Quantum error-correcting codes can be used to protect qubits involved in quantum computation. This requires that logical operators acting on protected qubits be translated to physical operators (circuits) 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 C N×N as a partial 2m × 2m binary symplectic matrix, where N = 2 m . We state and prove two theorems that use symplectic transvections to efficiently enumerate all binary symplectic matrices that satisfy a system of linear equations. As an important corollary of these results, we prove that for an [[m, m − k]] stabilizer code every logical Clifford operator has 2 k(k+1)/2 symplectic solutions. The desired physical circuits are then obtained by decomposing each solution as a product of elementary symplectic matrices, each corresponding to an elementary circuit. Our assembly of the possible physical realizations enables optimization over the ensemble with respect to a suitable metric. Furthermore, we show that any circuit that normalizes the stabilizer of the code can be transformed into a circuit that centralizes the stabilizer, while realizing the same logical operation. However, the optimal circuit for a given metric may not correspond to a centralizing solution. Our method of circuit synthesis can be applied to any stabilizer code, and this paper provides a proof of concept synthesis of universal Clifford gates for the [[6, 4, 2]] CSS code. We conclude with a classical coding-theoretic perspective for constructing logical Pauli operators for CSS codes. Since our circuit synthesis algorithm builds on the logical Pauli operators for the code, this paper provides a complete framework for constructing all logical Clifford operators for CSS codes. Programs implementing the algorithms in this paper, which includes routines to solve for binary symplectic solutions of general linear systems and our overall circuit synthesis algorithm, can be found at https://github.com/nrenga/symplectic-arxiv18a.

The Clifford hierarchy of unitary operators is a foundational concept for universal quantum computation. It was introduced to show that universal quantum computation can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. (Phys. Rev. A 95, 012329) recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal unitaries whose action on a computational basis qudit state is described by a 2 k -th root of unity raised to some polynomial function of the state, and they established the level of such unitaries in the hierarchy as a function of k and the degree of the polynomial. For qubit systems, we consider k-th level diagonal unitaries that can be described just by quadratic forms of the state over the ring Z 2 k of integers modulo 2 k . The quadratic forms involve symmetric matrices over Z 2 k that can be used to efficiently describe all two-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal unitaries from lower levels. The result involves symplectic matrices over Z 2 k and hence our perspective unifies a subgroup of diagonal gates in the Clifford hierarchy with the binary symplectic framework for gates in the Clifford group. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to expensive magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+T set of gates, perhaps leading to shorter depth circuits. Our results suggest that some non-diagonal gates in the hierarchy might also be understood by generalizing other binary symplectic matrices to integer rings. *

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For space-based laser communications, when the mean photon number per received optical pulse is much smaller than one, there is a large gap between communications capacity achievable with a receiver that performs individual pulse-by-pulse detection, and the quantum-optimal “joint-detection receiver” that acts collectively on long codeword-blocks of modulated pulses; an effect often termed “superadditive capacity”. In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a qubit, described by an inner product that is a function of the mean photon number per pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of “belief-propagation with quantum messages” (BPQM). We quantify its performance improvement over the Dolinar receiver that performs optimal pulse-by-pulse detection, which represents the best “classical” approach. We analyze the scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating 8 (BPSK) codewords of a length-5 binary linear code with a tree factor graph. Our result suggests that a BPQM receiver might attain the Holevo capacity of this BPSK-modulated pure-loss channel. Moreover, our receiver circuit provides an alternative proposal for a quantum supremacy experiment, targeted at a specific application that can potentially be implemented on a small, special-purpose, photonic quantum computer capable of performing cat-basis universal qubit logic.

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. In this article, 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 C N×N as a 2m × 2m binary symplectic matrix, where N = 2 m . 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 2 r(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 article, 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.