Logical Clifford Synthesis for Stabilizer Codes

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. 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.

Approach for the construction of non-Calderbank-Steane-Shor low-density-generator-matrix–based quantum codes

“…After this short introduction to the stabilizer formalism, we move towards to the description of stabilizer coding principles from the perspective of group theory. Those familiar with these concepts will realize that this interpretation deviates from conventional representations and that the employed notation differs slightly from the one usually found in the literature [88,75,129,130,131]. This is meant to facilitate the comprehension of certain topics in the field of quantum stabilizer codes that are sometimes misunderstood.…”

Degeneracy and Its Impact on the Decoding of Sparse Quantum Codes

“…Next, we describe the principles of stabilizer codes by making use of the group theoretical framework explained in the previous section. The description provided in this work deviates somewhat from the conventional representation of stabilizer coding concepts and previous work on the structure of the Pauli group [75], [76]. We do this in an attempt to facilitate the comprehension of certain topics in the field of quantum stabilizer codes that are sometimes misunderstood.…”

Approximate Unitary 3-Designs from Transvection Markov Chains

“…The number of 1s in Q and P directly relates to number of gates involved in the circuit realizing the respective unitary operators (see [14,23,Appendix I]). The N coordinates are indexed by binary vectors v ∈ F m 2 , and ev denotes the standard basis vector in C N with an entry 1 in position v and all other entries 0.…”

“…It is well-known that there is no proper subgroup of the Clifford group that can form a unitary 3-design [11]. In this paper, we combine our (Kerdock) 2-design with symplectic transvections [12][13][14] to construct a Markov process that results in an approximate unitary 3-design. Hence, our work demonstrates how one can "smoothly" turn the Kerdock 2-design into a 3-design.…”