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, acting on protected qubits, to be translated into 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 ${\mathbb{C}}^{N\times N}$ as a partial $2m\times 2m$ 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 a corollary of these results, 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 of the code 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

Main content 12 pages, double column. This is primarily a shortened version of the paper arXiv:1803.06987 based on feedback received, and also has a much more improved introduction. It was presented at ISIT 2018 as well as posters in TQC 2018 and QIP 2019. All comments are welcome!

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• GNarsil: Splitting Stabilizers into Gauges
• Enlarging the GKP stabilizer group for enhanced noise protection
• Fault Tolerant Quantum Simulation via Symplectic Transvections
• Weakly Fault-Tolerant Computation in a Quantum Error-Detecting Code
• Clifford Transformations for Fermionic Quantum Systems: From Paulis to Majoranas to Fermions