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IEEE Information Theory Workshop (ITW)

Entanglement assisted binary quantum tensor product codes

Priya J. Nadkarni, Shayan Srinivasa Garani

Published Nov 1, 2017• 8 citations
Publisher

Abstract

We propose the construction and error correction procedures of an entanglement assisted binary quantum tensor product code. We devise an efficient procedure to construct the code with complexity O(max{ρ21, ρ22}) compared to O(ρ21ρ22) using symplectic Gram-Schmidt orthogonalization, where ρ1and ρ2are the number of parity bits of the component codes. Our error correction procedures can correct quantum burst errors without destroying the quantum state.

DOI
10.1109/itw.2017.8277961

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We propose the construction and error correction procedures of an entanglement assisted binary quantum tensor product code. We devise an efficient procedure to construct the code with complexity O(max{ρ21, ρ22}) compared to O(ρ21ρ22) using symplectic Gram-Schmidt orthogonalization, where ρ1and ρ2are the number of parity bits of the component codes. Our error correction procedures can correct quantum burst errors without destroying the quantum state.
Entanglement-assisted quantum error correcting codes have the nice feature that they can be constructed from classical additive codes that do not need to be self-orthogonal - an advantage over stabilizer codes. In the literature, these codes were investigated for finite fields, mostly binary fields. A generalization of the Pauli basis to nice error bases indexed by rings allows one to consider alphabet sizes that are not restricted to powers of a prime. The main goal of this paper is to show how entanglement-assisted quantum error correcting codes over nice rings can be constructed. We develop the rudiments of symplectic geometry over rings and prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs.
We present a general framework for the construction of quantum tensor product codes (QTPC). In a classical tensor product code (TPC), its parity check matrix is constructed via the tensor product of parity check matrices of the two component codes. We show that by adding some constraints on the component codes, several classes of dual-containing TPCs can be obtained. By selecting different types of component codes, the proposed method enables the construction of a large family of QTPCs and they can provide a wide variety of quantum error control abilities. In particular, if one of the component codes is selected as a burst-error-correction code, then QTPCs have quantum multiple-burst-error-correction abilities, provided these bursts fall in distinct subblocks. Compared with concatenated quantum codes (CQC), the component code selections of QTPCs are much more flexible than those of CQCs since only one of the component codes of QTPCs needs to satisfy the dual-containing restriction. We show that it is possible to construct QTPCs with parameters better than other classes of quantum error-correction codes (QECC), e.g., CQCs and quantum BCH codes. Many QTPCs are obtained with parameters better than previously known quantum codes available in the literature. Several classes of QTPCs that can correct multiple quantum bursts of errors are constructed based on reversible cyclic codes and maximum-distance-separable (MDS) codes.
This article develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error-correction performance, high rates, and low decoding complexity. The proposed method produces several infinite families of codes with a wide variety of parameters and entanglement requirements. Our framework encompasses the previously known entanglement-assisted quantum LDPC codes having the best error-correction performance and many other codes with better block error rates in simulations over the depolarizing channel. We also determine important parameters of several well-known classes of quantum and classical LDPC codes for previously unsettled cases.