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Physical Review A

Quantum annealing approach to the minimum distance problem of quantum codes

Anonymous, Ashish Kakkar, Anatoly Y. Dymarsky

Published Sep 4, 2026• 0 citations
Publisher
DOI
10.1103/573l-6cc2

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A quantum annealing approach to the minimum distance problem of quantum codes

Ismail, Refaat, Ashish Kakkar, Anatoly Y. Dymarsky

Quantum error-correcting codes (QECCs) are at the heart of fault-tolerant quantum computing. As the size of quantum platforms is expected to grow, one of the open questions is to design new optimal codes of ever-increasing size. A related challenge is to certify the quality of a given code by evaluating its minimum distance, a quantity characterizing code's capacity to preserve quantum information. This problem is known to be NP-hard. Here we propose to harness the power of contemporary quantum platforms to address this question, and in this way help design quantum platforms of the future. Namely, we introduce an approach to compute the minimum distance of quantum stabilizer codes by reformulating the problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem and leveraging established QUBO algorithms and heuristics as well as quantum annealing (QA) to address the latter. The reformulation as a QUBO introduces only a logarithmic multiplicative overhead in the required number of variables. We demonstrate the formulation using the D-Wave Advantage 4.1 quantum annealer and the hybrid quantum-classical algorithm Qbsolv, and benchmark the resulting QUBO-based implementations against specialized classical distance-finding methods. The QA- and SA-assisted implementations show broadly comparable performance, while the classical comparison clarifies the relative strengths of heuristic optimization and exact distance certification. Our results establish a solver-independent QUBO formulation with only a logarithmic multiplicative overhead in the number of variables and characterize connectivity and minor embedding as central limitations of its present implementation on quantum-annealing hardware.

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In the forthcoming 6G era, driven by the escalating demands for enhanced performance and reliability in communication systems, Low-Density Parity-Check (LDPC) codes emerge as a robust error correction coding scheme poised to effectively meet these requirements. In recent years, with the advancement of quantum computing, a proposal has emerged to integrate classical and quantum computing for the decoding of LDPC. The LDPC decoding can be treated as a Quadratic Unconstrained Binary Optimization (QUBO) problem, thereby allowing the utilization of the D-Wave 2000Q quantum annealer for solving it. Expanding on this groundwork, the paper proposes an approach to enhance quantum annealing performance using classical algorithms. Specifically, it involves performing quantum annealing on the problem first, followed by post-processed using Maximum Likelihood Estimation (MLE) for the multiple outputs, known as QAML. Furthermore, to fully exploit the advantages of quantum annealing, this paper incorporates the energy and frequency of occurrence of candidate codewords as prior information into the Maximum Likelihood Estimation. These are called QAMLE and QAMLF respectively, further enhancing decoding performance. The proposed method is evaluated and compared with direct Quantum Annealing (QA) and Quantum Annealing with post-processing using Minimum Distance (QAMD). Research findings indicate that QAML exhibits a lower bit error rate compared to QA and QAMD.

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Optimal design, decoding, and minimum distance analysis of Goppa codes using heuristic method

Bouchaib Aylaj, Saïd Nouh, Mostafa Belkasmi

Error-correcting codes are crucial to ensure data reliability in communication systems often affected by transmission noise. Building on previous successful applications of our heuristic method degenerate quantum simulated annealing (DQSA) to Bose–Chaudhuri–Hocquenghem (BCH) and quadratic residue (QR) codes. This paper proposes two algorithms designed to address two coding problems for Goppa codes. DQSA-dmin computes the minimum distance (dmin) while DQSA-Dec, serves as a hard decoder optimized for additive white gaussian noise (AWGN) channels. We validate DQSA-dmin comparing its computed minimum distances with theoretical estimates for algebraically constructed Goppa codes, showing accuracy and efficiency. DQSA-dmin further used to find the optimal Goppa codes that reach the lower bound of dmin for linear codes known in the literature and stored in Marcus Grassl's online database. Indeed, we discovered 12 Goppa codes reaching this lower bound. For DQSA-Dec, experimental results show that it obtains a bit error rate (BER) of 10-5 when SNR=7.5 for codes with lengths less than 65, which is very interesting for a hard decoder. Additionally, a comparison with the Paterson algebraic decoder specific to this code family shows that DQSA-Dec outperforms it with a 0.6 dB coding gain at BER=10-4. These findings highlight the effectiveness of DQSA-based algorithms in designing and decoding Goppa codes.

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Akseli Partanen

•Sep 15, 2017•0 citations
Quantum error-correcting codes (QECCs) are at the heart of fault-tolerant quantum computing. As the size of quantum platforms is expected to grow, one of the open questions is to design new optimal codes of ever-increasing size. A related challenge is to certify the quality of a given code by evaluating its minimum distance, a quantity characterizing code's capacity to preserve quantum information. This problem is known to be NP-hard. Here we propose to harness the power of contemporary quantum platforms to address this question, and in this way help design quantum platforms of the future. Namely, we introduce an approach to compute the minimum distance of quantum stabilizer codes by reformulating the problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem and leveraging established QUBO algorithms and heuristics as well as quantum annealing (QA) to address the latter. The reformulation as a QUBO introduces only a logarithmic multiplicative overhead in the required number of variables. We demonstrate the formulation using the D-Wave Advantage 4.1 quantum annealer and the hybrid quantum-classical algorithm Qbsolv, and benchmark the resulting QUBO-based implementations against specialized classical distance-finding methods. The QA- and SA-assisted implementations show broadly comparable performance, while the classical comparison clarifies the relative strengths of heuristic optimization and exact distance certification. Our results establish a solver-independent QUBO formulation with only a logarithmic multiplicative overhead in the number of variables and characterize connectivity and minor embedding as central limitations of its present implementation on quantum-annealing hardware.
In the forthcoming 6G era, driven by the escalating demands for enhanced performance and reliability in communication systems, Low-Density Parity-Check (LDPC) codes emerge as a robust error correction coding scheme poised to effectively meet these requirements. In recent years, with the advancement of quantum computing, a proposal has emerged to integrate classical and quantum computing for the decoding of LDPC. The LDPC decoding can be treated as a Quadratic Unconstrained Binary Optimization (QUBO) problem, thereby allowing the utilization of the D-Wave 2000Q quantum annealer for solving it. Expanding on this groundwork, the paper proposes an approach to enhance quantum annealing performance using classical algorithms. Specifically, it involves performing quantum annealing on the problem first, followed by post-processed using Maximum Likelihood Estimation (MLE) for the multiple outputs, known as QAML. Furthermore, to fully exploit the advantages of quantum annealing, this paper incorporates the energy and frequency of occurrence of candidate codewords as prior information into the Maximum Likelihood Estimation. These are called QAMLE and QAMLF respectively, further enhancing decoding performance. The proposed method is evaluated and compared with direct Quantum Annealing (QA) and Quantum Annealing with post-processing using Minimum Distance (QAMD). Research findings indicate that QAML exhibits a lower bit error rate compared to QA and QAMD.
Error-correcting codes are crucial to ensure data reliability in communication systems often affected by transmission noise. Building on previous successful applications of our heuristic method degenerate quantum simulated annealing (DQSA) to Bose–Chaudhuri–Hocquenghem (BCH) and quadratic residue (QR) codes. This paper proposes two algorithms designed to address two coding problems for Goppa codes. DQSA-dmin computes the minimum distance (dmin) while DQSA-Dec, serves as a hard decoder optimized for additive white gaussian noise (AWGN) channels. We validate DQSA-dmin comparing its computed minimum distances with theoretical estimates for algebraically constructed Goppa codes, showing accuracy and efficiency. DQSA-dmin further used to find the optimal Goppa codes that reach the lower bound of dmin for linear codes known in the literature and stored in Marcus Grassl's online database. Indeed, we discovered 12 Goppa codes reaching this lower bound. For DQSA-Dec, experimental results show that it obtains a bit error rate (BER) of 10-5 when SNR=7.5 for codes with lengths less than 65, which is very interesting for a hard decoder. Additionally, a comparison with the Paterson algebraic decoder specific to this code family shows that DQSA-Dec outperforms it with a 0.6 dB coding gain at BER=10-4. These findings highlight the effectiveness of DQSA-based algorithms in designing and decoding Goppa codes.