Faster quantum computation with permutations and resonant couplings
FFaster quantum computation with permutations and resonant couplings
Yingkai Ouyang,
1, 2, 3, ∗ Yi Shen,
4, † and Lin Chen
4, 5, ‡ Department of Physics & Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 School of Mathematics and Systems Science, Beihang University, Beijing 100191, China International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China (Dated: July 9, 2019)Recently, there has been increasing interest in designing schemes for quantum computations that are robustagainst errors. Although considerable research has been devoted to quantum error correction schemes, muchless attention has been paid to optimizing the speed it takes to perform a quantum computation and developingcomputation models that act on decoherence-free subspaces. Speeding up a quantum computation is important,because fewer errors are likely to result. Encoding quantum information in a decoherence-free subspace is alsoimportant, because errors would be inherently suppressed. In this paper, we consider quantum computationin a decoherence-free subspace and also optimize its speed. To achieve this, we perform certain single-qubitquantum computations by simply permuting the underlying qubits. Together resonant couplings using exchange-interactions, we present a new scheme for quantum computation that potentially improves the speed in which aquantum computation can be done.
I. INTRODUCTION
By harnessing the powers of quantum mechanics, quan-tum computers can potentially unleash algorithms of unprece-dented power, improve the precision of metrology, and enablenovel cryptographic schemes. However the inherent fragilityof quantum information frustrates the construction of anyquantum computer that is to compute correctly. Designingschemes for quantum computation that can proceed with highfidelity is thus an important problem.Performing a quantum computation within a decoherence-free subspace (DFS) is a natural approach towards combatingdecoherence, and is applicable in a wide range of physical sys-tems. Conventionally in an artificial atom, a qubit has its basisstates assigned to be a ground state and an excited state re-spectively. Because of this, the qubit is vulnerable to pickingup an unwanted phase that results entirely from the physicalsystem’s natural dynamics. To avoid this problem, several ar-tificial atoms could encode a logical qubit in a constant energysubspace by assigning the logical basis states to have a con-stant number of excitations. Spurious phase errors that wouldhave resulted from the natural dynamics of the system’s evolu-tion can thereby be intrinsically avoided. Such encodings intoDFSs have been explored both in the context of quantum com-putation [1–3] and quantum error correction [4–7]. Combinedwith carefully tailored active control on the underlying physi-cal system [8, 9], decoherence can be substantially mitigated.An observation that one could make is that the longer ittakes to perform a quantum computation, the more likely itis for quantum information to decohere. It is therefore im-portant to speed up a quantum computation to maximize thecomputation’s fidelity. This has been discussed in the contextof performing quantum computations within DFSs [2], wherethe speed of a spin-based quantum computation is optimized ∗ y.ouyang@sheffield.ac.uk (corresponding author) † [email protected] (corresponding author) ‡ [email protected] (corresponding author) by sidestepping the need to use slow single qubit gates. In Ref[2], a logical qubit comprises of three physical qubits, and allgates are driven entirely by resonant couplings between pairsof physical qubits. Because these coupling strengths are large,the computation can be fast. In the work of DiVincenzo et al. ,the speed of the computation can be quantified by the numberof timesteps the computation requires. With respect to this,single-qubit gates and a CNOT gate can be performed in 3 and11 timesteps respectively.Inspired by the possibility of performing quantum compu-tations entirely by exchange interactions in Ref [2], Levy de-vised a scheme for quantum computation using only exchangeinteractions, but which requires far fewer timesteps [10]. InLevy’s scheme, just two physical qubits form a logical qubit,and single-qubit gates and an entangling two-qubit gate areperformed in 3 and 2 timesteps respectively. However, Levy’sscheme no longer computes within a DFS, and its logicalqubits are hence vulnerable to phase noise. Similarly, while su-perconducting qubits [11] can have quantum computation thatresults from strong resonant couplings and are hence fast, theresultant computation also does not lie within a DFS. Thereinarises the question: can one reduce the number of timesteps re-quired in a quantum computation that proceeds within a DFS?The subject of how one can perform a quantum computa-tion has been extensively studied. Of specific interest to us aremodels of quantum computation based on the quantum cir-cuit model, where the computation is composed from certainsingle-qubit and two-qubit gates [1, 12, 13].In a markedly different setting, one can indeed imple-ment quantum computations in one of the simplest imagin-able ways, which is solely by performing permutations of theunderlying particles. One related model of quantum compu-tation relies on braiding particles [14]; the braiding group for n objects is different and richer than the permutation groupof n objects. However, quantum computation by braiding usesnonabelian anyons, and remains challenging to implement ina laboratory setting. In view of this, one might wonder ifquantum computation can be performed simply by permut-ing regular qubits. However in most physical implementations,permuting the qubits is not an easy operation. What we ac- a r X i v : . [ qu a n t - ph ] J u l ually recommend is that physical qubits need not be actu-ally permuted in practice; a classical computer keeps track ofall the permutations that take place, essentially relabeling thequbits whenever a permutational operation has to take place.Non-permutational gates then proceed by interacting the cor-responding relabeled qubits. This allows permutational gatesto be performed by a classical computer, and thereby allowfaster quantum computation by this precompiling technique.The question of permutational quantum computation hasbeen studied by Jordan, who studied the extent in which per-mutations alone can perform an interesting family of quantumcomputations on rather specific problems [15]. It is howeverunlikely that permuting qubits can effect an arbitrary quantumcomputation, because there is only a finite number of permu-tations while the number of possible quantum computations isinfinite. For example, Jordan gives in [15, Section 9] a count-ing argument explaining why it is unlikely that permutationalquantum computing is equal to BQP, the class of decisionproblems solvable by a quantum computer in polynomial timewith a bounded probability of error. Hence, to achieve a qubit-based universal quantum computation, permutations must beaugmented by non-permutational gates.Could one perform quantum computations in a DFS us-ing only permutations and resonant couplings? In this paper,we answer this question in the affirmative. We show that anon-trivial set of permutations augmented by realistic resonantcouplings can indeed allow for a universal quantum computa-tion to proceed within a DFS. Such a scheme could reducethe number of timesteps required in the quantum computation.This is because a classical computer could keep track of howthe underlying qubits are permuted and determine where andwhen resonant couplings are applied between pairs of physicalqubits. Because of this, parts of a quantum computation can beoffloaded to a classical computer, which allows the quantumcomputation to proceed with fewer timesteps than the schemeof Ref [2].The caveat of requiring at least 32 physical qubits to en-code each logical qubit might not be too severe, because thepotential speed ups offered when the quantum computation tobe performed is reasonably complicated, such as that in fault-tolerant quantum computation, might be worth the while.The possibility of performing the ‘classical part’ of a quan-tum computation by permutations, while intuitive is far fromobvious. The Clifford group of single qubit gates, generatedby a Hadamard and a complex phase gate, is a finite group,and hence is by Cayley’s theorem a subgroup of a symmet-ric group of some size [16]. While the explicit structure ofthe single-qubit Clifford group has been elucidated by Planat[17], its realization in terms of permuting qubits is far from re-solved. This is because even though the matrix representationof the single-qubit Clifford group in terms of permutations ex-ists, there might not exist a state space on which these permu-tations act faithfully. For example, while the matrix algebraof the single-qubit Clifford gates can be made to be correct,its action is entirely dependent on the basis chosen for qubit’slogical encoding. An arbitrary selection of the basis would in-variably lead to an incorrect group action on the chosen ba-sis states. We show that certain subgroups of the single-qubitClifford group can be performed entirely by permutations, andgive a lower bound for the number of qubits required to realize single-qubit Clifford computations by permutations.The scheme that we propose to augment present can be seento be a discretized version of the dual-rail encoding [1]. Whenour scheme encodes 32 physical qubits into a logical qubit, itallows the π -gate and bit flip gates to be performed by per-mutations. Exploiting the fact that the bit-flip gate is a prod-uct of disjoint swaps, we show that the Hadamard gate andCNOT can be performed using physically realistic resonantcouplings. To illustrate the potential of our scheme, we calcu-late the number of timesteps required to implement an impor-tant gate in quantum computation, the Toffoli gate. Our pro-posed scheme implements the Toffoli gate within a DFS in82 timesteps, which is slightly faster than the 85 timesteps re-quired in the scheme of DiVincenzo et al. .The resonant couplings that we require in our scheme are ar-guably physically realistic. Their purpose is to implement thequantum Fredkin gate and exchange interactions on the phys-ical qubit level. Exchange interactions can potentially be im-plemented accurately even at a non-zero temperature [18] inan experimental setting. The quantum Fredkin gate is a con-trolled swap gate, and has been extensively studied [19–21]since its introduction [22]. While the quantum Fredkin gateis more challenging to implement than simple exchange, re-cent work suggests that it can be implemented using physicallyrealistic resonant interactions on superconducting qubits in asingle timestep [21].Our results are theoretical in nature, and do not deal withany specific experimental system. This allows our approachto be potentially applied in a wide range of physical systemsonce the specifics of these systems are taken into account. InSection II, we present our first scheme which allows universalquantum computation. This scheme is an extended dual-railscheme which uses 32 qubits to encode a single logical qubit.We further analyze the performance of our extension of thedual-rail encoding scheme with respect to generating a Toffoligate in Section II B. In Section II C, we present our secondscheme, which show how to implement the Hadamard gate ( H ) and the phase-flip ( Z ) simultaneously using only permuta-tions. This demonstrates for the first time how a permutationalHadamard could be implemented with other Pauli permuta-tional gates. In Section II D, we investigate the possibility ofpermutational single-qubit Clifford gates, specifically on theminimum number M of physical qubits to generate the fullset of single-qubit Clifford gates by permuting the underlyingqubits. In this section we present a non-trivial bound of M ,ruling out schemes that can perform permutational Cliffordson too few qubits. We also give a set of necessary and suffi-cient conditions to obtain M , which paves the way forward infinding permutational gates that implement the Clifford gates.Finally we summarize and discuss our work in Section III. II. RESULTS
To explain how our scheme works in a DFS, we first de-fine the system’s Hamiltonian. We are interested in a systemwith N modes, each mode of which can be described by iden-tical quantum harmonic oscillators. Explicitly, we can writethe Hamiltonian as H = ∑ Ni = a † i a i , where a i denotes the low-ering operator for the i th mode. Such a Hamiltonian for exam-2le is compatible with a physical system comprising of pho-tons of identical frequencies within a quantum bus, and canalso be engineered from any coupled quantum harmonic os-cillator system via application of dynamical decoupling pulsesequences [9]. Note that we can denote | x (cid:105) ⊗ · · · ⊗ | x N (cid:105) as aquantum state with x i excitations in the i th mode. Then wecall x + · · · + x N denote the total excitation number of such astate. If the number of excitations per mode is at most one, then ( x , . . . , x N ) are just binary vectors, and the corresponding ex-citation number is just their Hamming weight. It is easy to seethat the eigenspaces of H are spanned by states | x (cid:105)⊗· · ·⊗| x N (cid:105) with a constant total excitation number. Schemes based onsuch constant-excitation subspaces have also recently beenstudied [6, 7].Up to a global phase, single-qubit Clifford gates are givenby 24 matrices A i j . When j = , , , A j = (cid:18) i j (cid:19) and A j = (cid:18) i j (cid:19) . Moreover A j = √ ( A j + iA j ) and A j = √ ( A j − iA j ) . Furthermore A = √ ( A + A ) , A = √ ( A + A ) , (II.1) A = √ ( A + A ) , A = √ ( A + A ) , (II.2)and A = √ ( A − A ) , A = √ ( A − A ) , (II.3) A = √ ( A − A ) , A = √ ( A − A ) . (II.4)Note that these 24 gates can be generated by the Hadamardgate H = A and the phase gate P = A . In what follows,we will investigate the extent to which these 24 gates can beimplemented by permutations. A. An extended dual-rail encoding scheme
In the dual-rail encoding scheme previously considered, thestates | (cid:105)| (cid:105) and | (cid:105)| (cid:105) encode the logical states of a qubit [1].In this scheme, the only non-trivial permutation possible is onethat swaps the first qubit with the second. We denote this per-mutation by ( , ) q , where the subscript q indicates that ( , ) q applies to qubits. The permutation, ( , ) q is equivalent to thebit-flip operation on the space, because ( , ) q | (cid:105)| (cid:105) = | (cid:105)| (cid:105) and ( , ) q | (cid:105)| (cid:105) = | (cid:105)| (cid:105) . Since there is no other non-trivialpermutation available on two qubits, only the bit-flip gate canbe performed using permutations in the dual-rail encodingscheme.By extending the dual-rail encoding scheme, it becomespossible to implement more gates by permuting qubits.Namely, we map each physical qubit in the dual-rail encod-ing scheme to a 2 n -qubit state with two rows of n qubits each.By extending the dual-rail encoding scheme, we can not onlyimplement the logical bit flip, but also logical gates that per-form logical (cid:18) e π i / n (cid:19) gates in the logical basis. Before we proceed to describe our scheme, we wish to em-phasize we do not recommend that permutational gates be ac-tually implemented on the physical system. This is becausepermutational gates are challenging to implement in practice,and may take too long to perform, thereby exposing the sys-tem to decoherence. In our scheme, the permutational gatesare only meant to be carried out by a classical computer.The classical computer computes the permutations of the la-bels on the underlying physical qubits. The non-permutationalgates, which can be carried out by resonant couplings, are thenperformed between appropriate pairs or subsets of relabeledqubits. This can be achieved in physical systems where long-range interactions may be possible, such as in ion-trappedquantum architectures or carefully designed superconductingquantum circuits.The basis states in our extended dual-rail encoding schemewill comprise of 4 n qubits, with four rows with n qubitseach. The basis states can be defined in terms of the states | ( j ) n (cid:105) = X j | (cid:105) ⊗ n , where X j denotes the n -qubit matrix that ap-plies the bit-flip operator on the j th qubit and leaves the re-maining qubits unchanged. In particular, instead of using twophysical qubits in the dual-rail encoding scheme, we use 4 n qubits to encode a logical qubit. Denoting x = e π i / n to be aroot of unity, we denote the states | ψ (cid:105) = √ n n ∑ j = x j − | ( j ) n (cid:105) ⊗ | (cid:105) ⊗ n , (II.5)and | ψ (cid:105) = | (cid:105) ⊗ n ⊗ √ n n ∑ j = x j − | ( j ) n (cid:105) . (II.6)We interpret both of these states to have qubits on two rows,where each row has n qubits. The orthonormal basis states ofour scheme are then given by | XTL (cid:105) = | ψ (cid:105)| ψ (cid:105) , | XTL (cid:105) = | ψ (cid:105)| ψ (cid:105) , (II.7)which are states on 4 n qubits arranged in 4 rows with n qubitseach.To implement the bit-flip operation, it suffices to apply thepermutation α = β ⊗ β , where β is a permutation that swapsthe rows in | ψ (cid:105) and | ψ (cid:105) . This is depicted in Figure 1(a).Formally, β = ( , n + ) q ( , n + ) q ... ( n , n ) q is a product ofswaps, where ( j , n + j ) q denotes a swap between the j th qubitwith the ( n + j ) th qubit.To implement a logical gate that induces a phase on | XTL (cid:105) but not on | XTL (cid:105) , it suffices to cycle qubits on the third row ofthe logical qubit leftwards. We can achieve this formally withthe permutation γ = I n ⊗ ( n , n − , ..., ) q , where I k denotesa size k identity matrix, and the permutation ( n , n − , ..., ) q cycles qubits 1 to n , where the j th qubit is mapped to the ( j − ) th qubit for all j = , . . . , n , and the first qubit is mapped tothe n th qubit. We depict this cyclic permutation γ in Figure1(c). Then γ | XTL (cid:105) = | XTL (cid:105) (II.8)and γ | XTL (cid:105) = x | XTL (cid:105) . (II.9)3o understand this more explicitly, note that every qubit onthird row of qubits for the logical | XTL (cid:105) are all identically equalto | (cid:105) , and are thereby left unchanged by any permutation onthem. On the other hand, the third row of qubits in the logical | XTL (cid:105) have the joint state √ n ∑ nj = x j − | ( j ) n (cid:105) , which picks upa phase of x when cycled leftwards.Thus, on the logical basis, γ implements the (cid:18) x (cid:19) , whichis equivalent to the T gate when n =
8, because in this scenario, x = e π i / = e π i / . β : 1 2 3 nn +1 n +2 n +3 2 n n +1 2 n +2 2 n +3 3 n n +1 3 n +2 3 n +3 4 nβ : α : (a)The permutation α that implementsa bit-flip gate. nn +1 n +2 n +3 2 n n +1 2 n +2 2 n +3 3 n n +1 3 n +2 3 n +3 4 n n +2 2 n +3 3 n n +1 3 n +2 3 n +3 4 nU β : (b)The Hadamard uses U β , whereresonances occur in parallel. nn + 1 n + 2 n + 3 2 n n + 1 2 n + 2 2 n + 3 3 n n + 1 3 n + 2 3 n + 3 4 nγ : (c)The permutation γ that implements a | XTL (cid:105)(cid:104) XTL | + e π i / n | XTL (cid:105)(cid:104) XTL | gate. FIG. 1:
Some quantum gates constructed from permutations.
Therefore, with 32 qubits and with the logical states inEq. (II.7), we can use the permutations α and γ to implementthe bit-flip gate and the T gate respectively. When n = n is,the more gates we can implement by permutations. TABLE I:
The eight single-qubit Clifford gates A i j generated by P gate and X gate when n is a multiple of 4.i A i j j 1 2 3 41 P P P P PX ( PX ) ( PX ) ( PX ) In order to allow for arbitrary single-qubit operations, wealso need to implement the Hadamard gate on the basis statesgiven by Eq. (II.7). Here, we can no longer rely on permut-ing the underlying physical qubits. Instead, we must utilizeresonant couplings which induce Rabi oscillations betweenspecific two-level systems of our choosing. Specifically, the main idea of how to implement the logical Hadamard arisesfrom a simple observation with respect to the dual-rail encod-ing scheme. For the dual-rail encoding scheme, it suffices toimplement the Hamiltonian X X and appropriate phase gates.The Hamiltonian X X in turn is equivalent to implementingthe Hamiltonian X sandwiched between two CNOT gates.The permutation β which acts as a bit-flip gate between thestate | ψ (cid:105) and | ψ (cid:105) is essentially a swap of two rows of qubits.It is thus a many-body Hamiltonian, and is challenging to im-plement in practice. Fortunately, as we will show, a Hamilto-nian β is effectively simulated by pairwise resonant couplingsbetween pairs of qubits on a subspace our states reside in. Thefirst type of resonant coupling we use must induce Rabi oscil-lations between the two-qubit states | (cid:105)| (cid:105) and | (cid:105)| (cid:105) . This canbe achieved using the Heisenberg exchange interaction ( , ) q An important technical observation that we use is that a many-body Pauli-type interaction with exchange interactions on asingle-excitation subspace B k = { X | (cid:105) ⊗ k , . . . , X k | (cid:105) ⊗ k } can beparallelized as given in Theorem 1. Theorem 1.
Let | ψ (cid:105) be any state in the span of B n , and let θ be any real number. Thene i θπ e i θπ . . . e i θπ ( n − )( n ) | ψ (cid:105) = e ( n − ) i θ e i θπ π ... π ( n − )( n ) | ψ (cid:105) , where π i j = ( i , j ) q is an operator that swaps qubit i and qubitj. We prove Theorem 1 in Appendix A. Because of this theo-rem, we can decompose U β as illustrated in Figure 1(b) into aproduct U β = e − ( n − ) i π e i ( , n + ) q π . . . e i ( n , n ) q π ⊗ I n . (II.10)It is therefore possible to implement U β in a single timestep byparallel use of Heisenberg exchange couplings, which makesit possible to implement in realistic physical settings. We willsoon see that the Hadamard gate relies on sandwiching the uni-tary U β = e i β π ⊗ I n between CNOT-like gates on the states | ψ i (cid:105)| ψ j (cid:105) , which we denote as C β . In particular we would like C β | ψ i (cid:105) ⊗ | ψ j (cid:105) = | ψ i (cid:105) ⊗ | ψ ( i + j mod 2 ) (cid:105) , i , j = , . (II.11)We now propose how C β can be implemented using quan-tum Fredkin gates, i.e. controlled swap operations. An ad-vantage of relying on quantum Fredkin gates is that they canbe implemented using an effective Hamiltonian g | (cid:105)(cid:104) | ( , ) q .Similar interactions can be implemented using superconduct-ing qubits for example [21]. This effective Hamiltonian al-lows Rabi oscillations to be induced between a pair of states | (cid:105)| (cid:105) and | (cid:105)| (cid:105) conditioned on the state of the control qubit,while leaving | (cid:105)| (cid:105) and | (cid:105)| (cid:105) unchanged. Now denote F a , ( b , c ) as a quantum Fredkin gate with the a th qubit as the control-ling qubit, and the swap occurring between the ( n + b ) th and ( n + c ) th qubits respectively. Specifically, we propose that C β = n ∏ j = C j , β (II.12)where C j , β = F n + j , ( , n + ) . . . F n + j , ( n , n ) (II.13)4s an operator that swaps the third and fourth rows within thelogical qubit conditioned on the value of the j th qubit in thesecond row. Note that C j , β takes n timesteps to proceed, and C β can be arranged such that n gates occur in n timesteps.Then it follows that γ − C β U β C β acts as a Hadamard on thebasis states given by Eq. (II.7) because for j = , γ − C β U β C β | j XTL (cid:105) = | XTL (cid:105) + ( − ) j | XTL (cid:105)√ . (II.14)Hence to implement the Hadamard gate, we require 2 n + n timesteps arise from C β , one time steparises from U β , and another n arise from C β .We now explain why the U β and C β gates can be imple-mented within a decoherence-free subspace of the physicalHamiltonian. Explicitly, the physical Hamiltonian that we con-sider is a sum of identical artificial atoms (or quantum har-monic oscillators). What this means is that this Hamiltoniancommutes with any swap operator on qubits. Since U β relieson Hamiltonians that are swaps on distinct pairs of qubits, U β must proceed with the DFS of the physical system. For a simi-lar reason, the operator | (cid:105)(cid:104) | ( , ) q commutes with the phys-ical Hamiltonian on the subspace where the quantum compu-tation takes place. This is because the control part | (cid:105)(cid:104) | isjust diagonal in the basis of the physical Hamiltonian, and thephysical Hamiltonian being permutation-invariant commuteswith any swap operation.To make universal quantum computation possible with ourextended dual-rail scheme, we need to show how to performan entangling gate. It turns out that we can perform a CNOTbetween two logical qubits defined on the basis states (II.7), byappropriate controlled swaps of the underlying rows of qubits.Now we consider quantum Fredkin gates with controls withinthe first logical qubit and swaps occurring on the second logi-cal qubit. In particular, let C i , j , k denote a quantum Fredkin gatethat swaps the j th and k th qubits on the target logical qubit.conditioned on the value of the i th qubit in the control logicalqubit. Then the logical CNOT can be achieved by doing thefollowing.1. Apply C n + i , j , n + j for i , j = , . . . , n .2. Apply C n + i , j , n + j for i , j = , . . . , n .This procedure swaps the first and second rows, and the thirdand fourth rows of the target logical qubit. These swaps areconditioned to swapped conditioned on whether the secondand third rows in the control logical qubit are all in the all | (cid:105) state. Hence our logical CNOT uses 2 n quantum Fredkingates in total, and can be achieved in n timesteps by runningappropriate quantum Fredkin gates in parallel. B. Timesteps required for the Toffoli gate
Here, we analyze the performance of our extension of thedual-rail encoding scheme with respect to generating a Toffoligate. This Toffoli gate can be decomposed into CNOTs and Tgates, and it is illustrated in Figure 2.Two H gates and six CNOTs are required in this setup. Thetotal number of S , T and T † gates, which are permutational in • • • • T • • T † T † SH T † T T † T H
FIG. 2:
The Toffoli gate constructed from CNOTs and T gates our scheme, is 8. Hence the number of timesteps required inour extended dual-rail encoding scheme is 6 ( n ) + ( n + ) = n + =
82 for n =
8. We again emphasize that the permuta-tions corresponding to the T , T † and S gates in our scheme arekept track of by a classical computer, and subsequently, andthe resonant couplings are carried out between appropriatelypermuted pairs of qubits.In the scheme of DiVincenzo et al. [2] which operateswithin a DFS, each single qubit gate requires at least one timestep, and each CNOT gate requires 13 timesteps. Hence thisscheme requires at least 7 + ( ) =
85 timesteps are requiredfor a single Toffoli gate, which takes longer than that of ourscheme.
C. Hadamard gates by permutations
It is considerably more complicated to implement aHadamard by permutations. Extending the results of the pre-vious section would surely fail, because H and T generate anset of infinite size, while the number of permutations on anyfixed number of qubits is always finite. Hence, any schemewhich implements a Hadamard gate by permutations neces-sarily has to be quite different from the extended dual-rail en-coding scheme.Here, we show the gates generated by the Hadamard ( H )and the phase-flip ( Z ) can be implemented using only permu-tations. From Table II we can implement 8 such Clifford gates A i j by using H and Z gates simultaneously. We first consider TABLE II:
The eight single-qubit Clifford gates A i j generated by H gate and Z gatei A i j j 1 2 3 41 Z Z ZHZH Z HZH H H ZH H ZH HZ the construction of permutational Hadamard gates that act onthe basis states | H (cid:105) = √ ( | x , y (cid:105) + | x , y (cid:105) ) , (II.15) | H (cid:105) = i √ ( | x , y (cid:105) − | x , y (cid:105) ) . (II.16)Here, we require the vectors | x (cid:105) , | x (cid:105) , | y (cid:105) , | y (cid:105) to have unitnorm, and (cid:104) x | x (cid:105) = (cid:104) y | y (cid:105) =
0. A permutation H acts as aHadamard gate if H | H (cid:105) = | H (cid:105) + | H (cid:105)√ and H | H (cid:105) = | H (cid:105)−| H (cid:105)√ .Equivalently, we require H | x , y (cid:105) = w − | x , y (cid:105) , where w = π i . One can verify that these equations holds whenever H = U ⊗ Q , where U , Q are permutations that are Hermitian andalso satisfy the equations U | x (cid:105) = | x (cid:105) and Q | y (cid:105) = w − | y (cid:105) .In our construction, the logical basis vectors are spanned by | x (cid:105)| y (cid:105) and | x (cid:105)| y (cid:105) where | x k (cid:105) = w k √ ∑ j = w ( − ) k ( j − ) | ( j ) (cid:105) , | y (cid:105) = √ ∑ j = ( − w ) ( j − ) | ( j ) (cid:105) , | y (cid:105) = w | x (cid:105) . (II.17)Let us consider the permutations U =( , ) q ( , ) q ( , ) q ( , ) q , Q = ( , ) q ( , ) q , and R = ( , ) q ( , ) q ( , ) q . Disregarding the effects ofglobal phases, the permutational Hadamard and phase-flips are H = U ⊗ Q and Z = R ⊗ Q respectively.Similarly, the permutational bit-flip operator and XZ are X = URU ⊗ Q = ( , ) q ( , ) q ( , ) q ⊗ Q and Y = URUR ⊗ I = ( , , , ) q ( , , , ) q ⊗ I respectively. D. On the possibility of permutational single-qubit Cliffordgates
So far, we have shown that it is possible to implement asubgroup of the single-qubit Clifford gates by permuting theunderlying physical qubits. But can we implement the entireset of single-qubit Clifford gates using permutations alone?Here, we supply bounds on M , where M denotes the minimumnumber of physical qubits for which the full set of single-qubitClifford gates can be performed just by permuting the under-lying qubits.It is shown in [17] that the set of all single-qubit Cliffordgates modulo the global phase is isomorphic to S , which is asymmetric group of size 4. Now we argue that M ≥
12. Let M k denote the subspace spanned by | x (cid:105) for x of Hamming weight k . Without loss of generality, the subspace C spanned by ourlogical qubit lies within M k . Because H and P must be permu-tations on qubits, they induce orbits on appropriate subspacesof M k . Because HP and P are isomorphic to a 3-cycle and a4-cycle respectively on S , there must be bases W q = {| q i (cid:105)} and W p {| p i (cid:105)} within M k of cardinality 3 n and 4 n respec-tively where (1) n and n are positive integers, (2) q i and p i are binary vectors of weights k , and (3) HP | q i (cid:105) = | q i mod 3 n (cid:105) and P | p i (cid:105) = | p i mod 4 n (cid:105) . Let C HP denote the span of W q and C P denote the span of W p . So C must lie within the intersec-tion of C HP and C P . But if the intersection of C HP and C P notequals C P , then P does not stabilize C which is a contradic-tion. Hence we must have C = C HP = C P and hence M mustbe a multiple of 12 lcm ( n , n ) . We proceed to give a set of necessary and sufficient con-ditions for implementing the full set of single-qubit Cliffordgates on M qubits. Theorem 2.
If the full set of single-qubit Clifford gates canbe implemented by M qubits, then there exist two permutationmatrices P and H with size M and complex numbers z and z of modulus one such that rank (cid:18) AA (cid:48) (cid:19) < M + , whereA = (cid:18) P − z I − i P − z I (cid:19) , A (cid:48) = (cid:18) √ H − z I − z I − z I √ H + z I (cid:19) . (II.18)The above can be easily shown because the gates P and H which generate the set of single-qubit Clifford gates must sat-isfy the equations Pu = z u , Pv = iz v , Hu = z u + v √ , Hv = z u − v √ . Then we can write the above as a matrix equation as (cid:18) P P (cid:19) (cid:18) uv (cid:19) = z (cid:18) u i v (cid:19) , (cid:18) H H (cid:19) (cid:18) uv (cid:19) = z √ (cid:18) u + vu − v (cid:19) . (II.19)This is equivalent to (cid:18)(cid:18) P − i P (cid:19) − z (cid:18) I I (cid:19)(cid:19) (cid:18) uv (cid:19) = , (cid:18)(cid:18) H H (cid:19) − z √ (cid:18) I II − I (cid:19)(cid:19) (cid:18) uv (cid:19) = . (II.20)It follows that any non-zero solution for u and v yields a logi-cal basis for our single-qubit Clifford gates. Since we only fo-cus on the non-zero solutions, A and A (cid:48) are not unique. For ex-ample, one can also suppose A = (cid:18) P − z I P − iz I (cid:19) . Henceit suffices to find a non-zero intersection of kernels of A and A (cid:48) . This non-zero intersection in turn occurs if and only ifrank (cid:20) AA (cid:48) (cid:21) < M + . (II.21)which is a standard fact in matrix analysis [23, Fact 2.11.3].We however leave the problem of obtaining an upper boundon M open. III. DISCUSSION
Expediting quantum computation on schemes that are in-herently protected against noise is a tantalizing prospect. Suchschemes have been explored by DiVincenzo et al. [2] on spin-based quantum computers, and more recently, also on topo-logical quantum computers [14]. Utilizing permutations is oneapproach that can potentially speed up quantum computationsthat has been explored recently [17, 24–28]. However thesequantum computations either operate on non-abelian anyons[24–26], focus on just the group structure of permutationalsubgroups related to quantum computations [17, 28], or workon the magic state model of quantum computation related topermutations [27]. In this sense, prior work has neither ad-dressed the effect of permutations on the underlying qubitsnor addressed the case when the qubits are regular fermions orbosons which are more abundant in an experimental setting.In this report, we fill this gap by showing explicitly howto perform certain single-qubit quantum computations by sim-ply permuting the underlying qubits. Together with exchange-interactions and other resonant couplings, our scheme allowsa faster implementation of the Toffoli gate. Our scheme can6e seen to be a permutational extension of the dual-rail encod-ing scheme [1], and by virtue of being supported on encodedqubits with a low excitation number, allows universal gate setby using simple resonant interactions. We also explore the pos-sibility of implementing other single qubit gates by permuta-tions, and give necessary and sufficient conditions for their re-alization.We believe that determining if all the single-qubit Cliffordgates can be realized on a DFS with only a single-excitationis an important problem. This is because if this were possi-ble, exchange type couplings can implement a CNOT gate inparallel, and make possible an arbitrary Clifford computationon any number of logical qubits without requiring use of theFredkin gate. Given that Clifford computations are known tobe hard under reasonable computation assumptions [29], thiswould give rise to a way to realize speedy Clifford computa-tions using a simple scheme, and could bring us closer to thedemonstration of quantum supremacy.
ACKNOWLEDGEMENTS
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Appendix A: A parallel construction of an effective many-body Hamiltonian
Here, we will prove Theorem 1 in the main text. First we present a supporting lemma as follows.
Lemma 3.
Let θ be any real number, and P i j be the permutation matrix which swaps the ith and jth rows. Then e i θ P e i θ P = e i θ e i θ P P . Moreover for every integer n for which n ≥ we also have e i θ P ... P ( n − )( n − ) e i θ P ( n − )( n ) = e i θ e i θ P ... P ( n − )( n ) .Proof. One can verify (cid:2) P P . . . P ( n − )( n − ) , P ( n − )( n ) (cid:3) = n ≥
2. We first show the following identity. P P . . . P ( n − )( n − ) + P ( n − )( n ) = P P . . . P ( n − )( n − ) P ( n − )( n ) + I n , ∀ n ≥ . (A.1)By computing we obtain P P . . . P ( n − )( n ) = σ ⊕ · · · ⊕ σ (cid:124) (cid:123)(cid:122) (cid:125) n , where σ = (cid:20) (cid:21) . Therefore, we obtain P P . . . P ( n − )( n − ) + P ( n − )( n ) = ( σ ⊕ · · · ⊕ σ (cid:124) (cid:123)(cid:122) (cid:125) n − ⊕ I ) + ( I n − ⊕ σ )= ( σ ⊕ · · · ⊕ σ (cid:124) (cid:123)(cid:122) (cid:125) n ) + I n = P P . . . P ( n − )( n ) + I n . (A.2)Then it follows that e i θ P e i θ P = e i θ P + i θ P = e i θ P P + i θ I = e i θ e i θ P P , (A.3)where the first equality uses (A.1) and the second equailty uses (A.2). Similarly the result for n ≥ Proof.
Using Lemma 3 and the fact that the swap operators π i j on qubits correspond precisely to the permutation of qubit labels,we obtain e i θπ e i θπ . . . e i θπ ( n − )( n ) = e i θ e i θπ π e i θπ . . . e i θπ ( n − )( n ) = e i θ e i θπ π π e i θπ . . . e i θπ ( n − )( n ) = · · · = e ( n − ) i θ e i θπ π ... π ( n − )( n ) ..