 9.transfer is possible. We conclude that the perfect transfer is lost very quickly upon adding on extra link inbetween for signed as well as unsigned version.Figure 3.13: Non-cyclic signed graph. The task was to check by the construction of examples that signed corona graphs can support perfect statetransfer which can be seen from these special examples. Around 30 examples were constructed while these10 were important. In summary: • We see that perfect state transfer is possible for certain signed graphs which preserve unity fidelityunder Corona product for certain pair of nodes. Thereby showing that under signed graphs we canrecover the PST in Coronas. • This indicates a class of possible graphs which sustain perfect transfer in contrast to Theorem 4.1 in[19] which forbids perfect transfer in unsigned graphs • Fidelity for some node pairs increase while for others it decreases (this needs to be classified analytically) • Symmetries in the graph allow to choose the pair of nodes for perfect transfer33 hapter 4
Scalable and routing enabled networkfor Perfect State Transfer
In the light of section 1.7 and section 1.9, there are limitations to routing and transfer distance. Transferdistance was tackled in [20] as presented in section 1.10 as Cartesian product resulting for PST for the pairof antipodal points on the hypercube of any order. This is limited to only antipodal points and cost of theconstructing such large number of edges is high just to enable PST over two given vertices on the graph. Inthis chapter we aim to propose a solution to both of these problems. In our state transfer scheme, • Arbitrary number of vertices n (intrinsically qubits) is allowed for the qubit network • Perfect state transfer is enabled from all-to-all vertices on the graph in at most time 2 t with the samefidelity of unity • Enables a growing network architecture for scalability of quantum network while preserving both theabove propertiesWe assume that a quantum communication network is a connected graph, allowing perfect quantum statetransfer between any two vertices. The above three features can be enabled when we have the freedomof edge switching, that is, we are allowed to switch off and on the couplings (edges) in the graph. Thiscorresponds to switching off and on the interaction between the qubits. We justify this requirement boththeoretically and experimentally (in chapter 6). A connected graph is described in a graph theoretic fashion,which has a path that is a sequence of vertices and edges between any two vertices. Therefore, the graphoffers a classical platform for a quantum mechanical operation. There is no graph other than K allowingperfect state transfer between any two vertices. In general, the perfect state transfer is possible between afew specific vertices, in a larger graph. Increasing the number of attempts for state transfer makes is limitedbetween two specific vertices only. It leads us to the conclusion that only quantum mechanical process is notsufficient to fulfill our requirements, that is the perfect state transfer between any two vertices of a graph.Therefore, we propose a hybrid of combinatorial and quantum information theoretic method, such that, aperfect quantum state transfer is possible between any two vertices of the graph. Our results in this workhold both for XY as well as the Laplacian coupling Hamiltonian. Let G = ( V ( G ) , E ( G )) be a graph with V vertices. We label the vertices by the integers 0 , , , . . . ( | V | − v ∈ { , , . . . ( | V | − } has a ( k + 1) term binary representation bin( i ), where 2 k < | V | ≤ ( k +1) ,for k = 1 , , . . . . Now, | bin( v ) (cid:105) represents a quantum state vector in C ( k +1) = C ⊗ C ⊗ · · · ⊗ C (( k + 1)-times). For example if v = 2, then bin( v ) = 10 and | bin( v ) (cid:105) = | (cid:105) = | (cid:105) ⊗ | (cid:105) , where | (cid:105) = (cid:20) (cid:21) and | (cid:105) = (cid:20) (cid:21) are the standard basis vectors. This coincides with the first excitation subspace of the XY and LaplacianHamiltonian. 34orresponding to the vertex v we also associate a state vector | v (cid:105) ∈ C | V | . If we denote a vector in C | V | as | m (cid:105) = ( m , m , . . . m | V |− ) T then the vector | v (cid:105) is given by m u = 0 for u (cid:54) = v and m v = 1. We define a lineartransformation R : C k → C ( k +1) by R | m (cid:105) = ( m , m , . . . m | V |− , , , . . . ( k +1) − | V | ) -times)) T whichwill help us to extend over to a larger Hilbert space by appending extra fixed labels for a state. Therefore,now R | m (cid:105) belongs to C k +1 . When we have n number of vertices, where 2 k ≤ n ≤ k +1 , then we adopt thelabeling for 2 k +1 vertex graph. Cartesian product was presented in section 2.3.2. Here, for hypercubes, we are concerned with the Cartesianproduct of a graph with itself. The Cartesian product of G with itself is denoted by G (cid:3) = G (cid:3) G . Similarly,for any natural number k we denote the k -th Cartesian product as G (cid:3) k . The Cartesian product is associative.In general, it is commutative when the graphs are not labelled. Also, the graphs G (cid:3) H and H (cid:3) G are naturallyisomorphic.A hypercube Q k of dimension k is a graph with 2 k vertices for k = 0 , , , . . . . For k = 0 the graph Q consists of a single vertex. For k = 1 we have two vertices and an edge in the hypercube Q , which can alsobe described as the complete graph K with two vertices. When k ≥ Q k = ( K ) (cid:3) n . Hence,the Cartesian product of two hypercubes is another hypercube, that is Q i (cid:3) Q j = Q i + j [41].Let the vertices of K are given by 0 and 1. Then the vertices of Q k are represented by the elements inthe set { , } × k = { , } × { , } × · · · × { , } ( k -times). Note that the elements of { , } × k are the k -termbinary representations of the natural numbers 0 , , . . . k . Hence bin( v ) denotes the label of the vertex v inthe hypercube. An important property for the hypercubes is that any two vertices u and v in Q k are adjacentwhen the Hamming distance between bin( u ) and bin( v ) is 1, for k > Definition 4. Antipodal points : Two vertices u and v which are labeled by the binary sequences bin( u ) =( u j ) ( k − j =0 and bin( v ) = ( v j ) ( k − j =0 in the hypercube Q k are called the antipodal points if u j (cid:54) = v j , for all j .For example, the antipodal points of Q are 0 and 1. The antipodal points of 00 in Q is 11 and 01 for10. In case of Q , we can write the antipodal points as pairs (000 , , (001 , , (101 , , Q k we have C | V ( Q k ) | = C ( k +1) . Therefore, the linear operator T isthe identity function for this case.The hypercube of dimension 0 is a single vertex labeled by 0 only. The hypercube of dimension 1 isdenoted by Q which is depicted as follows 0 1Note that Q is the complete graph with two vertices K . The hypercube Q = K (cid:3) K has four verticeswhich is represented by 0001 1011Also, the hypercube Q = Q (cid:3) K has 8 vertices which is given by000001 010011101 111110100 35n general, the hypercube Q k +1 = Q k (cid:3) K for k ≥
2. The vertex labels of Q k are the distinct binarysequences of length k . The Cartesian product between Q k and K makes the number of vertices doubled aswell as add an additional index to the vertex labeling.A hypercube of dimension k consists of smaller hypercubes of dimension i for i = 0 , , . . . ( k − i are unique upto isomorphism. But, all the hypercubes Q i embedded in Q k have different vertex labelling. The number of distinct hypercubes Q i embedded in Q k is given by (cid:0) ki (cid:1) ( k − i ) [42]. The next lemma suggests how to distinguish a particular subhypercube which is embedded in a largerhypercube. Lemma 1.
Let the 2 k vertices of the hypercube Q k be labeled by the binary sequences bin( v ) = ( v j ) ( k − j =0 , v j ∈{ , } . For some i with 1 ≤ i < k , consider ( k − i ) integers { m t : t = 1 , , . . . ( k − i ) } , such that0 ≤ m < m < · · · < m ( k − i ) ≤ ( k − M = ( M t ) ( k − i ) t =1 . Corresponding to theset of indices { m t } and the binary sequence M construct a set of vertices V i = { v : bin( v ) = ( v j ) ( k − j =0 , v m t = M t for t = 1 , , . . . ( k − i ) } ⊂ V ( Q k ). Then the induced subgraph of Q k generated by V i is isomorphic to thehypercube Q i Proof.
To label a vertex in Q i we need a binary sequence of length k . For constructing the vertex set V i wekeep ( k − i ) terms in the sequence constant, which are equal to the elements of M . Therefore, number ofelements in V i is 2 i , which is the number of vertices in Q i .Let H = ( V ( V i ) , E ( H )) be the induced subgraph of Q k generated by V i . Clearly, V ( Q i ) = V i . Wewrite bin( v ) = ( v j ) ( k − j =0 = ( v , v , . . . v ( k − ). Given the set of indices { m t } define bin( v ) (cid:9) { m t } =( v , v , . . . v m − , v m +1 , . . . v m − , v m +1 , . . . v k ), that is we remove the terms of bin( v ) corresponding tothe indices in { m t } . Clearly, after removing ( k − i ) terms from bin( v ) we find a new binary sequences oflength i . Now, define a function f : V i → V ( Q i ), such that, f ( v ) = bin( v ) (cid:9) { m t } and prove that it is bijectivefunction. For any two different u and v in V i , we have u m t = v m t . When these equal entries are removedwe get two different binary sequences. Hence, the function f is injective. Consider any element w ∈ V ( Q i ).Note that, bin( w ) = ( w j ) ( i − j =0 , w i ∈ { , } . This binary sequence of i terms can be extended to a binarysequence of k terms by including the elements M t of M at the m t -th index. It concludes that f is surjective.Therefore f is a bijective mapping. This function does the reverse of the function R | . (cid:105) defined previously.Consider two adjacent vertices u and v in H ( V i ). As H ( V i ) is a subgraph of Q k the binary sequencesbin( u ) and bin( v ) has Hamming distance 1. The construct of V i suggests that m t -th entries of bin( u ) andbin( v ) are equal, which are removed by the function f . Therefore, the sequences f ( u ) and f ( v ) has Hammingdistance 1. As f ( u ) and f ( v ) represents two vertices in Q i , they are adjacent.Alternatively consider two adjacent vertices in Q i which are labeled by binary sequence of length i . Thesesequences have Hamming distance 1. We add equal entries at equal indexed position to get their inverse in V i ⊂ V ( Q k ). The inverses also have Hamming distance 1. Hence, they are adjacent in Q k . As H ( V i ) is aninduced subgraph of Q k , they are also adjacent. Therefore, f is a graph isomorphism. Corollary 1.
Consider two hypercubes Q p and Q q with p > q . Then there is an induced subgraph of Q p with 2 q vertices, which is isomorphic to Q q . Proof.
The vertices of Q p can be labeled by the sequence of binary digits bin( v ) = ( v j ) ( p − j =0 = ( v , v , . . . v ( p − )of length p . Similarly, the vertices of Q q can be given by the sequences bin( u ) = ( u j ) ( q − j =0 = ( u , u , . . . u ( q − ).Now we construct a set of vertices V q = { v : v i = 0 for i = 0 , , , . . . ( p − i − } . Clearly, V q has 2 q vertices,which is the number of the vertices in Q q . Consider the induced subgraph of Q p generated by V q which isisomorphic to Q q . It can be easily shown by considering that the adjacent vertices have hamming distanceone.Recall that, a Hamiltonian path is a path in a graph that visits each vertex only once. A Hamiltoniancycle is a Hamiltonian path which is a cycle. Every hypercube Q n with n > Corollary 2.
Let the vertices v of a hypergraph Q i are labeled by the binary sequences bin( v ) = ( v j ) ( k − j =0 where k > i , such that, the Hamming distance between the labels of any two adjacent vertices is one. Then,there are a sequence of non-negative integers { m t } of length ( k − i ), such that, 0 ≤ m < m < · · · < Sir, please find if the operation is standard in the literature of coding theory or Boolean functions. ( k − i ) ≤ ( k − M = ( M t ) ( k − i ) t =1 which determine a set of vertices V i = { v : bin( v ) =( v j ) ( k − j =0 , v m t = M t for t = 1 , , . . . ( k − i ) } ⊂ V ( Q k ). Then the induced subgraph of Q k generated by V i hasthe same vertex labeling as of the hypercube Q i Proof.
Assume that v = v , v , . . . v i , v i +1 = v is a Hamiltonian cycle starting and ending at a vertex v ∈ V ( Q i ). Let the vertex labels v p are given by bin( v p ) = ( v pj ) ( k − j =0 for p = 1 , , . . . i . The Hammingdistance between bin( v ) and bin( v ) is 1. Therefore, there is an index q such that v j = v j when j (cid:54) = q .Similarly there are indices q , q , . . . q i , such that v j = v j for j (cid:54) = q ; v j = v j for j (cid:54) = q , and so on.Note that, the binary sequence bin( v ) has length k . Therefore, q , q , . . . q i may not be all distinct. Inthe sequence bin( v ) the element v j represents 0 or 1. Therefore, to represent q , q , . . . q n we need only i positions in the sequence bin( v ), which are given by q (cid:48) , q (cid:48) , . . . q (cid:48) i . Now define the entries of { m t } , suchthat, 0 ≤ m < m < · · · < m ( k − i ) ≤ ( k −
1) and m t / ∈ { q (cid:48) , q (cid:48) , . . . q (cid:48) i } for any t . Construct the sequence M = { M t } ( k − i ) t =1 , such that M t = v m t . Note that, for any t we have M t = v m t = v m t = · · · = v i m t ,otherwise the condition of unite Hamming distance between the vertex labeling of adjacent vertices will beviolated.Now, in the hypergraph Q k we construct the set of vertices V i = { v : v m t = M t for t = 1 , , . . . ( k − i ) } with respect to the sequences { m t } ( k − i ) t =1 and M . Clearly, the induced subgraph G ( V i ) of Q k generated by V i is a hypercube Q i . The vertex labeling of Q i considered in the statement and vertex labellings of G ( V i ) areequal because of the particular choice of { m t } and M . To define quantum walk and state transfer on the graphs we associate a basis vector of C | V | to the individualvertex v ∈ V ( G ). A continuous time quantum walk on a graph G is defined using the Schr¨odinger equationwith the A ( G ) as the Hamiltonian [22]. If | ζ ( t ) (cid:105) ∈ C | V | is a time-dependent quantum state, then the evolutionof the quantum walk is given by | ζ ( t ) (cid:105) = exp( − itA ( G )) | ζ (0) (cid:105) , (4.1)where | ζ (0) (cid:105) is the initial state vector. The probability for getting the quantum state localised at the vertex v at time t is given by | (cid:104) v | ζ ( t ) (cid:105) | . We say G has a perfect state transfer from vertex u to vertex v at time t if | (cid:104) v | exp( − it A ( G )) | u (cid:105) | = 1 . (4.2)This is the same condition for perfect state transfer expressed in graph theoretic fashion [43] and impliesequation 1.11. When {| v (cid:105) : v ∈ V ( G ) } represents the computational basis of C | V | , we say that the graph G allows a perfect state transfer from the vertex u to v if the ( u, v )-th term of exp( − itA ( G )) has magnitude 1.Besides in [27], a necessary and sufficient condition is proved for PST. The well-known examples of graphsallowing perfect state transfer over long distances are described below [20][21]1. The complete graph K with two vertices allow perfect state transfer between its vertices in time t = π/ P has perfect state transfer between its end vertices in time t = π/ √ π/
2. And any order of Cartesian product of P has PST between its antipodal vertices in the sametime π/ √ Recall the known result, which will be applicable for proving the next lemma and its corollary.
Lemma 2.
Let A = diag { B , B , . . . B k } be a block diagonal matrix, where B i are square matrices ofarbitrary order for i = 1 , , . . . k , then exp( A ) = diag { exp( B ) , exp( B ) , . . . exp( B k ) } .37et G = ( V ( G ) , E ( G )) and H = ( V ( H ) , E ( H )) be two graphs. The union of G and H is denoted by G ∪ H = ( V ( G ∪ H ) , E ( G ∪ H )) where V ( G ∪ H ) = V ( G ) ∪ V ( H ) and E ( G ∪ H ) = E ( G ) ∪ E ( H ) [36]. Lemma 3.
Let G be a connected graph with perfect state transfer between two vertices u and v at time τ .Also, let H be a connected graph with perfect state transfer between two vertices p and q , at time τ . Thenthe graph G ∪ H has state transfer between u and v as well as p and q , at time τ . Proof.
Let the graph G has | V | vertices. The graph G has perfect state transfer between the vertices u and v at time τ . It indicates | (cid:104) u | exp( − iτ A ( G )) | v (cid:105) | = 1, where | u (cid:105) and | v (cid:105) are the state state vectors in C | V | corresponding to the vertices u and v , respectively. Similarly, if H contains | V | vertices, we have | (cid:104) p | exp( − iτ A ( G )) | q (cid:105) | = 1, where | p (cid:105) and | q (cid:105) are the state vectors in C | V | corresponding to p and q respec-tively. We know that the graph G ∪ H has | V | + | V | vertices. Corresponding to the vertices u, v, p , and q definestate vectors in C | V | + | V | as | u (cid:48) (cid:105) = (cid:20) | u (cid:105) (0) | V |× (cid:21) , | v (cid:48) (cid:105) = (cid:20) | v (cid:105) (0) | V |× (cid:21) , | p (cid:48) (cid:105) = (cid:20) (0) | V |× | p (cid:105) (cid:21) and | q (cid:48) (cid:105) = (cid:20) (0) | V |× | q (cid:105) (cid:21) ,respectively. Note that A ( G ∪ H ) = (cid:20) A ( G ) (0) | V |×| V | (0) | V |×| V | A ( H ) (cid:21) or exp( − iτ A ( G ∪ H )) = (cid:20) exp( − iτ A ( G )) (0) | V |×| V | (0) | V |×| V | exp( − iτ A ( H )) (cid:21) . (4.3)Now, | (cid:104) u (cid:48) | exp( − iτ A ( G ∪ H )) | v (cid:48) (cid:105) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) u (cid:48) | (cid:20) exp( − iτ A ( G )) (0) | V |×| V | (0) | V |×| V | exp( − iτ A ( H )) (cid:21) (cid:20) | v (cid:105) (0) | V |× (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:2) (cid:104) u | (0) ×| V | (cid:3) (cid:20) exp( − iτ A ( G )) | v (cid:105) (0) | V |× (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | (cid:104) u | exp( − iτ A ( G )) | v (cid:105) | = 1 . (4.4)Similarly, | (cid:104) p (cid:48) | exp( − iτ A ( G ∪ H )) | q (cid:48) (cid:105) | = 1. Therefore the graph G ∪ H has state transfer between u and v as well as p and q , at time τ . Corollary 3.
Let G be a graph with state transfer between the vertices u and v at time τ . Then the graph G ∪ { v } ∪ { v } · · · ∪ { v k } has perfect state transfer between u and v , where v , v , . . . v k are isolated verticesat time τ . Proof.
Proof follows trivially.
Definition 5.
Hopping: In our formalism, the hopping on a graph G = ( V ( G ) , E ( G ) is a combination of aClassical (C), a true Quantum hopping (Q), and again a Classical process (C) which consists of the followingsteps:1. Switch off the selected edges (Classical):
Construct a subgraph H = ( V ( H ) , E ( H ) of G such that V ( H ) = V ( G ) and E ( H ) ⊂ E ( G ). Let S = E ( G ) − E ( H ).2. Perfect state transfer (Quantum):
Perform quantum operations on H , such that, the quantumstate can be transferd from vertex u to vertex v in the graph H .3. Switch on the edges (Classical):
Construct the graph G form the graph H by adding the edgesfrom S in the graph G .We call these processes together as CQC-hopping. So, CQC-hopping is a process of a classical switching,followed by true quantum evolution (hopping), followed by another switching of edges. Now we have thefollowing important result. 38 .5.1 Perfect State Transfer in hypercubes from all-to-all vertices in single CQC-hopping In this section, we prove that, given any hypercube of any dimension, we enable the perfect state transferfrom all-to-all vertices of the graph. This is in contrast to the main result of [9] presented in literature section1.11 where only pair of antipodal vertices are PST enabled. This is accomplished with the edge switching.
Theorem 6.
The perfect state transfer is possible between any two vertices of a hypercube Q k by a singlehopping CQC process. Proof.
Consider any two vertices u and v in the hypercube Q k which are labeled by the binary sequencesbin( u ) = { u j } ( k − j =0 and bin( v ) = { v j } ( k − j =0 , respectively. If u and v are the antipodal points of the hypercubethen there is a state transfer between u and v . In this case, we need no edge to switch on or switch off.Let u and v are not the antipodal points of the hypercube Q k . Then there is a sequence of indices { m t } ,such that, 0 ≤ m < m < · · · < m ( k − i ) ≤ ( k −
1) and u m t = v m t = M t holds. Corresponding to the setof indices { m t } and the binary sequence M construct a set of vertices V i = { v : bin( v ) = ( v j ) ( k − j =0 , v m t = M t for t = 1 , , . . . ( k − i ) } ⊂ V ( Q k ). Denote the induced subgraph of Q k generated by V i as G ( V i ). Usinglemma 1, we find that G ( V i ) is isomorphic to the hypercube Q i .Now we perform a CQC hopping on Q k . First, we switched off all the edges which are not includedin the induced subgraph G ( V i ) that is in E ( Q k ) − E ( G ( V i )). The new graph can be expressed as H = Q i ∪ ( V ( Q k ) − V i ), where ( V ( Q k ) − V i ) denotes the set of isolated vertices. The considered vertices u and v are the antipodal points of G ( V i ) or Q i . Hence, there is a perfect state transfer between u and v . Bycorollary 3 we find that there is state transfer between the vertices u and v in the graph H . After statetransfer we switched on the edges in E ( Q k ) − E ( G ( V i )). In this way we can transfer the sate between anytwo vertices of a hypercube Q k .This theorem suggests the number of edges to switch off and switch on in CQC process. When u and v are two vertices belonging to the hypercube Q k , following this theorem we construct a sub-hypercube Q i which is essential for the state transfer. Now Q k has k k − edges and Q i has i i − edges. Therefore, we needto switched off k k − − i i − = 2 i − ( k k − i − i ) edges, in the hypercube Q k . We can also present this resultas algorithm 1. Example 1.
Consider two arbitrary verticees u and v in Q where bin( u ) = 00101101 and bin( v ) = 10011000.Note that, u j = v j for j = 1 , m = 1 , m = 4, and m = 6, as well as M = 0 , M = 1 and M = 0. Following lemma 1 we construct a set of vertices V and an induced subgraph H ( V ) of Q whichis isomorphic to Q . The nodes u and v correspond two antipodal points in H ( V ). To make state transferbetween u and v we switched off and switched on all the edges in E ( Q ) − E ( H ( V )). Given any natural number n there is a natural number k , such that, 2 k ≤ n < ( k +1) and n has a ( k + 1)-term binary representation. For any natural number n there is a graph G which allows perfect state transferbetween any two nodes in two CQC-hoppings. The graph G can be constructed as follows. Procedure 1. Constructing graphs allowing perfect state transfer:
Let the natural number n canbe written as n = a p ( k − p ) + a p k − p + a p k − p + . . . , where p = 0 and a p = a p = · · · = 1.1. Use 2 k vertices to construct Q k and label its nodes v with a ( k + 1)-term sequence v = ( v j ) kj =0 where v j ∈ { , } and v = 0.2. Now fix v = 1 for all remaining constructions.3. For every p i with 0 < p < p < . . . construct a hypercube Q k − p i with 2 k − p i nodes. First label thevertices with a ( k − p i )-term sequence u = ( u j ) k − p i − j =0 where u j ∈ { , } . A vertex can be included inat most one hypercube.4. Re-label the vertices of Q k − p i with a ( k + 1)-term sequence v = ( v j ) kj =0 where v j ∈ { , } with v = 1 , v j = 0 for j = 1 , , . . . ( p i + 1) and v p i +1+ j = u j for j = 0 , , . . . ( k − p i − lgorithm 1 Find the sub-hypercube for perfect state transfer between two arbitrarily chosen vertices.
Require:
Vertices u and v of Q k for state transfer. Ensure: bin( u ) = { u j } ( k − j =0 where u j ∈ { , } for all u ∈ V ( Q k ). if u j (cid:54) = v j for all j = 0 , , . . . ( k − then Perfect state transfer between u and v . return Q k else Count = 0m = [] { List of indices m t such that u m t = v m t . } M = [] { List of common indices in bin( u ) and bin( v ). } for j ← k − doif u j = v j then Count = Count + 1m.insert(j)M.insert( u j ) end ifend for Construct V ( Q i ) = { v : bin( v ) = ( v j ) ( Count − j =0 } { Here the vertices are labelled by the binary sequencesof length i . } G = ( V ( G ) , E ( G )). for v ∈ V ( Q i ) do v : bin( v ) = ( v (cid:48) j ) ( k − j =0 { We want to label the vertices by the binary sequences of length k . } for j ← k − do C = 0 if j = m t then v (cid:48) m t = M t C = C + 1 else v (cid:48) j = v j + C end ifend forend for V ( G ) = V ( Q i ). E ( G ) = Set of edges in the induced subgraph of Q k generated by V ( Q i )Perfect state transfer between u and v . return G end if
40. Join pairs of nodes u and v with an edge where u ∈ V ( Q k − p i ) and v ∈ V ( Q k ) as well as hammingdistance between their labeling is 1.6. Fix v p i +1 = 1 for all remaining constructions.7. Similarly, relabel all other hypercubes and create edges between them.Note that, a single node is a hypercube of dimension 0. Lemma 4.
Let n = 2 k + a k − + a k − + . . . a k where a , a , . . . a k ∈ { , } and there are integers p < p < . . . . such that a p = a p = · · · = 1. Then, the graph G with n vertices which is constructed byfollowing the procedure 1 has T ( n ) edges, where T ( n ) = (cid:80) p i (cid:2) ( k − p i )2 ( k − p i − + i × ( k − p i ) (cid:3) . Proof.
Recall that any hypercube Q k has k k − edges. For every p i , with a p = a p = a p = · · · = 1 weconstruct a hypercube Q k − p i . Therefore we add (cid:80) a pi (cid:54) =0 ( k − p i )2 ( k − p i − edges in the graph G .Consider a vertex u ∈ Q ( k − p ) . As a vertex of G , the labeling of u is given by the sequence bin( u ) =( u j ) kj =0 . Note that u = 1. Therefore, there is exactly one vertex v = ( v j ) kj =0 in Q k such that, v = 0 and v j = u j for j = 1 , , . . . k . Clearly the Hamming distance between bin( u ) and bin( v ) is 1. Hence, u and v are adjacent. Therefore every vertex of Q ( k − p ) is adjacent to only one vertex of Q k . Therefore, there are2 ( k − p ) edges ( u, v ) ∈ E ( G ) such that u ∈ V ( Q k ), and v ∈ V ( Q ( k − p ) ).In a similar fashion, we can justify that there are 2 k − p edges ( u, v ) ∈ E ( G ) such that u ∈ V ( Q k ), and v ∈ V ( Q ( k − p ) ). In addition, there are another 2 k − p edges ( u, v ) ∈ E ( G ) such that u ∈ V ( Q ( k − p ) ), and v ∈ V ( Q ( k − p ) ). Adding we get there are 2 × k − p edges whose one end vertex in V ( Q ( k − p ) ) and anotherend vertex is either in V ( Q k ) or in V ( Q ( k − p ) ).Extending we get, there are 3 × k − p edges whose one end vertex in V ( Q ( k − p ) and another end vertexis either in V ( Q k ) or in V ( Q ( k − p ) ) or in V ( Q ( k − p ) ). Collecting all the edges we get the total number ofedges, which is mentioned in the statement. Theorem 7.
Let u and v be two vertices in the graph G of order n which is constructed following theprocedure 1. Then there is a perfect state transfer between u and v by at most two CQC-hoppings. Proof.
According to the construction procedure there are hypercubes containing the vertices u and v . If u and v belongs to same hypercube then by theorem 6 state transfer from u to v is possible by one CQC-hoppingonly.Let u and v belong two two different hypercubes. The procedure 1 indicates that no two hypercubes canbe of equal size. For simplicity let u ∈ Q p and v ∈ Q q where p > q . Now corollary 1 suggests that Q p has aninduced subgraph isomorphic to Q q . Therefore, there is a vertex w in Q p which is equivalent to v in Q q .The vertices u and v are represented by binary sequences of length ( k +1). Note that, the binary sequencesrepresenting v and w have Hamming distance 1. Therefore, there is an edge ( v, w ) in the graph G .Now the first CQC-hop allows state transfer from v to w . The second CQC-hop allows state transferfrom w to u . Hence, the proof.This theorem suggests the number of edges to switch off and switch on in CQC-hopping process. Let u and v belong to two different hypercubes. For first CQC we need only one edge. Therefore, we need toswitched off T ( n ) − Q q . Recall that number ofedges in Q q is q q − . We need to switched on only these edges. When the state transfer is done (state hasbeen recovered at the desired vertex), we shall switched on all the edges in G as desired.Construction of this graph can be seen as a growing network. This tool allows us for constructing a graph G with an additional node from a given graph G allowing state transfer in two-CQCs. The motivation forsuch an argument is due to the fact that experimentally only a few number of qubits are added with everytechnological improvement in quantum technologies, therefore, suggesting the need of an architecture whichallows the growth as addition of one qubit each time. In the following procedure, we propose the growing network architecture where each new qubit can be addedto the existing network. It suffices to start with a 2 k -vertex hypercube and grow it to the next hypercube of2 k +1 vertices. 41 rocedure 2. Growing network method Let Q k be a hypercube with 2 k vertices which are labeled byall possible binary sequences of length k . Follow the steps below to construct a graph G with number ofnodes < k +1 allowing state transfer with two CQC-hoppings. Note that, we can add at most (2 k −
1) newvertices with Q k to construct the graph G .1. Relabel all the vertices of Q k with binary sequences of length ( k + 1) such that the left-most elementof the sequence is 0 and the others are equal to the old labeling of Q k .2. The vertex labeling of the l -th new vertex will be given by a ( k + 1) term binary sequence initiated by1. The remaining k -terms are the k -term binary representation of l .3. The new vertex will be adjacent to all other vertices with Hamming distance 1.Each application of this procedure will increase the number of vertices in the initial graph Q k for anyinteger k >
0. In each step the new graph G will allow perfect state transfer in two CQCs. Equivalent definition:
The above procedure can be see in terms of binary addition (denote it as ⊕ )where the one bit is carried to the left. For any n in the range given above, the first node is | ... (cid:105) , whichis (2 k + 1)th node. For all subsequent nodes just add a 1 via binary addition to the existing labeling of thelast added node with any possible carry to the left. Hence, the next node will be | ... ⊕ (cid:105) = | ... (cid:105) .The next node will be | ... ⊕ (cid:105) = | ... (cid:105) where one carry is taken over to the second index from theright and so on until the desired vertex. The second last node will be | ... (cid:105) and adding 1 further to thefirst index will give give the last node | ... (cid:105) which completes the 2 k +1 -vertex hypercube. For furtheraddition of nodes simply append one more index and start the same set of steps. Therefore, it can easily beseen that any (2 k + 1 + m )th node is simply | ... (cid:76) m − j =1 (cid:105) , where 2 k + m = n , and this determines the fullconnectivity of any new node with the existing graph, which defines a simple graph upon addition of everynew node. In our CQC-hopping scheme we have given the optimal number of hoppings required to transfer a statefrom any vertex to any other vertex in the graph for arbitrary number of total vertices. At least one ofthe two CQC-hoppings for a dual hop is a hypercube jump, where the involved hypercube maybe of a largedimension. This state transfer will be ensured by ensuring the correct set of edges where the total numberof edges are indeed large. If the cost of creating a large number of edges is high, experimentally, then it maypose a new problem to the architecture. It can be asked whether there exists another architecture whichminimizes the number of edges instead of CQC-hoppings. Ideally, the least number of edges are there for thecase of a path graph P n for n vertices. But then the question of finding a network is trivial. Therefore, weneed a trade-off between both of these extremes because creating and manipulating a large number of edgeswill also leave the system prone to more error in state transfer resulting in low fidelity.A trade-off between both of these extremes can be can be discussed for a neighborhood zone for twogiven vertices. The possible moves allowed under two CQC-hopping are: one-link jump ( t = π/
2) and thetwo-link jump ( t = π/ √ √ π using a total of four edges only. Incontrast, had these two vertices separated by a distance of 4 were the antipodal vertices, we would have tocreate 4 × = 32 edges in total. However, this would perform the task in single CQC-hopping in time π/ | x (cid:105) by partition into the followingsets. These sets also do not contain any vertices that are missing due to an incomplete hypercube, we assumeall these vertices to not be in these sets by construction. For any arbitrary 2 k ≤ n < k +1 , we have • Set of all adjacent vertices:
Any vertex has some i th adjacent vertex as the i th flipped index. So, theset of all these adjacent nodes is {| α xi (cid:105) ≡ | x , x , ..., x i ⊕ , ..., x n (cid:105)} ∀ i ∈ { , , ...n } . Total number ofsuch distinct nodes at most is k + 1. Each of these nodes can be reached by one unique path.42 Set of all next-adjacent vertices:
Set of next-adjacent nodes can be reached in one CQC-hopping by thetwo-link jump in time π/ √
2. All available vertices accessible via | α xi (cid:105) except trivially | x (cid:105) itself. Hencethis is the set {| β xij (cid:105) ≡ | x , x , ..., x j ⊕ , ..., x i ⊕ , ..., x k (cid:105)} ∀ i, j ∈ { , , ..., k } & i (cid:54) = j . Total numberof such distinct nodes is ( k + 1) k/ k +1 − vertex hypercube. • Set of all next-to-next adjacent vertices:
These can be reached by one-link jump followed by a two-linkjump or vice versa with time π/ π/ √
2. Using the previous argument it is simply the set defined as {| γ xijp (cid:105) ≡ | x , x , ..., x p ⊕ , ..., x j ⊕ , ..., x i ⊕ , ..., x k (cid:105)} ∀ i, j, p ∈ { , , ..., k } & i (cid:54) = j (cid:54) = p . Total numberof such distinct nodes is at most ( k + 1) k ( k − /
6. Each of these nodes can be reached by three uniquepaths if no hyperjump is involved and all the adjacent vertices are available relative to the completehypercube. • Set of all next-to-next-to-next adjacent vertices:
These can be reached by two-link jump followed byanother two-link jump with time √ π . It is the set defined as {| δ xijpq (cid:105) ≡ | x , x , ..., x q ⊕ , ..., x p ⊕ , ..., x j ⊕ , ..., x i ⊕ , ..., x k (cid:105)} ∀ i, j, p, q ∈ { , , ..., k } & i (cid:54) = j (cid:54) = p (cid:54) = q . Total number of such distinctnodes is at most ( k + 1) k ( k − k − /
24. Each of these nodes can be reached by four unique pathsif no hyperjump is involved if all vertices are available.This lets us minimize the number of edges for small distance of 4 between any two given vertices.Another argument from the physical point of view is for the decoherence error. During the quantumevolution, the system is most prone to error. In contrast, the edge deletion and creation are purely classicalphenomenon and do not introduce any direct error quantum mechanically into the system. Minimising thisevolution time by minimising the number of CQC-hoppings will be more robust approach for the implemen-tation for state transfer.
For any pair of two chosen nodes for arbitrary number of nodes n (cid:54) = 2 k for the graph, we have two subgraphscorresponding to the action-space of two adjacency matrices A and A that describe the first and second hoprespectively over the entire graph as described before. We perform the first CQC-hop from some vertex | u (cid:105) toa vertex | v (cid:105) keeping all edges switched off that belong to the second hop, followed by another CQC-hop from | v (cid:105) to | w (cid:105) while all edges corresponding to first hop are switched off. So, both the adjacency matrices haveone common element of action which is the intermediate vertex | v (cid:105) such that A | v (cid:105) (cid:54) = 0 & A | v (cid:105) (cid:54) = 0. Thismeans we act with exp( − iA t ) followed by exp( − iA t ) over the entire graph as continuous quantum walk. A and A corresponds to the first and second CQC-hop adjacency matrices respectively, resulting from theswitching. Is it possible to capture these two unitary evolution as one single unitary evolution? In the lightof Baker–Campbell–Hausdorff (BCH) formula, let us look at the commutator [ A , A ] for all possible verticeson this graph. We may classify all nodes in three sets: All nodes | u (cid:48) (cid:105) which can be reached by the repeatedaction of A (except | v (cid:105) ), the intermediate node | v (cid:105) itself and all the nodes | w (cid:48) (cid:105) that can be reached by therepeated action of A (except | v (cid:105) ). Any arbitrary chosen initial graph state linear combination of all of theseelements (which act as a basis for the action space of the two adjacency matrices), at most. Let us evaluatethe action of the commutator on a general state (un-normalised)[ A , A ]( | u (cid:105) + | v (cid:105) + | w (cid:105) ) = ( A A − A A )( | u (cid:105) + | v (cid:105) + | w (cid:105) (4.5)= (cid:88) { u (cid:48) } , { w (cid:48) } ( α u (cid:48) | u (cid:48) (cid:105) + α w (cid:48) | w (cid:48) (cid:105) ) + α v | v (cid:105) (4.6)which is not necessarily zero, except for a special initial state. Here, α λ ∈ Z are the coefficients. Therefore, A and A do not commute in general and hence we cannot write down a general evolution exp( − iA t ) · exp( − iA t ) = exp [ − i ( A + A ) t ] and it implies they have to be treated strictly as non-simultaneous unitaryevolution in time which is also expected otherwise as it is a strictly two step task, which does not allow theleakage of state out of the action space of one adjacency matrix.43enerally, for the part of our protocol which requires a dual CQC-hop when we are away from a perfecthypercube, let us define the overall adjacency matrix as a step-function in time as follows: A ( G ( t )) = (cid:40) A , for 0 < t ≤ t ,A , for t < t ≤ t (4.7)Hence, for any given pair of nodes we quickly identify A and A and perform the PST from all-to-all nodesin at most time 2 t . At time t we perform the switching and the adjacency matrix of the graph is switchedfrom A to A . In the ideal scenario, it is assumed to take no time which is realistically not the case. Forthose cases and to quantify the error introduced in the evolution due to switching by approximating the stepfunction by some function close to it and see how much error it introduces into the fidelity. Switching willintroduce error in the quantum system as it will be probed by a classical system to switch the couplingswhich is close to a classical task. The original idea [8] of perfect state transfer was to transport a state from one location to another usinga channel like mechanism which is non-computational for quantum computing usage. For example, a spinlattice channel fabricated between two quantum processors can be used to transport arbitrary state from oneprocessor to another for large scale quantum architecture. So, perfect state transfer has been seen as a meansof communication between two or more quantum computational units. This idea has been experimentallydemonstrated in [44]. The implication of such as idea involves transferring state from one specific qubit fromthe first processor to a specific qubit in the second processor. Within each processing unit, conventionalquantum SWAP gates have to be used transport the state from one qubit to another (given that we do notcare about the state of the other qubit). In fact, a large amount of cascaded SWAP operations have to beused in order to accomplish this task within each processor, if the qubits are distant, assuming one SWAPgate is one operation quantum mechanically.The possibility of application of two-qubit gates between for given qubits depends upon whether thesetwo qubits are allowed to interact, that is, that are coupled by some coupling Hamiltonian which definesan exchange interaction. Two qubits which are not directly coupled to one another cannot have two-qubitinteractions directly. This means we cannot have two qubit gates, such as SWAP or CNOT, acting betweenthese qubits. This connectivity on the network of qubits is what we impose by defining a graph scheme.Hence, in a quantum circuit, we cannot have two-qubit quantum gates between the qubits which are physicallydistant on the real processor. In theoretical quantum circuits, we assume that two qubit interactions arepossible for every pair of qubits, but the disconnectedness imposes even more number of gates to be used forthe same task. When two qubits are distant and cannot interact directly, we may have cascaded two qubitgates. For example, on a linear chain we are allowed to perform SWAP operation for every ( i, i + 1) qubitpair. To transfer a state | (cid:105) from one end to another we have to apply ( l −
1) SWAP gates, where l is thelength of the chain, and all other qubits are intialized to | (cid:105) . Similarly, we have to use a large number ofcascaded SWAP gates for to perform state transfer to distant qubits. We deduce that the minimum numberof SWAP gates required for the state transfer is equal to the minimum edge distance between two givenvertices.In this section we aim to present perfect state transfer as a quantum operation in the conventionalquantum circuit model. In chapter 6, we present a physical model which serves as high fidelity quantumprocessor along with the ability of perfect state transfer all over the computational qubits. This model servesas the physical implementation of our scalable architecture proposed in this section. Having realised inchapter 6 that same quantum computing hardware can be used for perfect state transfer, we look at perfectstate transfer from a another perspective of a quantum gate. We propose the idea of quantum computingassisted with perfect state transfer on graphs.We can take the advantage of perfect state transfer as a SWAP gate operation. Consider a quantumcircuit with qubit u in some state | ψ (cid:105) while all other qubits are initialized to zero. Notice the action of aCQC-hop state transfer from a given qubit at site ue − iA t | u (cid:105) ψ = e iφ | v (cid:105) Phase correction( ϕ ) −−−−−−−−−−−−−→ | v (cid:105) ψ (4.8)for some arbitrary phase φ , which can be corrected post the state transfer. Furthermore, if we are not at a44erfect hypercube we require another state transfer, due to equation (4.7), we have e − iA t | v (cid:105) ψ = e iφ | w (cid:105) Phase correction( ϕ ) −−−−−−−−−−−−−→ | w (cid:105) ψ (4.9)which takes us to a final vertex w . Each of these CQC-hoppings’ action is equivalent to the SWAP gatebetween the qubits at those vertices.The reported time for a iSWAP gate in [45] is reported to be around τ = 35 ns (differs for differentnature of decoherence errors incorporated in the model. However, to show the advantage of CQC-hop statetransfer scheme, only the order is important). For the same values of expermental data in [45], we showthat the total time 2 t for our scheme is of the order of 1.2 ns only. Moreover, SWAP and CNOT gateson a physical architecture are not a single step operations, but usually involves a series of steps with timecontrolled quantum evolution with various parameters of control (such as the famous Cirac-Zoller CNOTgate in trapped ion [46] and similar implementation in C-QED Jaynes-Cummings interaction [2]). However,for the purpose of quantum circuits, all two-qubit gates are seen as single operation for the circuit. Incontrast to a SWAP gate, a perfect state transfer requires less time if parameters are chosen correctly, for ourarchitecture (see chapter 6). In contrast, perfect state transfer is simply the time controlled free quantumevolution of the whole network and is more robust to errors and decoherence because there is no quantummanipulation into the network during the process.Figure 4.1: Quantum circuit for our CQC-hopping schemeFor a d -dimensional hypercube, the shortest path length between two given vertices is at most d . Hence,at most d SWAP gates will be required to transfer a state between these two vertices. Whereas CQC-hoppingcan make this task possible in at most 2 operations for arbitrarily chosen qubits in the network. For example,consider the case of (cid:3) K − {| (cid:105)} (which is not a perfect hypercube with 31 qubits). The vertices canbe labeled as | x x x x x (cid:105) . Let us say we want to the state transfer from | (cid:105) ( u ) to | (cid:105) ( w ). Noticethat they are not antipodal because the vertex | (cid:105) is missing. There are multiple shortest paths for thisgiven pair. One of these is | (cid:105) u → | (cid:105) → | (cid:105) → | (cid:105) → | (cid:105) → | (cid:105) w . (4.10)This requires us to use five SWAP gates in this desired sequence. Whereas for the CQC-hopping scheme wehave | (cid:105) u e − iA t −−−−−→ | (cid:105) v e − iA t −−−−−→ | (cid:105) w (4.11)which is just a two step task. For any order of hypercube and any large arbitrary large number of qubits n ,this holds. In this example, the first CQC-hopping is one-link jump whereas the second CQC-hopping is ahypercube (of dimension 4) jump. Hence, the computational advantage. Refer to figure 4.1 for the quantumcircuit for CQC-hopping scheme for general two hoppings. If it so happens that u and w are antipodalvertices of a hypercube then only one hopping is required. Phase correction gates ϕ and ϕ are applied to45ecover the original state | ψ (cid:105) at the desired qubits. This phase correction can be easily performed at anygiven qubit when our state transfer qubits are themselves the computational qubits (see chapter 6). Notethat the ket states in 4.1 denote the excitation states at different qubits and this is different from the binaryket representation of the same. Therefore, whenever a quantum circuit has evaluated a segment of circuitand is brought to halt with a result stored in one of qubit’s state, it can be easily transferred to any otherqubit purely by our state transfer scheme in contrast to multiple SWAP gates. The only condition that needsto be ensured is to initialize all other qubits to ground state except the one we want to transfer.46 hapter 5 Perfect State Transfer for qudit(d-level systems) networks
Instead of a qubit (a two-level quantum system), we may require to transfer the state of a d − level system,called a qudit. If a qudit is thought of as stacked qubits with equally spaced levels (like finite harmonicoscillator), then there is a finite possibility that the d − level excitation can split and distribute as a sum oflower order excitations (such as different Γ k in section 1.2) which splits this qudit state while preserving thetotal excitation. However, we want to perfectly transfer the full qudit state from one site to another in thenetwork (graph) of qudits. This requires special conditions. We want to transfer any general qudit statefrom one vertex to another. We denote a general qudit state | ψ d (cid:105) as | ψ d (cid:105) = d (cid:88) j =0 α j | j (cid:105) (5.1)with α i ∈ C and (cid:80) dj =0 | α j | = 1, as the normalization condition. Similar to equation (1.1), the transfer Hamiltonian for the XY model can be formulated using Lie algebra.This was proposed in [47] for chains. Analogous to the Pauli operators for SU (2) group for qubit, Lie algebragenerators can be defined for SU ( d ) group for the d − level system transfer dynamics. First step is to definethe corresponding projectors ( P k,j ) µ,ν = | k (cid:105)(cid:104) j | = δ µ,j δ ν,k , ≤ µ, ν ≤ d (5.2)We want to obtain all the d − SU ( d ) group. The first set of these are the projectorsΘ k,j = P k,j + P j,k , β k,j = − i ( P k,j − P j,k ) , (5.3)with 1 ≤ k ≤ j ≤ d . And the remaining d − η r,r = (cid:115) r ( r + 1) r (cid:88) j =1 P j,j − rP r +1 ,r +1 (5.4)with 1 ≤ r ≤ ( d − d − SU ( d ) group that are necessaryto define the dynamics for qudit chains. For d = 3, these sets give the Gell-Mann matrices, and so on. Fordefining the dynamics for the XY model, only the off-diagonal operators of the first set are enough.For defining the qudit network we assume that all the qudits are identical. Then we define the XY Hamiltonian as H dXY = (cid:88) ( i,i +1) ∈L ( G ) J i (cid:16) Θ k,j ( i ) Θ k,j ( i +1) + β k,j ( i ) β k,j ( i +1) (cid:17) (5.5)47here L ( G ) denotes a line graph of G . And the couplings are weighted couplings with J i = (cid:112) i ( n − i ) / ≤ k ≤ j ≤ d . This is analogous to equation (1.78) deduced earlier. This Hamiltonian preserves a quantitysimilar to the total z − spin in the SU (2) model. The conservation relation is H dXY , n (cid:88) j =1 η r,r ( j ) = 0 , for 1 ≤ r ≤ ( d − . (5.6)The dynamics of Laplacian can be similarly defined. The transfer dynamics of this Hamiltonian for entan-glement transfer with chains has been shown in [47]. Spin-1/2 (fermionic) systems can be seen as qubits ( d = 2). Similarly, arbitrary d -level system can be realisedby an appropriate spin. Here we investigate perfect state transfer of the bosonic spin particles on bosonicnetwork lattice - the Bose-Hubbard model. This work was proposed in [26]. First, we present the incorrectcalculation reported in [26]. Equation (cid:80) l U l ˜ b † l = (cid:80) l ˜ b † l following after equations 2-8 on page 174 in [26] isincorrect. This equation has been used to derive equation 2-9 and finally the condition 2-12 for perfect statetransfer which is incorrect. This relation implies that every entry in the first column of U is unity. This isbecause U is a unitary matrix after all where a complete column cannot have each entry as unity. This wouldimply a non-unitary transformation over each adjacency matrix A k .We reformulate this method to derive the right condition for the perfect state transfer of qudits. TheHamiltonian governing the Bose-Hubbard model is H d = n − (cid:88) k =1 J k (cid:16) b † k b k + b k b † k (cid:17) + n (cid:88) k =1 (cid:15) k b † k b k (5.7)where the first term governs the hopping from one site to its adjacent site and second term is the local energyof that site. b † k and b k are the bosonic creation and annihilation operators for the site k whose action on thebasis kets | (cid:105) , | (cid:105) (for each site) is well known. We are mostly concerned with the first coupling term. Thismodel is applicable on general arbitrary finite graphs also. This can be generalised to an arbitrary graph in aweighted scheme. Suppose Ω is a connected graph. For each coupling strength J k , k = 0 , , ..., d , we can forma graph Ω k in which vertices are adjacent if their coupling in Ω equals k . Let A k be the adjacency matrix ofΩ k . For instance, A is the adjacency matrix A of Ω. Also, let A = I , the identity matrix. This gives us d + 1 matrices A , A , ..., A d , called the adjacency matrices of Ω. Their sum is the matrix J in which everyentry is 1. In the other words, we assume that the dynamics of bosons, in a system with n sites (associatedwith the nodes of a finite group), is governed by the following off-diagonal Bose-Hubbard Hamiltonian H d = n (cid:88) i,j =1 d (cid:88) k =0 J k ( A k ) ij b † i b j (5.8)The ground assumption is that the matrices A k for k = 0 , , ..., d commute with each other such that allof them are simultaneously by the matrix U . The same matrix U pointed above for a calculation mistakein the original work. The diagonalization follows as U A k U † = D k , where D k = diag( λ ( k )1 , λ ( k )2 , ..., λ ( k ) n ) is adiagonal matrix with eigenvalues of A k on its diagonal. Using this relation in the Hamiltonian expression weget H d = n (cid:88) i,j =1 d (cid:88) k =0 J k ( U † D k U ) ij b † i b j = d (cid:88) k =0 n (cid:88) l =1 J k λ ( k ) l (cid:32)(cid:88) i U † il b † i (cid:33) (cid:88) j U lj b j (5.9)with l = 1 , , ..., n . The term in the brackets can be treated as Bogoliubov transformation which is a lineartransformation on creation/annihilation operators. Define this change of basis as˜ b l = (cid:88) j U lj b j , b l = (cid:88) j U ∗ jl ˜ b j (5.10)48efine the number operator in the new tilde basis as ˜ n l = ˜ b † l ˜ b l and the effective coupling as ˜ J l = (cid:80) dk =0 J k λ ( k ) l .Then Hamiltonian simply takes the diagonal form H d = n (cid:88) l =1 ˜ J l ˜ n l . (5.11)Without loss of generality, at t = 0, we start with an initial state localised at the first qudit | ψ d (0) (cid:105) = d (cid:88) j =0 α j ( b † ) j | (cid:105) (5.12)where | (cid:105) := | ... (cid:105) and ( b † k ) j | mathbf (cid:105) = | ...j k ... n (cid:105) = | j (cid:105) (which is understood to be a action on site k by context). This is a harmonicoscillator (energy levels equally spaced) like d -level state. All the qudits are identical in the network. For d = 2, analogous to 1.13, we will have the evolution of the initial state as | ψ d ( t ) (cid:105) = e − iH d t | ψ d (0) (cid:105) = α | (cid:105) + α n (cid:88) j =1 f j ( t ) b † j | (cid:105) (5.13)where f j ( t ) = (cid:104) mathbf | e − iH d t (cid:80) nl =1 ˜ J l ˜ n l b † | (cid:105) (5.14)and for the perfect state transfer from forst qubit to some m th qubit, we impose | f m ( t ) | = 1 for some finite t = t . We derive a condition for qubit using this Hamiltonian and generalise it over d -levels. Change ofbasis in first part of 5.13 yields | ψ d ( t ) (cid:105) = α | (cid:105) + α e − it ( (cid:80) nl =1 ˜ J l ˜ b † l ˜ b l ) n (cid:88) m =1 U m ˜ b † m | (cid:105) (5.15)= α | (cid:105) + α n (cid:88) l =1 m =1 e − it ˜ J l | ˜ l (cid:105)(cid:104) ˜ l | U m | ˜ m (cid:105) (Spectral decomposition)= α | (cid:105) + α n (cid:88) l =1 m =1 e − it ˜ J l U m | ˜ l (cid:105) δ ˜ l ˜ m = α | (cid:105) + α n (cid:88) l =1 e − it ˜ J l U l ˜ b † l | (cid:105) . (5.16)After reverting back to the initial basis we obtain | ψ d ( t ) (cid:105) = α | (cid:105) + α n (cid:88) l,j =1 e − it ˜ J l U l U ∗ lj b † j | (cid:105) (5.17)in contrast to equation 2-9 in [26] which we report as incorrect. To extract a more compact form define thecolumn vector ˜ J = (cid:16) U e − it ˜ J U e − it ˜ J ... U n e − it ˜ J n (cid:17) T (5.18)then equation (5.17) can be re-expressed as | ψ d ( t ) (cid:105) = α | (cid:105) + α n (cid:88) j =1 (cid:32) n (cid:88) l =1 U † jl ˜ J l (cid:33) b † j | (cid:105) = α | (cid:105) + α n (cid:88) j =1 (cid:16) U † ˜ J (cid:17) j b † j | (cid:105) . (5.19)49omparing with equation (5.13) we have that f j ( t ) = (cid:16) U † ˜ J (cid:17) j = n (cid:88) l =1 (cid:0) U † (cid:1) jl U l e − it ˜ J l (5.20)and the condition for perfect state transfer to the m th site imposes (cid:16) U † ˜ J (cid:17) j = e iφ δ jm (5.21)for some arbitrary phase φ which can always be corrected, post the transfer.To have estimation about the entries of the matrix U , define the product b † i b j := E ij . (5.22)Here the action of b † i b j is b † i b j | k (cid:105) = δ jk | i (cid:105) as understood by the operator action on the basis. We can deducethat E ij is a matrix with all entries zero except the ( i, j ) entry ( (cid:104) i | E ij | j (cid:105) = 1), that is, ( E ij ) kl = δ ik δ jl . Thenfor the indices which are adjacent in graph Ω k , denoted as i ∼ k j (cid:88) i ∼ k j = (cid:88) i ∼ k j E ij = A k . (5.23)Then the Hamiltonian can be expressed in terms of adjacency matrices A k as H d = d (cid:88) k =0 J k (cid:88) i ∼ k j E ij = d (cid:88) k =0 J k A k . (5.24)The matrices A k can always be diagonalized as A k = (cid:80) l λ ( k ) l | l (cid:105)(cid:104) l | = (cid:80) l λ ( k ) l E l , where E l are the correspond-ing projectors of the l th subspace spanned by the eigenvectors corresponding to the eigenvectors of λ ( k ) l .Using this observation for spectral decomposition we have f j ( t ) = (cid:104) | b j e − iHt b † | (cid:105) = (cid:88) l e − it ˜ J l (cid:104) | b j E l b † | (cid:105) (5.25)= (cid:88) l e − it ˜ J l (cid:104) j | E l | (cid:105) = (cid:88) l e − it ˜ J l U l U † jl . (5.26)Last step is done using equation (5.20). This gives us the information about the entries of U as follows U l U † jl = (cid:104) j | E l | (cid:105) = ( E l ) j . (5.27)This procedure for two levels can be applied to arbitrary d levels treating each level as some higher levelexcitation of the ground level. Label 1 can be replaced by any arbitrary label x which is initial site to beginthe transfer from. For simplicity, let us stick with the initial site as the first site. The free evolution of aqudit will be as follows for each arbitrary level i , and all levels evolve independently as they are the basiskets, e − iH d t ( b † ) i | (cid:105) = e − iH d t (cid:88) l ,...,l i ˜ b † l ˜ b † l ... ˜ b † l i | (cid:105) (5.28)= (cid:88) l ,...,l i e − it (cid:80) ik =1 ˜ J lk ˜ b † l ˜ b † l ... ˜ b † l i | (cid:105) (5.29)= (cid:88) k ,...,k i (cid:32)(cid:88) l e − it ˜ J l U l U ∗ l k (cid:33) (cid:32)(cid:88) l e − it ˜ J l U l U ∗ l k (cid:33) ... (cid:32)(cid:88) l i e − it ˜ J li U l i U ∗ l i k i (cid:33) × b † k b † k ...b † k i | (cid:105) (5.30)which can again be written in the compact form similar to the previous analysis as e − iH d t ( b † ) i | (cid:105) = (cid:88) k ,...,k i (cid:16) U † ˜ J (cid:17) k (cid:16) U † ˜ J (cid:17) k ... (cid:16) U † ˜ J (cid:17) k i × b † k b † k ...b † k i | (cid:105) . (5.31)50his transfers each excitation level of the qudit independently from one site to another. The final state ofthe system will be | ψ d ( t ) (cid:105) = e − iH d t | ψ d (0) (cid:105) = α | (cid:105) + α (cid:88) k (cid:16) U † ˜ J (cid:17) k b † k | (cid:105) + α (cid:88) k ,k (cid:16) U † ˜ J (cid:17) k (cid:16) U † ˜ J (cid:17) k b † k b † k | (cid:105) + ... + α d (cid:88) k ,...,k d (cid:16) U † ˜ J (cid:17) k (cid:16) U † ˜ J (cid:17) k ... (cid:16) U † ˜ J (cid:17) k d × b † k b † k ...b † k d | (cid:105) . (5.32)The condition for perfect state transfer applies to each excitation level which exactly remains the samecondition as equation (5.21) for t = t . The final state when transfer is accomplished to the m th site issimply | ψ d ( t ) (cid:105) = d (cid:88) j =0 α j (cid:0) b † m (cid:1) j | (cid:105) . (5.33)Special graphs can be studied under this model of perfect state transfer for qudits. PST for qudits has beendemonstrated experimentally on superconducting transmon qudits in [48]. Qudit PST for pseudo-regularnetworks is explored in [49]. An architecture for arbitrary long distances qudit perfect state transfer usingmultiple hoppings is presented in [25]. 51 hapter 6 Physical implementation for qubitnetworks with superconductingcircuits
Our proposed architecture in chapter 4 addresses the problem of scalability of quantum processors whilepreserving the perfect fidelity for state transfer with full control on routing of initial states. Scalability ofquantum processors is a deep concern in the development of quantum computing hardware [50][51][52]. In theNISQ (Noisy Intermediate-Scale Quantum) [33] era quantum processors, high-fidelity quantum operationsand attempt of scalable quantum networks are key features. These features are essential for determining howgood is an architecture for quantum information processing [53]. One of the main challenges for scalablequantum network is the imperfect two-qubit interaction. For perfect state transfer with maximum fidelity, thepairwise interaction should be improved for large-scale quantum processors [54]. Different physical systemsfor quantum computation have different advantages, such as, high-fidelity and control in ion-traps [46][4]versus the scalability of superconducting circuits [55][2][56]. Our focus in this work will be with the high-fidelity, state control and scalability of the superconducting quantum computing architecture and provingthat our two-hopping all-to-all state transfer scheme can be perfectly realised with this approach.For the usual perfect state transfer scheme, the underlying assumption is that all the edges in the graphnetwork are precisely engineered at the desired coupling strength. If this is violated, we will have to com-promise a lot with the fidelity of the state transfer. PST Hamiltonians like XY and Heisenberg model arenothing but the sum of the pairwise interactions between the connected qubits that are allowed to interact inthese two defined schemes. The coupling Hamiltonian for an isolated pair in PST scheme is simply the gateHamiltonian for the quantum computation. This implies that PST is nothing but the complete free quantumevolution (for a desired time interval which is the control parameter) of the entire network in contrast tothe controlled quantum evolution of an isolated pair of qubits (which is a quantum gate operation). Improv-ing two qubit gate fidelity will improve the perfect state transfer fidelity. Connectivity of the qubits (as agraph structure) in the quantum circuit model will determine which pair of qubits can have two-qubit gateoperations between them mutually. Disconnected qubits on the physical architecture will prohibit two-qubitinteractions between these qubits. This is how the graph determines the architecture. More connectivity willenable more two-qubit interactions. However, there is also a physical limitation for the maximum nearestneighbour interactions a qubit can sustain [57].There are two sources of two-qubit interaction errors: decoherence (stochastic) and nonideal interactions(deterministic). The latter includes parasitic coupling, leakage to non-computational states, and controlcrosstalk. As one example of parasitic coupling, the next-nearest-neighbor (NNN) coupling is a phenomenoncommonly seen in many systems, including Rydberg atoms [58], trapped ions [3][4], semiconductor spin qubits[59], and superconducting qubits [5][6]. Unwanted interactions (such as next-nearest neighbour) betweenqubits are meant to be unconnected. 52 .1 Requirements for our CQC-hopping scheme To implement our network on a real architecture we follow the physical architecture given in [45] for twoqubits and generalize it over arbitrary number of qubits n . Physical key requirements of our architecture arethe following: • Multiple nearest-neighbour (NN) interactions • Since distant qubits are connected, implementation is not possible in planar integration. Three-dimensional (3D) integration is needed [60][57] • Tunable (eventually switchable) edges as couplings for each pair of nodes (qubits) • Multiple qubit coupling controlled independently (each qubit independently be brought in dispersiveregime) [54] • Addition of a new node to the existing network and so on (addressing scalability) • High fidelity control over the processor [61]Most of these requirements cannot be fulfilled by the usual spin model where switching and tuning of edgecoupling becomes nearly impossible as the experimentalist control is negligible for spins on a fixed lattice.Moreover, such coupling is a function of the distance between two nodes which is not changeable in practice.Multiple edges are very hard to be tuned to the same coupling strength and the nodes that are physicallydistant are impossible to couple. We show that all these implementation problems can be addressed with thearchitecture based on tunable-coupling superconducting circuits. A tunable coupler can also help mitigatethe problem of frequency crowding that exacerbates the effect from nonideal interactions. However, theseadditional elements often add architectural complexity, as well as open a new channel for decoherence andcrosstalk. Many prototypes of a tunable coupler have been demonstrated in superconducting quantumcircuits, such as the the gmon design ([62], two-qubit gate fidelity limited by decoherence) and xmon design([63], qubit’s coherence time is decreased by the tunable coupler), are two examples in the literature. Additionand deletion of a new node (qubit) to the graph (network) has been treated as the inclusion and exclusionof that qubit in the computational network. That is, this is achieved by turning of every interaction of thatnode with the rest of the network and switching it on when this node is to be added. Building and fabricatinga new qubit into the existing processor is a matter of the available resources and depends upon lab to lab.We do not address that matter. For physical realization, the addition of nodes can be regarded as turningon interaction with more qubits that already exist by construction in the quantum processor.
In the superconducting transmon charge qubit architecture [64] all these limitations can be tackled andall the above requirements can be fulfilled as we demonstrate. Moreover, this architecture maybe used forthe conventional quantum computation model in addition to quantum state transfer application due to thehigh-fidelity gate operations that are possible [45]. This architecture allows high control over the couplingmacroscopically and decoherence times are quite longer. A gate (two qubit interaction) fidelity of 99.999%is reported in [45] in the absence of decoherence.We consider a generic system consisting of an arbitrary network of qubits with exchange coupling betweennearest qubits (which have an edge between them) and a coupler between these that couples to both thesequbits. The approach is based on a generic three-body system with exchange-type interaction. A centralcomponent, the coupler, frequency tunes the virtual exchange interaction between two qubits and featuresa critical bias point, at which the exchange interaction offsets the direct qubit-qubit coupling, effectivelyturning off the net coupling. Two-qubit interactions are executed for each pair of qubits by operating therespective couplers in the dispersive regime, strongly suppressing leakage to the coupler’s excited states. Thetwo qubits (with Zeeman splittings ω i and ω j ) each couple to a center tunable coupler ( ω cij ) with a couplingstrength g i ( i = 1 , , ..., n ), as well as to each other with a coupling strength g ij . The ancillary coupling isstronger than the direct coupling , g i , g j > g ij >
0. Ancillary coupler does not count as the part of ournetwork, it is only needed as the part of the implementation of architecture. We have the total number of53ncillary couplers equal to the number of edges in the network. Without loss of generality, we begin ouranalysis with a two-level Hamiltonian, H = 12 n (cid:88) i =1 ω i σ zi + 12 (cid:88) (cid:104) i,j (cid:105) ω cij σ zC ij + (cid:88) (cid:104) i,j (cid:105) g i (cid:16) σ + i σ − C ij + σ − i σ + C ij (cid:17) + (cid:88) (cid:104) i,j (cid:105) g ij (cid:0) σ + i σ − j + σ − i σ + j (cid:1) (6.1)where all the operators are defined for the respective modes and (cid:104) i, j (cid:105) means there is an edge between qubits i and j . First two terms are just the Zeeman splittings of the network qubits and the ancillary couplers forall edges. Third term (call it V , treated as a perturbation) describes the coupling of each qubit to couplerand the last term is the direct coupling between the network qubits which have an edge between them. H = H + V can be written where we treat the coupling to the ancilla as external coupling one wants toget rid of. In this setting, all qubits are negatively detuned from the resonance with the ancillary couplerwith ∆ j = ω j − ω cij < g j (cid:28) | ∆ j | ∀ j . Any twoconnected qubits interact through two channels, the direct nearest coupling and the indirect coupling via theancilla (which can be regarded as a virtual exchange interaction). The idea is to make these two couplingscompete against each other and tune the desired strength of coupling for each pair of qubit. We desire totune all couplings at equal magnitude to make an uniformly coupled network as proposed. (a) (b) Figure 6.1: The part (a) shows the isolated pair of coupled qubits with the ancillary coupler acting as theinter-mediator. The situation for every pair that has an edge between them is the same. Part (b) showsour qubit architecture for n = 6 qubits with the ancillary couplers involved. The control parameter for edgeswitching here is the external control on the capacitance for each coupler, denoted as yellow.To find the effective qubit-qubit coupling and eliminate the qubit-ancilla coupling, we take advantage ofthe Schrieffer-Wolff unitary transformation U SW = e η . The transformation if formally represented as˜ H = U SW HU † SW = H + [ η, H ] + 12! [ η, [ η, H ]] + ... (6.2)= H + V + [ η, H ] + [ η, V ] + 12! [ η, [ η, H ]] + 12! [ η, [ η, V ]] + ... (6.3)If one can find a transformation η such that V + [ η, H ] = 0, the transformed Hamiltonian becomes:˜ H = H + 12 [ η, V ] + O ( V ) (6.4)54ecause η ∝ O ( V ). The standard way to find this transformation requires to calculate the commutator[ H , V ] = (cid:88) (cid:104) i,j (cid:105) g i ( g ij σ zi − ∆ i ) (cid:16) σ + i σ − C ij − σ − i σ + C ij (cid:17) (6.5)and impose this operator ansatz form with free parameters α ± i asη = α + i σ + i σ − C ij + α − i σ − i σ + C ij . (6.6)Then we evaluate the commutator [ H , η ] and equate to V to find the free parameters α ± i = ± g i / ∆ i . Thisgives the transformation U SW for our case as U SW = e η = exp (cid:88) (cid:104) i,j (cid:105) g i ∆ i (cid:16) σ + i σ − C ij − σ − i σ + C ij (cid:17) . (6.7)For general SW-transformation within and beyond rotating wave approximation (RWA), see [65] and Ap-pendix: B of [45]. Performing the SW transformation up to second order we get the Hamiltonian˜ H = 12 n (cid:88) j =1 ω j σ zj + 12 n (cid:88) (cid:104) i,j (cid:105) ω cij σ zC ij + (cid:88) (cid:104) i,j (cid:105) (cid:18) g i g j ∆ ij + g ij (cid:19) (cid:0) σ + i σ − j + σ − i σ + j (cid:1) σ zC ij + (cid:88) (cid:104) i,j (cid:105) g i ∆ i (cid:16) σ zi σ − C ij σ + C ij + σ − i σ + i σ zC ij (cid:17) + (cid:88) (cid:104) i,j (cid:105)(cid:104) i,k (cid:105) g i ∆ i (cid:16) σ + i σ − i σ − C ij σ + C ik − σ − i σ + i σ + C ij σ − C ik (cid:17) . (6.8)Now we make an important assumption that the ancillary couplers always remain in their ground state for allthe edges and drop out the constant energy terms. This combines first and fourth term in the Hamiltonianand removes the last term which is exchange interaction between different ancillary couplers of the samequbit. This transformation finally decouples the ancillary coupler from the qubits up to second order in g i / ∆ i resulting in ˜ H = 12 n (cid:88) j =1 ˜ ω j σ zj + (cid:88) (cid:104) i,j (cid:105) (cid:18) g i g j ∆ ij + g ij (cid:19) (cid:0) σ + i σ − j + σ − i σ + j (cid:1) (6.9)where ˜ ω j = ω j + g j / ∆ j is the Lamb-shifted frequency revealed by the SW transformation and ∆ ij =2∆ i ∆ j / (∆ i + ∆ j ) < g ij , is theeffective tunable coupling between any two qubits that are coupled in the network wherever there is an edge.Identifying ˜ g ij as 2 J ij gives the identical coupling Hamiltonian in equation (1.1). However, now our J ij istunable to a range of values we desire by setting the desired couplings and detunings. The effective coupling˜ g ij in equation (6.9) can be adjusted by the ancilla coupler frequency through ∆ ij , as well as g i and g j , bothof which may be implicitly dependent on ω cij . Therefore, ˜ g ij is a function of ω cij in general. The first term(indirect coupling) in the expression of ˜ g ij is negative while the second (direct coupling) is positive and thisenables a competition between the two where ω cij can be taken to act as the tunable parameter since it canbe externally controlled in the experiment. ˜ g ij ( ω cij ) can be tuned negative when ancilla coupler frequency isdecreased or positive when this frequency is increased. And this is a continuous parameter, therefore we havesome ω cij off such that ˜ g ij ( ω cij off ) = 0 which should be permitted by the bandwidth of the ancilla coupler. Itis shown [45] that this cut-off frequency can be found even in weak dispersive regime with g j < | ∆ j | . Thus,in principle, we obtain the switchable edges with ω cij as the parameter. We can simply tune each frequency ω cij for each edge E ( i, j ) to switch it on or off when our protocol requires and this is essentially a classicaloperation in experiment. Therefore, some of the edges maybe switched off by selecting special cut-off values.The couplers remain in their ground state throughout the quantum evolution as the effective interaction isonly for one quantum exchange between the two qubits which are part of the network. Similar effectivecoupling Hamiltonians based on Cavity and Circuit-QED have been proposed in [66] (scalability has beenaddressed with experimental concerns using molecular architecture for qubits in superconducting resonators)and [2] (foundational reference for superconducting electrical circuits).55 .3 Circuit Hamiltonian quantization Now, the above general formalism can be applied to specific system Hamiltonian. For our case we use thetransmon qubits [67]. Josephson energy Hamiltonian with tunable energy is E J λ = ( E J λ ,L + E J λ ,R ) (cid:115) cos (cid:18) π Φ e,λ Φ (cid:19) + (cid:18) E J λ ,L − E J λ ,R E J λ ,L + E J λ ,R (cid:19) sin (cid:18) π Φ e,λ Φ (cid:19) (6.10)where λ = i, j, c ij ∀ i, j ( c ij is the labeling for the coupler connecting i and j qubits) and Φ = h/ e is thesuperconducting flux quantum. The coefficient of the sine term quantifies the asymmetry of the junction[68]. Refer to figure 6.2 for notation. E J λ,L ( R ) is the Josephson energy of the left(right) junction in mode λ . C λ is the dominant capacitance for that mode. C jc ij is the coupling capacitance between the qubit j and thecoupler (cid:104) ij (cid:105) . C ij is the directing coupling capacitance between the two qubits i and j , and φ λ is the reducedtotal flux for that node. Other couplings
Figure 6.2: Schematic circuit diagram for a pair of connected tunable transmon qubits. Each connected pairwhich forms an edge on the graph has this structure.This Hamiltonian can be canonically quantized in second quantization in the transmon regime with E J λ /E C λ (cid:29)
1, where E C λ = e / C λ for the corresponding mode [45]. The system is then described by thecoupled oscillators ( (cid:126) = 1): H = (cid:88) (cid:104) i,j (cid:105) (cid:0) H i + H j + H c ij + H ic ij + H jc ij + H ij (cid:1) (6.11)Let the corresponding creation and annihilation operators for the respective mode be b † λ and b λ respectively,with similar action as in 5.2. The terms of the Hamiltonian are H λ = ω λ b † λ b λ + α λ b † λ b † λ b λ b λ , (6.12)where α λ is the anharmonicity (energy difference between the first excitation energy and further secondexcitation energy) of the oscillator, H jc ij = g j (cid:16) b † j b c ij + b j b † c ij − b † j b † c ij − b j b c ij (cid:17) , (6.13) H ij = g ij (cid:16) b † i b j + b i b † j − b † i b † j − b i b j (cid:17) , (6.14)56here the second order effect has also been taken into consideration. The energies are given as ω λ = (cid:112) E J λ E C λ − E C λ , (6.15) g j = 12 C jc ij (cid:112) C j C c ij (cid:113) ω j ω cij (6.16)and g ij = 12 (1 + η ij ) C ij (cid:112) C i C j √ ω i ω j (6.17)where η ij = C ic ij C jc ij /C ij C c ij . Single quantum exchange is due to the Jaynes-Cumming type interactionwhile the double excitation and de-excitation effect arises due to counter rotating terms, without the rotatingwave approximation (RWA), which is important when the coupler frequency is higher than of the relatedqubit frequencies. Following the general SWT formulated in [65], the dynamics beyond the RWA for the above Hamiltonian isstraightforward. The right SWT is U SW = exp (cid:88) (cid:104) i,j (cid:105) (cid:20) g j ∆ j (cid:16) b † j b c ij − b j b † c ij (cid:17) − g j Σ j (cid:16) b † j b † c ij − b j b c ij (cid:17)(cid:21) (6.18)where the second term takes care for the counter-rotating terms and Σ j = ω j + ω cij . In the weak anharmoniclimit, α λ (cid:28) ∆ j (detuning of nearly the same order for all qubits). Expansion in the second order in couplings g j we obtain the effective qubit-qubit Hamiltonian˜ H = U SW HU † SW = (cid:88) j (cid:18) ˜ ω j b † j b j + ˜ α j b † j b † j b j b j (cid:19) + (cid:88) (cid:104) i,j (cid:105) ˜ g ij (cid:16) b † i b j + b i b † j (cid:17) (6.19)which is identical to equation (6.9), plus the anharmonic terms along with counter rotating contribution tothe coupling coefficients. Here, ˜ ω j ≈ ω j + g j (cid:18) j + 1Σ j (cid:19) , (6.20)˜ α j ≈ α j , (6.21)and the effective coupling as ˜ g ij ≈ g i g j (cid:18) i + 1∆ j − i − j (cid:19) + g ij . (6.22)Here, the same assumption of the coupler being strictly in its ground state has been taken into account. Notethat this is exactly similar to making the assumption that the cavity resonator remains in the constant photonnumber in the conventional cavity-quantum-electrodynamics Hamiltonians, while performing their Schrieffer-Wolff transformations. For the dispersive regime we simply have | ∆ j | ≈ | Σ j | , which is when counter rotatingterms contribute significantly. The computational states are | i c ij j (cid:105) and | i c ij j (cid:105) and they exchange theirenergy virtually through the non-computational coupler excited state | i c ij j (cid:105) by the virtue of the Jaynes-Cummings interaction ( b † j b c ij + b j b † c ij ). The counter-rotating term ( b † j b † c ij + b j b c ij ) involves exchange via thehigher non-computational state | i c ij j (cid:105) . Substituting the values for the couplings we obtain˜ g ij ≈ (cid:20) ω cij (cid:18) i + 1∆ j − i − j (cid:19) η ij + η ij + 1 (cid:21) × C ij (cid:112) C i C j √ ω i ω j . (6.23)For the case when all qubits are identically set in the dispersive regime with identical construction, we set ω i = ω j = ω . This results in ˜ g ij = 12 (cid:20) ω ∆ i Σ i η ij + 1 (cid:21) C ij (cid:112) C i C j ω (6.24)57he first term in the can be made arbitrarily small for very high values ω cij and very large values of η ij whichcan give zero effective coupling (a switched-off edge). This way, by just adjusting the coupler dynamics(which is externally under full control) we can turn off selected edges (couplings) in the network. The tunable couplers are used to turn off the graph edge interactions by biasing their frequency at ω cij off during the switching period. To activate the two-qubit interaction as the edge in the network graph, onetunes the couplers’ frequency to a desired value ω cij on , yielding a finite ˜ g ( ω cij on ). All the couplers are set tothe same strength of coupling. Then a PST can be performed by modulating only the coupler frequency to˜ g ( ω cij on ) for all the edges E ( i, j ) which have to be switched on while leaving the other qubits unperturbedduring the PST. The edges which are switched on and switched off are known from our formalism in Chap.4. By operating the couplers in the dispersive limit, parasitic effects from higher-order terms that are ignoredafter SWT are strongly suppressed, leading to higher two-qubit hopping fidelity. During this process, thecontrol Hamiltonian σ zC ij commutes with the qubits’ degrees of freedom within the dispersive approximation,causing reduced leakage to the non-computational (coupler) state. The non-adiabatic effect in this case issuppressed by the relatively large qubit-coupler detuning ∆ j , allowing a shorter PST time and therefore andreduced decoherence error. - - - - - - - - Δ i for all qubits in GHz Tun a b l ec oup li ng g ij ˜ i n M H z Figure 6.3: Variation of the dynamic tunable coupling ˜ g ij w.r.t. the detuning ∆ i for each qubit. There existsa cutoff value, in this case ∆ i = − .
426 GHz, corresponding to ω cij off = 5 .
426 GHz. For all configurations,such a cut-off value can always be obtained.Figure 6.3 shows the variation of the dynamic tunable coupling ˜ g ij with respect to the control parameter ω cij . The reasonable experimental values for the parameters used are [45]: C i = 70 fF, C j = 72 fF, C c ij = 200fF, C ic ij = 4 fF, C jc ij = 4 . C ij = 0 . ω j = ω j = ω = 4 GHz (because all qubits are identical).The fabrication defects and imperfection is accounted in the different values for the capacitances. However,for a quite good variation amongst these values, we can still guarantee a cut-off value existence.The perfect state transfer time for our CQC-hopping (2 t ) is plotted against the detuning of qubits. PSTtime is t if we are at a perfect hypercube or 2 t otherwise. For the same experimental values consideredabove, this is plotted in figure 6.4It can be deduced that the typical time scale for our CQC-hopping scheme is around 1.5 ns. This is lessthan the single iSWAP quantum gate time for the same experimental values which is reported to be around35 ns. Even if we consider that switching takes few nanoseconds, still we perform PST in less time relative to58 - - - - Δ i for all qubits in GHz PS T t i m e2 t i nn s Figure 6.4: Variation of the Perfect State Transfer time 2 t w.r.t. qubit detuning ∆ i for all qubits. Thetypical PST time is around 1.5 ns.the SWAP gate operation between two given qubits. We have to ensure in the experiment that all detuningsare onset to the same value to realise a uniformly coupled qubit network. If all detunings are not equalthis will actually realise a weighted coupled qubit network and introduce an error since our CQC-hoppingprotocol is not for weighted graph networks. This error can be estimated by calculation pairwise transferfidelity via two different couplings. The typical coupling can be tuned around to 2 MHz (for example), togive PST time of 1.55 ns, by setting the detuning to − ω cij = 8 GHz for all couplers.59 hapter 7 Conclusion and future work
We proposed a switching procedure on a memory-enhanced hypercube such that an induced hypercube canbe determined with a desired pair of antipodal vertices. A framework of superconducting qubits is defined forphysical implementation of the switching procedure under the XY coupling. This same physical architecturealso realises growing network scheme. It was shown that perfect state transfer between any pair of vertices ina hypercube or more generally in a network of arbitrary number of vertices is possible utilizing the proposedswitching scheme. We have proved the computational advantage of using PST assisted quantum computing.We showed counter-examples numerically for PST under Corona product of graphs and corrected an errorin a qudit PST paper which is a very rich resource for computational power in quantum computing.There are certain results which can further be extended. Such as finding a physical system for scalableLaplacian perfect state transfer for switchable hypercubes similar to our scheme. One can also look to findmore optimal graph operations, if they exist, for long distance PST to realize a growing network. Analyticalresults for signed Corona transfer may also be attempted to look for the class of graphs which support PSTunder Corona product. Qudit state transfer for large distances using least hoppings is still not reported inliterature, to this date. It would be a big result to find such an optimal network for qudits under a suitableHamiltonian so that large amount of quantum information can be transferred over long distances, similar toqubits. 60 ibliography [1] Charles H. Bennett and David P. DiVincenzo. Quantum information and computation.
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