Ising model formulation for highly accurate topological color codes decoding
Open accessAbstract
Quantum error correction is an essential ingredient for reliable quantum computation for theoretically provable quantum speedup. Topological color codes, one of the quantum error correction codes, have an advantage against the surface codes in that all Clifford gates can be implemented transversally. However, the hardness of decoding makes the color codes not suitable as the best candidate for experimentally feasible implementation of quantum error correction. Here we propose an Ising model formulation that enables highly accurate decoding of the color codes. In this formulation, we map stabilizer operators to classical spin variables to represent an error satisfying the syndrome. Then we construct an Ising Hamiltonian that counts the number of errors and formulate the decoding problem as an energy minimization problem of an Ising Hamiltonian, which is solved by simulated annealing. In numerical simulations on the (4.8.8) lattice, we find an error threshold of 10.36(5)% for bit-flip noise model, 18.47(5)% for depolarizing noise model, and 2.90(4)% for phenomenological noise model (bit-flip error is located on each of data and measurement qubits), all of which are higher than the thresholds of existing efficient decoding algorithms. Furthermore, we verify that the achieved logical error rates are almost optimal in the sense that they are almost the same as those obtained by exact optimizations by CPLEX with smaller decoding time in many cases. Since the decoding process has been a bottleneck for performance analysis, the proposed decoding method is useful for further exploration of the possibility of the topological color codes. Published by the American Physical Society 2024
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MLE can be mapped to a quadratic unconstrained binary optimization (QUBO) problem.
“In particular, one first maps MLE to a quadratic unconstrained binary optimization (QUBO) problem [213–215].”
From Quantum error correction and fault tolerance: A comprehensive tutorial (2026) · cites this paper as [215]
Section B, “Decoding and combinatorial optimization,” says the decoding problem is mapped to unconstrained optimization. But Section III.A formulates it as a three-body Ising model, not a quadratic unconstrained binary optimization (QUBO) problem. The cited paper therefore supports the unconstrained-optimization aspect, but not the claimed QUBO mapping. (LLM, conf=0.86)
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- Authors
- Takada, Y., Takeuchi, Y., & Fujii, K.
- Affiliations
- The University of Osaka (JP)
- RIKEN Center for Quantum Computing (JP)
- Venue
- Physical Review Research
- Publisher
- American Physical Society (APS)
- Published
- 2024-01-24
- Volume
- 6
- Issue
- 1
- ISSN
- 2643-1564 (ISSN-L)
- OpenAlex
- W4391167845
- Cited by
- 10
- References
- 44
- Citations by year
- 2026: 5 · 2025: 2 · 2024: 3
- FWCI
- 1.48
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- Physical Review Research (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- arXiv (Cornell University)
- Funders
- Ministry of Education, Culture, Sports, Science and Technology (JPMXS0118067394, JPMXS0120319794)
- Japan Science and Technology Corporation (JPMJPF2014, JPMJMS2061)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Keywords
- quantum error correction, Ising model, decoding, simulated annealing, error threshold, energy minimization, logical error rate, bit-flip noise, depolarizing noise
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