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IEEE Transactions on Quantum Engineering

Coding Analog of Superadditivity Using Entanglement-Assisted Quantum Tensor Product Codes Over $\mathbb {F}_{p^k}$

Priya J. Nadkarni, Shayan Srinivasa Garani

Published Jan 1, 2020• 10 citations
Publisher

Abstract

We provide a procedure to construct entanglement-assisted Calderbank-Shor-Steane (CSS) codes over qudits from the parity check matrices of two classical codes over Fq, where q = pk, p is prime, and k is a positive integer. The construction procedure involves the proposed Euclidean Gram-Schmidt orthogonalization algorithm, followed by a procedure to extend the quantum operators to obtain stabilizers of the code. Using this construction, we provide a construction of entanglement-assisted tensor product codes from classical tensor product codes over Fq. We further show that a nonzero rate entanglement-assisted tensor product code can be obtained from a classical tensor product code whose component codes yield zero rate entanglement-assisted CSS codes. We view this result as the coding analog of superadditivity.

DOI
10.1109/tqe.2020.3027035

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Coding Analog of Superadditivity Using Entanglement-Assisted Quantum Tensor Product Codes Over Fpk\mathbb {F}_{p^k}Fpk​
We provide a procedure to construct entanglement-assisted Calderbank-Shor-Steane (CSS) codes over qudits from the parity check matrices of two classical codes over Fq, where q = pk, p is prime, and k is a positive integer. The construction procedure involves the proposed Euclidean Gram-Schmidt orthogonalization algorithm, followed by a procedure to extend the quantum operators to obtain stabilizers of the code. Using this construction, we provide a construction of entanglement-assisted tensor product codes from classical tensor product codes over Fq. We further show that a nonzero rate entanglement-assisted tensor product code can be obtained from a classical tensor product code whose component codes yield zero rate entanglement-assisted CSS codes. We view this result as the coding analog of superadditivity.
We present the construction of standard entanglement-assisted (EA) qubit Reed-Muller (RM) codes and their tensor product variants from classical RM codes. We show that the EA RM codes obtained using the CSS construction have zero coding rate and negative catalytic rate. We further show that EA codes constructed from these same classical RM codes using the tensor product code (TPC) construction have positive coding rate and provide a subclass of EA RM TPCs that have positive catalytic rate, thus establishing the coding analog of superadditivity for this family of codes, useful towards quantum communications. We also generalize this analysis to obtain conditions for EA TPCs from classical codes to have positive catalytic rate when their corresponding EA CSS codes have zero rate.
Fp\mathbb {F}_pFp​-Linear and Fpm\mathbb {F}_{p^m}Fpm​-Linear Qudit Codes From Dual-Containing Classical Codes
Quantum code construction from two classical codes D1[n,k1,d1] and D2[n,k2,d2] over the field Fpm(p is prime and m is an integer) satisfying the dual containing criteria D1⊥⊂ D2using the Calderbank-Shor-Steane (CSS) framework is well-studied. We show that the generalization of the CSS framework for qubits to qudits yields two different classes of codes, namely, the Fp-linear CSS codes and the well-known Fpm-linear CSS codes based on the check matrix-based definition and the coset-based definition of CSS codes over qubits. Our contribution to this article are three-folds. 1) We study the properties of the \mathbb Fp-linear and Fpm-linear CSS codes and demonstrate the tradeoff for designing codes with higher rates or better error detection and correction capability, useful for quantum systems. 2) For Fpm-linear CSS codes, we provide the explicit form of the check matrix and show that the minimum distances dxand dzare equal to d2and d1, respectively, if and only if the code is nondegenerate. 3) We propose two classes of quantum codes obtained from the codes D1and D2, where one code is an Fpl-linear code (l divides m) and the other code is obtained from a particular subgroup of the stabilizer group of the Fpm-linear CSS code. Within each class of codes, we demonstrate the tradeoff between higher rates and better error detection and correction capability.
Quantum low-density parity-check (LDPC) codes based on the Calderbank-Shor-Steane construction have low encoding and decoding complexity. The sum-product algorithm (SPA) can be used to decode quantum LDPC codes; however, the decoding performance may be significantly decreased by the many four-cycles required by this type of quantum codes. All four-cycles can be eliminated using the entanglement-assisted formalism with maximally entangled states (ebits). The proposed entanglement-assisted quantum error-correcting code based on Euclidean geometry outperform differently structured quantum codes. However, the large number of ebits required to construct the entanglement-assisted formalism is a substantial obstacle to practical application. In this paper, we propose a novel class of entanglement-assisted quantum LDPC codes constructed using classical Euclidean geometry LDPC codes. Notably, the new codes require one copy of the ebit. Furthermore, we propose a construction scheme for a corresponding zigzag matrix and show that the algebraic structure of the codes could easily be expanded. A large class of quantum codes with various code lengths and code rates can be constructed. Our methods significantly improve the possibility of practical implementation of quantum error-correcting codes. Simulation results show that the entanglement-assisted quantum LDPC codes described in this study perform very well over a depolarizing channel with iterative decoding based on the SPA and that these codes outperform other quantum codes based on Euclidean geometries.
We propose fault-tolerant quantum encoders for Calderbank-Shor-Steane (CSS) and non-CSS codes over qudits by utilizing ggg-partite maximally entangled states. The qudits are grouped into ggg contiguous blocks such that entangled states are shared across the blocks, and multiqudit gates are applied locally to restrict the error propagation within each block. We also propose a transversal fault-tolerant block decoder to decode the blocks individually and recover the complete encoded quantum information. The optimum value of ggg to obtain an improvement in the probability of error propagation of the proposed encoders against non-fault-tolerant schemes is analytically determined for application in the encoding circuit. The proposed fault-tolerant scheme with entanglement assistance scales linearly with the number of ggg-partite entangled states and overcomes the limitations of diminishing coding rates for higher levels of concatenation within the concatenated coding scheme. We validate the proposed ideas by constructing fault-tolerant encoding circuits for the non-CSS [[5,1,3]]3{[[5,1,3]]}_{3}[[5,1,3]]3​ code and the [[7,1,3]]3{[[7,1,3]]}_{3}[[7,1,3]]3​ CSS code through illustrative examples. We reduced the probability of error propagation from 0.248 to 0.212 for the [[5,1,3]]3{[[5,1,3]]}_{3}[[5,1,3]]3​ code and from 0.207 to 0.174 for the [[7,1,3]]3{[[7,1,3]]}_{3}[[7,1,3]]3​ code. For example, over [[100,70]]2{[[100,70]]}_{2}[[100,70]]2​ quantum code without any fault tolerance, using three-partite entangled pairs, one can get two orders of improvement (1e-2) in the probability of error propagation compared to the non-fault-tolerant encoder by encoding over 190 physical qubits with the same number of stabilizers.
Several notions of code products are known in quantum error correction, such as hypergraph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalization of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both X parity checks and Z parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialize to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC 3 -fold product CSS code having parameters [[512,174,8]] are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarizing noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.