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Reinterpreting the Stabilizer Formalism

Vatsal Pramod Jha, Udaya Parampalli, Abhay Kumar Singh

Published Sep 1, 2021• 0 citations
Publisher

Abstract

Stabilizer codes, introduced in [2], [3], have been a prominent example of quantum codes constructed via classical codes. The paper [3], introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we reinterpret the stabilizer formalism by considering binary codes over the symbol-pair metric (see [9]). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters.

DOI
10.36227/techrxiv.16552878.v1

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Stabilizer codes, introduced in [2], [3], have been a prominent example of quantum codes constructed via classical codes. The paper [3], introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we reinterpret the stabilizer formalism by considering binary codes over the symbol-pair metric (see [9]). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters.
Stabilizer codes, introduced in \cite{gottesman}, \cite{GF(4)}, have been a prominent example of quantum codes constructed via classical codes. The paper \cite{GF(4)}, introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we present a few stabilizer code constructions considering binary codes over the symbol-pair metric (see \cite{sp}). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters [[15,[[15,[[15, 4,4,4, 3≤d≤7]]3\leq d\leq7]]3≤d≤7]], [[31,[[31,[[31, 10,10,10, 5≤d≤9]]5\leq d\leq9]]5≤d≤9]], [[62,[[62,[[62, 20,20,20, 5≤d]]5\leq d]]5≤d]], [[63,[[63,[[63, 18,18,18, 7≤d]]7\leq d]]7≤d]].
Stabilizer codes, introduced in \cite{gottesman}, \cite{GF(4)}, have been a prominent example of quantum codes constructed via classical codes. The paper \cite{GF(4)}, introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we present a few stabilizer code constructions considering binary codes over the symbol-pair metric (see \cite{sp}). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters [[15,[[15,[[15, 4,4,4, 3≤d≤7]]3\leq d\leq7]]3≤d≤7]], [[31,[[31,[[31, 10,10,10, 5≤d≤9]]5\leq d\leq9]]5≤d≤9]], [[62,[[62,[[62, 20,20,20, 5≤d]]5\leq d]]5≤d]], [[63,[[63,[[63, 18,18,18, 7≤d]]7\leq d]]7≤d]].
In order to construct quantum error correction code, we consider additive codes over Galois field GF(4), which are self-orthogonal with respect to a certain Hermitian product. In this paper, we first propose a new approach to the construction of Hermitian self-orthogonal linear codes, applying the extension to get a longer length, and prove the codes have good minimum distance. Then, we investigate the corresponding quantum stabilizer codes from the classical codes. Six optimal quantum stabilizer codes have been achieved to show the effectiveness of the proposed construction.
It is well known that n-length stabilizer quantum error correcting codes (QECCs) can be obtained via n-length classical error correction codes (CECCs) over GF(4), that are additive and self-orthogonal with respect to the trace Hermitian inner product. But, most of the CECCs have been studied with respect to the Euclidean inner product. In this paper, it is shown that n-length stabilizer QECCs can be constructed via 3n-length linear CECCs over GF(2) that are self-orthogonal with respect to the Euclidean inner product. This facilitates usage of the widely studied self-orthogonal CECCs to construct stabilizer QECCs. Moreover, classical, binary, self-orthogonal cyclic codes have been used to obtain stabilizer QECCs with guaranteed quantum error correcting capability. This is facilitated by the fact that (i) self-orthogonal, binary cyclic codes are easily identified using transform approach and (ii) for such codes lower bounds on the minimum Hamming distance are known. Several explicit codes are constructed including two pure MDS QECCs.
In [3], the relation between stabilizer codes and binary codes over the symplectic inner product and symplectic weight was established. In the current work, we present a relation between the symplectic weight and symbol-pair weight and use it to construct stabilizer codes of length n from binary codes of length n defined over the symbol-pair metric and Euclidean inner product. In particular, we use certain length n binary LCD codes over the symbol-pair metric to obtain stabilizer codes of length n. We also present the Modified CSS construction which outperforms the CSS construction in the given setup.
In this research, we propose a novel construction of quantum stabilizer code based on a binary formalism. First, from any binary vector of even length, we generate the parity-check matrix of the quantum code from a set composed of elements from this vector and its relations by shifts via subtraction and addition. We prove that the proposed matrices satisfy the condition constraint for the construction of quantum codes. Finally, we consider some constraint vectors which give us quantum stabilizer codes with various dimensions and a large minimum distance with code length from six to twelve digits.