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Register a new userSynthesis of Logical Clifford Operators via Symplectic Geometry
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Added by
Narayanan Rengaswamy
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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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 $mathbb{C}^{N times N}$ 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 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. Our assembly of the possible physical realizations enables optimization over them 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 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 our algorithms can be found at https://github.com/nrenga/symplectic-arxiv18a
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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
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