aa r X i v : . [ qu a n t - ph ] J a n Open Quantum Walks on Graphs
S. Attal
Universit´e de Lyon, Universit´e de Lyon 1, C.N.R.S., Institut Camille Jordan, 21 avClaude Bernard, 69622 Villeubanne cedex, France
F. Petruccione
Quantum Research Group, School of Chemistry and Physics and National Institute forTheoretical Physics, University of KwaZulu-Natal, Durban, 4001, South Africa
I. Sinayskiy ∗ Quantum Research Group, School of Chemistry and Physics and National Institute forTheoretical Physics, University of KwaZulu-Natal, Durban, 4001, South Africa
Abstract
Open quantum walks (OQW) are formulated as quantum Markov chains ongraphs. It is shown that OQWs are a very useful tool for the formulation ofdissipative quantum computing algorithms and for dissipative quantum statepreparation. In particular, single qubit gates and the CNOT-gate are imple-mented as OQWs on fully connected graphs. Also, dissipative quantum statepreparation of arbitrary single qubit states and of all two-qubit Bell-statesis demonstrated. Finally, the discrete time version of dissipative quantumcomputing is shown to be more efficient if formulated in the language ofOQWs.
Keywords:
Open quantum walk, dissipative quantum computing,dissipative state engineering
1. Introduction
Recently, the experimental realization of a quantum computer has beenthe focus of extensive research [1]. One of the main problems of the physical ∗ Corresponding author. Tel./Fax: +27-(0)31-260-8133/8090e-mail address: [email protected]
Preprint submitted to Physics Letters A June 27, 2018 mplementation of well known quantum algorithms [2] is the creation and ma-nipulation of entanglement between qubits. Any physical system is subjectto interaction with the environment, which inevitably leads to decoherenceand dissipation [3]. One way to compensate for this destructive environmen-tal influence in the unitary implementations of the quantum algorithms isto introduce error-correcting codes [4]. However, this approach treats theinteraction with the environment as an effect the influence of which needs tobe minimized.A paradigm shift in looking for alternative strategies to realize quantumcomputers was induced with the theoretical prediction that dissipation can beused to create complex entangled states [5] and to perform universal quantumcomputation [6]. This fundamental change is based on the assumption thatone can manipulate the coupling of a system to an environment in such away that the system is driven towards a thermal state, which is the solutionof a particular quantum computing task [6] or a target state in quantumstate engineering [5, 6]. The feasibility of this strategy was demonstrated byimplementing dissipative quantum state engineering with ensembles of atoms[7] and trapped ions [8].Quantum algorithms for universal quantum computing are convenientlyformulated in the language of quantum walks [9]. For example, a discrete timequantum walk implementation of the search algorithm on complex graphs hasbeen shown to be more efficient than other known implementations of thisquantum algorithm [10]. As for any other unitary implementation of quan-tum computing the efficiency of the quantum walk based realization decreasesdue to interaction with the environment. In view of the appeal of dissipa-tive quantum computing it seems natural to formulate a dissipative versionof the quantum walk so that algorithms for dissipative quantum computingand quantum state engineering are implemented efficiently. In other words,if dissipative quantum computing can make use of the interaction with theenvironment for performing universal quantum computation, can one intro-duce a framework which will use dissipative rather than unitary dynamics asa ”driving force” of the quantum walk?During the last few years attempts were made to take into account dis-sipation and decoherence in the description of quantum walks [11]. In theseapproaches decoherence is treated as an extra modification of the unitaryquantum walk scheme, the effect of which needs to be minimized and elimi-nated. In fact, the general framework of quantum stochastic walks was pro-posed [12], which incorporates unitary and non-unitary effects of the quan-2um Markovian dynamics. In particular, by adding extra decoherence inexperimental realizations of quantum walks, the transition from quantum toclassical random walks was observed [13].Recently, a formalism for discrete time open quantum walks (OQW),which is exclusively based on the non-unitary dynamics induced by the en-vironment was introduced [14]. OQWs are formulated in the language ofquantum Markov chains [15] and rest upon the implementation of appropri-ate completely positive maps [3, 16].
2. Formalism
To review briefly the formalism of OQWs, we consider a random walkon a set of nodes or vertices V with oriented edges { ( i, j ) ; i, j ∈ V} asillustrated in Fig. 1(a). The number of nodes is considered to be finite orcountable infinite. The space of states corresponding to the dynamics on thegraph specified by the set of nodes V will be denoted by K = C V and has anorthonormal basis ( | i i ) i ∈V . The internal degrees of freedom of the quantumwalker, e.g. the spin or n -energy levels, will be described by a separableHilbert space H attached to each node of the graph. More concisely, anystate of the quantum walker will be described on the direct product of theHilbert spaces H ⊗ K .To describe the dynamics of the quantum walker, for each edge ( i, j ) weintroduce a bounded operator B ij ∈ H . This operator describes the changein the internal degree of freedom of the walker due to the shift from node j to node i . By imposing for each j that, X i B ij † B ij = I, (1)we make sure, that for each vertex of the graph j ∈ V there is a corre-sponding completely positive map on the positive operators of H : M j ( τ ) = P i B ij τ B ij † . Since the operators B ij act only on H and do not perform tran-sitions from node j to node i , an operator M ij ∈ H ⊗ K can be introduced inthe following form M ij = B ij ⊗| i ih j | . It is clear that, if the condition expressedin Eq. (1) is satisfied, then P i,j M ij † M ij = 1. This latter condition defines acompletely positive map for density matrices on H ⊗ K , i.e., M ( ρ ) = X i X j M ij ρ M ij † . (2)3 igure 1: Schematic illustration of the formalism of the Open Quantum Walk. ( a ) Thewalk is realized on a graph with a set of vertices denoted by i, j, k ∈ V . The operators B ji describe transitions in the internal degree of freedom of the “walker” jumping fromnode ( i ) to node ( j ). ( b ) The simplest non-trivial example of the OQW on the finitegraph is a walk on a two-node network. In this case the walk is performed using fouroperators M ji ( i, j = 1 , M = B ⊗ | ih | and M = B ⊗ | ih | ; the operators describingchanges in internal degrees of freedom of the ”walker”, if the ”walker” does not jump, are M ii = C i ⊗ | i ih i | , ( i = 1 , . open quantum walk (OQW). For anarbitrary initial state the density matrix P i,j ρ i,j ⊗ | i ih j | will take a diagonalform after just one step of the open quantum walk. By the direct insertionof an arbitrary initial condition in Eq. (2) we get M X k,m ρ k,m ⊗ | k ih m | ! = X i,j,k,m B ij ⊗ | i ih j | ( ρ k,m ⊗ | k ih m | ) B ij † ⊗ | j ih i | = X i,j,k,m B ij ρ k,m B ij † ⊗ | i ih i | δ j,k δ j,m = X i X j B ij ρ j,j B ij † ! ⊗ | i ih i | . (3)Hence, we will assume that the initial state of the system has the form ρ = P i ρ i ⊗ | i ih i | , with P i Tr[ ρ i ] = 1. It is straightforward to give anexplicit formula for the iteration of the OQW from step n to step n + 1: ρ [ n +1] = M ( ρ [ n ] ) = P i ρ [ n +1] i ⊗ | i ih i | , where ρ [ n +1] i = P j B ij ρ [ n ] j B ij † . Thisiteration formula gives a clear physical meaning to the mapping M : thestate of the system on site i is determined by the conditional shift from allconnected sites j , which are defined by the explicit form of the operators B ij .Also, one can see that Tr[ ρ [ n +1] ] = P i Tr[ ρ [ n +1] i ] = 1. Generic properties ofOQWs have been discussed in [14].As a first illustration of the application of the formalism of open quantumwalks we consider the walk on a 2-node graph (see Fig. 1b). To be specific,the transition operators are M = B ⊗ | ih | and M = B ⊗ | ih | and theoperators describing changes in internal degrees of freedom of the ”walker”,if the ”walker” does not jump, are M ii = C i ⊗ | i ih i | , ( i = 1 , i = 1 ,
2) we have: B † i B i + C † i C i = I. (4)The state ρ [ n ] of the walker after n steps is given by, ρ [ n ] = ρ [ n ]1 ⊗ | ih | + ρ [ n ]2 ⊗ | ih | , (5)where the particular form of the ρ [ n ] i ( i = 1 ,
2) is found by recursion, ρ [ n ]1 = C ρ [ n − C † + B ρ [ n − B † , (6) ρ [ n ]2 = C ρ [ n − C † + B ρ [ n − B † . igure 2: Open quantum walk on Z . (a) A schematic representation of the OQW on Z :all transitions to the right are induced by the operator B i +1 i ≡ B , while all transitionsto the left are induced by the operator B i − i ≡ C (see Eq. (7)); Figures (b)-(e) show theoccupation probability distribution for the “walker” with the initial state I / ⊗ | ih | andtransition operators given by Eq. (7) after 10, 20, 50 and 100 steps, respectively. Twodistinctive behaviors of the walker are observed: a Gaussian wave-packet moving slowlyto the left (dots) and a deterministic trapped state propagating to the right at a speed of1 in units of cells per time step (cross). i are given by operators of the form B i +1 i ≡ B and B i − i ≡ C . Obviously, the operators B and C satisfy the condition B † B + C † C = I , as imposed in Eq. (1). Assuming the initial state ofthe system to be localized on site 0, i.e., ρ = ρ ⊗ | ih | , after one stepthe system will jump to sites ± ρ [1] = BρB † ⊗ | ih | + CρC † ⊗ | − ih− | . The procedure can easily beiterated. In Figs. 2(b)-(e) we show the probability to find a “walker” on aparticular lattice site for different numbers of steps. For this simulation wehave chosen the transition matrices B and C as follows, B = sin θ |−ih−| + | + ih + | , C = cos θ |−ih−| , (7)and cos θ = 4 /
5. One can see that already after 10 steps, there are twodistinctive behaviors of the “walker”. The first is a Gaussian wave-packetmoving slowly to the left and the second one is a completely deterministictrapped state propagating to the right at a speed of 1 in units of cells pertime step. Interestingly, the state of the “walker” in the “soliton like” part isgiven by | + ih + | , while in the other, Gaussian part is given by the p n |−ih−| ,where, of course | + i and | + i states are defined as |±i = ( | i ± | i ) / √ p n is the probability to find the “walker” on the site n . Even this simpleexample demonstrates a remarkable dynamical richness of OQW. Furtherexamples of OQW on Z which show a behavior distinct from the classicalrandom walk and the unitary quantum walk can be found in [14].
3. OQW for dissipative quantum computing and state engineering
To motivate the potential of the suggested approach for the formulation ofquantum algorithms for dissipative quantum computing and quantum stateengineering we consider the example of an OQW on the 2-node fully con-nected graph in Fig. 1 (b). This example, will show that it is possible toimplement all single-qubit gates and the CNOT-gate in the language of thesuggested formalism. To be specific, in order to realize an X-gate with OQWswe prepare the system in some initial state | ψ i in node 1 and we will readthe result of the computation in the node 2 (See Fig. 1(b)). If we choose B = √ pX , C = √ qI , B = √ qX and C = √ pI , where p and q are7 igure 3: OQW preparation of the Bell-states of two qubits. A generic OQW walk on a4-node graph is decomposed in two walks on two-nodes graphs: “left-right” and “down-up” walks. For an initial unpolarized state of two-qubits on any nodes ρ = I ⊗ | i ih i | the corresponding Bell-pairs are obtained by measuring the position of the walkers afterperforming the OQW. positive constants such that p + q = 1, then the stationary state of this walkwill have the following form ρ SS = q | ψ ih ψ | ⊗ | ih | + pX | ψ ih ψ | X ⊗ | ih | .Therefore, the OQW on this graph realizes the X-gate with probability p upon read-out of the presence of the “walker” in node 2. In a similar way, inorder to implement the CNOT-gate we initially place two “walkers” in node1. We choose B = √ pU CNOT , C = √ qI , B = √ qU CNOT and C = √ pI and if the presence of both “walkers” is measured in the node 2 then theOQW realized the CNOT-gate with probability p .On the same 2-node network we can also implement dissipative statepreparation (see Fig. 1(b)). To this end, we consider trivial transition ma-trices on the node 1, i.e., B = C = I / √ B = √ p | ψ (1) ih ψ (2) | and C = √ q | ψ (2) ih ψ (2) | + | ψ (1) ih ψ (1) | , where p and q are positive constants such that p + q = 1, | ψ (1) i = (cos α, sin αe − iβ ) † and | ψ (2) i = ( − sin α, cos αe − iβ ) † . With this choicean arbitrary initial state will converge to a unique steady state, namely ρ f = | ψ (1) ih ψ (1) | ⊗ | ih | . The probability of detecting the system in thesteady state after 2 m steps of the walk is given by P SS ∼ − ρ (1)22 (0) / m − ( ρ (1)11 (0) + ρ (2)11 (0))[min(1 / , q )] m , where ρ ( i ) jj (0) are the elements of the initialdensity matrix of the system, ρ ( i ) jj (0) = h i, ψ ( j ) | ρ (0) | i, ψ ( j ) i .OQWs on more complex graphs allow the dissipative preparation of en-tangled multi-qubit states. With two “walkers” on a 4-node network (seeFig. 3) we can prepare all two-qubits Bell-states. In this particular case,the OQW can be decomposed in a combination of two walks on two inde-8 igure 4: Efficient transport with Open Quantum Walk. Fig. (a): a scheme of the chainof the N nodes with neighbor-neighbor interaction. Fig. (b): occupation probabilitydistribution as a function of time and lattice sites. The initial state of the walker islocalized in the first node and given by ρ = I ⊗ | ih | . pendent 2-node networks. The first “walker” moves up and down, while thesecond one moves left and right (see Fig.3). We choose the transition op-erators to be B z = ( I − Z Z ), C z = ( I + Z Z ), B x = ( I − X X )and C x = ( I + X X ), where X i and Z i denotes Pauli matrices acting onthe corresponding qubit i ( i = 1 , ρ = I ⊗ | j ih j | , the OQW will con-verge to a state ρ = | ψ − ih ψ − | ⊗ | U, L ih U, L | + | φ − ih φ − | ⊗ | U, R ih U, R | + | ψ + ih ψ + | ⊗ | D, L ih D, L | + | φ + ih φ + | ⊗ | D, R ih D, R | . This means that mea-suring the position of the “walkers” will determine corresponding Bell-stateof their internal degrees of freedom [17].Another application of the OQWs is a description of a dissipatively drivenquantum bus between computational quantum registers. To this end weconsider a chain of nodes (see Fig. 4(a)). Initially, the first node of thechain is in the exited state, so that ρ = I ⊗ | ih | . To be specific, wechose transition operators B and C as follows, B = √ p | ψ (1) ih ψ (2) | and C = √ q | ψ (2) ih ψ (2) | + | ψ (1) ih ψ (1) | . It is very interesting to note that in this case thestate | ψ (1) i propagates through the chain with velocity almost equal to 1 (inunits of cells per time step): the initial excitation in node (1) is completelytransferred to the last node (N) in N+2 steps. In Fig. 4 (b) we consider9 igure 5: OQW formulation of dissipative quantum computing (DQC). The initial state isprepared in the time-register 0. After performing the OQW, the results of the algorithmcan be readout from the time-register N . The internal state of the ”walker” in the register T will be given by | ψ T i = U T . . . U | ψ i . The positive constants ω and λ satisfy ω + λ = 1. a 100 node chain with √ p = 4 / | ψ (1) i = | + i , | ψ (2) i = |−i and showthat the initial excitation reaches the final node (100) in 102 steps. Thehigh performance of transport of excitations in the OQW formalism opensup new avenues of research into the understanding of quantum efficiency inopen systems.OQWs include the discrete time version of the dissipative quantum com-puting (DQC) introduced by Verstraete et al. [6]. In the original setup a lin-ear chain of time registers is considered and the initial state is prepared in thetime register 0. A quantum computation is performed by the dissipative evo-lution of the system into its unique steady state ρ = T +1 P t | ψ t ih ψ t | ⊗ | t ih t | [6]. The result of the quantum computation can be read-out by measur-ing the state of the system in the last time register T which is given by | ψ T i = U T U T − . . . U U | ψ i , where { U t } Tt =1 is an appropriate sequence ofunitary operators [6]. The probability of a successful read-out is 1 / ( T + 1).A discrete time version of DQC can be realized as an OQW on a linear chainof time registers (Fig. 5) by choosing the transition operators as it is shown inFig. 5 and constants ω = λ = 1 /
2. However, with the same number of stepsin the OQW formulation of DQC the probability of successful read-out canbe increased arbitrarily close to one. In order to understand this dramaticimprovement in efficiency we recall that in the original DQC formulation theprobability of read-out of the final state is determined by the form of “jump-ing” operators between time registers, i.e., L t = U t ⊗ | t ih t + 1 | + U † t ⊗ | t + 1 ih t | .The probability to ”jump” forward and backward in the time register is thesame. In the OQW formulation of the DQC we have the freedom of choos-ing ω > λ which induces the steady state of the OQW with probability ofread-out of the final state between 1 / ( T + 1) and 1.10 . Conclusion In conclusion, we have shown that the recently introduced formalism ofOQWs is a very useful tool for the formulation of dissipative quantum com-puting algorithms and for dissipative quantum state preparation. OQWs areto dissipative quantum computing what Hadamard quantum walks are to cir-cuit based quantum computing. In particular, we have shown OQW imple-mentation of circuit and dissipative models of quantum computing. Remark-ably, the OQW discretisation of dissipative quantum computing increases theprobability of successful implementation of the quantum algorithm with re-spect to the original formulation. It is to be expected, that OQWs will leadto the optimal formulation of certain classes of quantum algorithms. Fur-thermore, we have indicated that OQW can be used to explain non-trivialhighly efficient transport phenomena not only in linear but also in morecomplex topologies of the underlying graphs. This implies that OQW asquantum walks which are driven by dissipation and decoherence are one ofthe candidates for understanding the remarkable transport efficiency in pho-tosynthetic complexes [18]. We expect the potential of this framework to besoon revealed in the realms of quantum computing and quantum biology.
Acknowledgements
This work is based upon research supported by the South African Re-search Chair Initiative of the Department of Science and Technology andNational Research Foundation. Work supported by ANR project “HAM-MARK” N ◦ ANR-09-BLAN-0098-01
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States, Effects and Operations: Fundamental Notions ofQuantum Theory , Springer Verlag (1983).[17] Please note that this is not a unique steady state of the system and it isdepends on initial conditions. It is clear that this type of the random walkwill not affect an initial state of the following form, ρ = α | ψ − ih ψ − | ⊗| U, L ih U, L | + β | φ − ih φ − | ⊗ | U, R ih U, R | + γ | ψ + ih ψ + | ⊗ | D, L ih D, L | + δ | φ + ih φ + | ⊗ | D, R ih D, R | , where α, β, γ, δ ≥ α + β + γ + δδ