Logical Clifford Synthesis for Stabilizer Codes
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 <inline-formula><tex-math notation="LaTeX">$\mathbb {C}^{N \times N}$</tex-math></inline-formula> as a <inline-formula><tex-math notation="LaTeX">$2m \times 2m$</tex-math></inline-formula> binary symplectic matrix, where <inline-formula><tex-math notation="LaTeX">$N = 2^m$</tex-math></inline-formula>. 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 <inline-formula><tex-math notation="LaTeX">$[\![m,k ]\!]$</tex-math></inline-formula> stabilizer code every logical Clifford operator has <inline-formula><tex-math notation="LaTeX">$2^{r(r+1)/2}$</tex-math></inline-formula> symplectic solutions, where <inline-formula><tex-math notation="LaTeX">$r = m-k$</tex-math></inline-formula>, 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 <inline-formula><tex-math notation="LaTeX">$[\![6,4,2 ]\!]$</tex-math></inline-formula> 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 <uri>https://github.com/nrenga/symplectic-arxiv18a</uri>