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

Quantum error-correcting codes (QECCs) is 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 practical viability of our method by comparing the performance of purely classical algorithms with the D-Wave Advantage 4.1 quantum annealer as well as hybrid quantum-classical algorithm Qbsolv. We found that the hybrid approach demonstrates competitive performance, on par with the best available classical algorithms to solve QUBO. In a practical sense, the QUBO-based approach is currently lagging behind the best deterministic minimal distance algorithms, however this advantage may disappear as the size of the platforms grows.