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Logical Clifford Synthesis for Stabilizer Codes

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Logical Clifford Synthesis for Stabilizer Codes Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000. Digital Object Identifier Logical Clifford Synthesis for Stabilizer Codes NARAYANAN RENGASWAMY1, (Member, IEEE), ROBERT CALDERBANK1, (Fellow, IEEE), SWANAND KADHE2, (Member, IEEE), and HENRY D. PFISTER1, (Senior Member, IEEE) 1Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 USA (e-mail: { narayanan.rengaswamy, robert.calderbank, henry.pfister }@duke.edu) 2Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720 USA (e-mail: [email protected]) Corresponding author: Narayanan Rengaswamy (e-mail: [email protected]). Part of this work has been presented at the 2018 IEEE International Symposium on Information Theory [1]. This work was supported in part by the National Science Foundation (NSF) under Grant Nos. 1718494, 1908730 and 1910571. Any opinions, findings, conclusions, and recommendations expressed in this material are those of the authors and do not necessarily reflect the views of these sponsors. ABSTRACT Quantum error-correcting codes are used to protect qubits involved in quantum computation. This process requires logical operators to be translated into physical operators acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in CN×N as a 2m×2m binary symplectic matrix, where N = 2m. We prove two theorems that use symplectic transvections to efficiently enumerate all binary symplectic matrices that satisfy a system of linear equations. As a corollary, we prove that for an [[m; k]] stabilizer code every logical Clifford operator has 2r(r+1)=2 symplectic solutions, where r = m − k, up to stabilizer degeneracy. The desired physical circuits are then obtained by decomposing each solution into a product of elementary symplectic matrices, that correspond to elementary circuits. This enumeration of all physical realizations enables optimization over the ensemble with respect to a suitable metric. Furthermore, we show that any circuit that normalizes the stabilizer can be transformed into a circuit that centralizes the stabilizer, while realizing the same logical operation. Our method of circuit synthesis can be applied to any stabilizer code, and this paper discusses a proof of concept synthesis for the [[6; 4; 2]] CSS code. Programs implementing the algorithms in this paper, which includes routines to solve for binary symplectic solutions of general linear systems and our overall LCS (logical circuit synthesis) algorithm, can be found at https://github.com/nrenga/symplectic-arxiv18a. INDEX TERMS Clifford group, Heisenberg-Weyl group, logical operators, stabilizer codes, binary symplectic group, transvections I. INTRODUCTION arXiv:1907.00310v2 [quant-ph] 18 Aug 2021 realizing error-resilient quantum computation in practice. T is expected that universal fault-tolerant quantum com- The Clifford hierarchy of unitary operators was defined I putation will be achieved by employing quantum error- to help demonstrate that universal quantum computation can correcting codes (QECCs) to protect the information stored be realized via the teleportation protocol [9]. The first level in the quantum computer and to enable error-resilient com- C(1) in the hierarchy is the Pauli group of unitary operators, putation on that data. The first QECC was discovered by and subsequent levels C(`); ` ≥ 2, are defined recursively as Shor [2], and subsequently, a systematic framework was de- those unitary operators that map the Pauli group into C(`−1), veloped by Calderbank, Shor and Steane [3], [4] to translate under conjugation. By this definition, the second level is the (pairs of) classical error-correcting codes into QECCs. Codes normalizer of the Pauli group in the unitary group, and hence produced using this framework are referred to as CSS codes. C(2) is the Clifford group [5]. It is well-known that the levels The general class of stabilizer codes includes CSS codes as a C(`) do not form a group for ` ≥ 3, but that the Clifford group special case and was introduced by Calderbank, Rains, Shor along with any unitary in C(3) can be used to approximate an and Sloane [5], and by Gottesman [6]. These codes, and their arbitrary unitary operator up to any desired precision. (Note variations [7], [8], still remain the preferred class of codes for that using a simple inductive argument it can be proven that VOLUME 4, 2016 1 N. Rengaswamy et al.: Logical Clifford Synthesis for Stabilizer Codes each level in the hierarchy is closed under multiplication by The primary contributions of this paper are the four theo- Clifford group elements.) Therefore, the standard strategy rems that we state and prove in Section III-B, and the main for realizing universal computation with QECCs is to first LCS algorithm (Algorithm 3) which builds on the results of synthesize1 logical Paulis, then logical Cliffords, and finally these theorems. These results form part of a larger program some logical non-Clifford in the third level of the Clifford for fault-tolerant quantum computation, where the goal is hierarchy. In this paper, we will be primarily concerned to achieve reliability by using classical computers to track with logical Cliffords because specific QECCs, such as tri- and control physical quantum systems, and perform error orthogonal codes [10], can be used to distill magic states [11] correction only as needed. for a non-Clifford gate in C(3), and these states can then We note that there are several works that focus on exactly be “injected” into the computation via teleportation in order decomposing, or approximating, an arbitrary unitary operator to realize the action of that gate at the logical level [9]. as a sequence of operators from a fixed instruction set, such Hence, any circuit implemented on the computer equipped as Clifford + T [18]–[23]. However, these works do not with error-correction might be expected to consist only of consider the problem of circuit synthesis or optimization over Clifford gates, augmented with ancilla magic states, and Pauli different realizations of unitary operators on the encoded measurements. space. We also note that there exists several works in the For the task of synthesizing the logical Pauli operators for literature that study this problem for specific codes and oper- stabilizer codes, the first algorithm was introduced by Gottes- ations, e.g., see [6]–[8], [24]–[27]. However, we believe our man [6, Sec. 4] and subsequently, another algorithm based work is the first to propose a systematic framework to address on a symplectic Gram-Schmidt procedure was proposed by this problem for general stabilizer codes, and hence enable Wilde [12]. The latter is closely related to earlier work by automated circuit synthesis for encoded Clifford operators. Brun et al. [13], [14]. Since the logical Paulis are inputs to This procedure is more systematic in considering all degrees our algorithm that synthesizes logical Clifford operators for of freedom than conjugating the desired logical operator by stabilizer codes, we will consider the above two procedures the encoding circuit for the QECC. to be “preprocessors” for our algorithm. Recently, we have used the LCS algorithm to translate Given the logical Pauli operators for an [[m; k]] stabilizer the unitary 2-design we constructed from classical Kerdock QECC, that encodes k logical qubits into m physical qubits, codes into a logical unitary 2-design [28], and in general physical Clifford realizations of Clifford operators on the log- any design consisting of only Clifford elements can be trans- ical qubits can be represented by 2m×2m binary symplectic formed into a logical design using our algorithm. An imple- matrices, thereby reducing the complexity dramatically from mentation of the design is available at: https://github.com/ 22m complex variables to 4m2 binary variables (see [15], nrenga/symplectic-arxiv18a. This finds direct application in [16] and Section II). We exploit this fact to propose an the logical randomized benchmarking protocol proposed by algorithm that efficiently assembles all 2r(r+1)=2, where r = Combes et al. [29]. This protocol is a more robust pro- m − k, symplectic matrices representing physical Clifford cedure to estimate logical gate fidelities than extrapolating operators (circuits) that realize a given logical Clifford oper- results from randomized benchmarking performed on phys- ator on the protected qubits. We will refer to this procedure ical gates [30]. Now we discuss some more motivations and as the Logical Clifford Synthesis (LCS) algorithm. Here, potential applications for the LCS algorithm. each symplectic solution represents an equivalence class of Clifford circuits, all of which “propagate” input Pauli opera- A. NOISE VARIATION IN QUANTUM SYSTEMS tors through them in an identical fashion (see Section III). Although depth or the number of two-qubit gates might Moreover, as we will discuss later in the context of the appear to be natural metrics for optimization, near-term quan- algorithm, the other degrees of freedom not captured by our tum computers can also benefit from more nuanced metrics algorithm are those provided by stabilizers (see Remark 12). depending upon the physical system. For example, it is now But, at the cost of some increased computational complexity, established that the noise in
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