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Eurasian Journal of Mathematical and Computer Applications

DECODING M-ARY LINEAR CODES WITH THE QUANTUM APPROXIMATE OPTIMIZATION ALGORITHM

Markel Epelde, Elías F. Combarro, I. F. R ́ua

Published Mar 1, 2022• 0 citations
Publisher

Abstract

Abstract The NP-hardness of the Minimum Distance Decoding Problem (MDDP) is the core of the McEliece cryptosystem. The difficulty of decoding a received word to the closest codeword in a given arbitrary code is key to its security. Related to the MDDP is the Coset Leader Problem (CLP), which consists in finding a word of a given syndrome and minimum Hamming weight. Both can be modelled as optimization problems, and solved using the Quantum Approximate Optimization Algorithm (QAOA), a well-known hybrid quantum- classical algorithm. In this paper, we model both the MDDP and CLP for linear codes over arbitrary m−ary alphabets, we make the theoretical analysis of the first level for the binary CLP problem, and introduce some experiments to test its performance. The experiments were carried out on both quantum computer simulators and real quantum devices, and use codes of different lengths and different depths of the QAOA.

DOI
10.32523/2306-6172-2022-10-1-4-25

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Abstract The NP-hardness of the Minimum Distance Decoding Problem (MDDP) is the core of the McEliece cryptosystem. The difficulty of decoding a received word to the closest codeword in a given arbitrary code is key to its security. Related to the MDDP is the Coset Leader Problem (CLP), which consists in finding a word of a given syndrome and minimum Hamming weight. Both can be modelled as optimization problems, and solved using the Quantum Approximate Optimization Algorithm (QAOA), a well-known hybrid quantum- classical algorithm. In this paper, we model both the MDDP and CLP for linear codes over arbitrary m−ary alphabets, we make the theoretical analysis of the first level for the binary CLP problem, and introduce some experiments to test its performance. The experiments were carried out on both quantum computer simulators and real quantum devices, and use codes of different lengths and different depths of the QAOA.
The security of a broad family of coding-based cryptographic techniques relies on the hardness of the Syndrome Decoding Problem (SDP). In this problem, the aim is to find a word with a given syndrome and of Hamming weight smaller than a prefixed bound. If this last condition is replaced by “of minimum weight,” then we have the Coset Leader Problem (CLP), being Finding Low Weight Codewords (FLWC) a particular case (when the zero syndrome is considered). An algorithm that has been proposed in order to obtain approximate solutions of problems of these kind (NP-complete) is the Quantum Approximate Optimization Algorithm (QAOA), a variational hybrid quantum-classical algorithm. In this paper, we apply the QAOA to the CLP for binary linear codes. We model the problem, make the theoretical analysis the case of the first level, and introduce some experiments to test its performance. The experiments have been carried out on quantum computer simulators with codes of different lengths and QAOA of different depth.
Abstract The syndrome decoding problem is known to be NP-complete. The goal of the decoder is to find an error of low weight that corresponds to a given syndrome obtained from a parity-check matrix. We use the quantum approximate optimization algorithm (QAOA) to address the syndrome decoding problem with elegantly-designed reward Hamiltonians based on both generator and check matrices for classical and quantum codes.We evaluate the level-4 check-based QAOA decoding of the [7,4,3] Hamming code, as well as the level-4 generator-based QAOA decoding of the [[5,1,3]] quantum code.Remarkably, the simulation results demonstrate that the decoding performances match those of the maximum likelihood decoding. Moreover, we explore the possibility of enhancing QAOA by introducing additional redundant clauses to a combinatorial optimization problem while keeping the number of qubits unchanged. Finally, we study QAOA decoding of degenerate quantum codes.Typically, conventional decoders aim to find a unique error of minimum weight that matches a given syndrome. However, our observations reveal that QAOA has the intriguing ability to identify degenerate errors of comparable weight, providing multiple potential solutions that match the given syndrome with comparable probabilities.This is illustrated through simulations of the generator-based QAOA decoding of the [[9,1,3]] Shor code on specific error syndromes.
Motivated by the recent advancement of quantum processors, we investigate quantum approximate optimization algorithm (QAOA) to employ quasi-maximum-likelihood (ML) decoding of classical channel codes. QAOA is a hybrid quantum-classical variational algorithm, which is advantageous for the near-term noisy intermediate-scale quantum (NISQ) devices, where the fidelity of quantum gates is limited by noise and de-coherence. We first describe how to construct Ising Hamiltonian model to realize quasi-ML decoding with QAOA. For level-1 QAOA, we derive the systematic way to generate theoretical expressions of cost expectation for arbitrary binary linear codes. Focusing on [7], [4] Hamming code as an example, we analyze the impact of the degree distribution in associated generator matrix on the quantum decoding performance. The excellent performance of higher-level QAOA decoding is verified when Pauli rotation angles are optimized through meta-heuristic variational quantum eigensolver (VQE). Furthermore, we demonstrate the QAOA decoding performance in a real quantum device.
This thesis lies within the framework of post-quantum cryptography, whose goal is to conceive cryptosystems that resist classical and quantum attacks. Our focus here is on the Decoding Problem (DP) and the Short Codeword Problem (SCP) in code-based cryptography. We study here how quantum algorithms can exploit the structure of these problems. In a first part, we generalized the information-theoretic achievability threshold for the Quantum Decoding Problem (QDP) to all memoryless noise models. By combining this result with Regev's reduction, we obtain a quantum algorithm that samples dual codewords whose weight approaches the minimum at the achievability threshold. In a second part, we formulate a general framework for adapting to coding theory the lattice reduction from Learning With Errors (LWE) to the Extrapolated Dihedral Coset Problem (EDCP). We study this framework in the binary Hamming case and show that the resulting parameter regime can already be decoded classically in polynomial time.
Decoding Low-Density Parity-Check (LDPC) codes is a fundamental problem in coding theory, and Belief Propagation (BP) is one of the most popular methods for LDPC code decoding. However, BP may encounter convergence issues and suboptimal performance, especially for short-length codes and in high-noise channels. The Quantum Approximate Optimization Algorithm (QAOA) is a type of Variational Quantum Algorithm (VQA) designed to solve combinatorial optimization problems by minimizing a problem-specific cost function. In this paper, we present a QAOA-based decoding framework for LDPC codes by formulating a decoding cost function that incorporates both parity-check constraints and soft channel reliability information. The resulting optimization problem is solved using QAOA to search for low-energy configurations corresponding to valid codewords. We test the proposed method through extensive numerical experiments and compare its performance with BP decoding. The experimental results demonstrate that the QAOA-based decoder achieves a higher probability of correctly recovering the transmitted codeword than BP across multiple experimental settings.
We consider a version of the nearest-codeword problem on finite fields Fq\mathbb{F}_qFq​ using the Manhattan distance, an analog of the Hamming metric for non-binary alphabets. Similarly to other lattice related problems, this problem is NP-hard even up to constant factor approximation. We show, however, that for q=pmq = p^mq=pm where ppp is small relative to the code block-size nnn, there is a quantum algorithm that solves the problem in time poly(n){\rm poly}(n)poly(n), for approximation factor 1/n21/n^21/n2, for any ppp. On the other hand, to the best of our knowledge, classical algorithms can efficiently solve the problem only for much smaller inverse polynomial factors. Hence, the decoder provides an exponential improvement over classical algorithms, and places limitations on the cryptographic security of large-alphabet extensions of code-based cryptosystems like Classic McEliece.