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This paper proves that a broad class of topological quantum error-correcting codes can tolerate a fixed amount of coherent rotation errors. Coherent errors are systematic misrotations of qubits that keep a phase relation and are not well described by random error models. The authors show that, under a natural error model of small single-qubit Z rotations and with ideal syndrome measurements, a standard decoding rule called maximum-likelihood Pauli recovery reduces the logical error (measured by entanglement infidelity) exponentially as the code distance grows, provided the rotation angles stay below a fixed constant that does not shrink with code size.
The result applies to Calderbank–Shor–Steane (CSS) quantum low-density parity-check (QLDPC) codes with a bounded number of logical qubits. This family includes the surface code and many other topological stabilizer codes used in proposals for fault-tolerant quantum computing. The paper proves that the entanglement infidelity after recovery is suppressed exponentially in the code distance, up to a prefactor that is linear in the number of physical qubits. If the code distance grows faster than a logarithm of the number of physical qubits, the bound implies the infidelity goes to zero as the code gets larger. The fixed constant on the rotation angle gives a rigorous lower bound on the coherent-error threshold in this idealized setting.
Proving this required keeping track of quantum interference among many error patterns. A single-qubit rotation expands into many possible Pauli error patterns with complex amplitudes. Unlike random (stochastic) errors, these amplitudes can add or cancel because of their phases, so simply replacing amplitudes by their absolute values gives a pessimistic bound. The authors use Fourier analysis to keep the interference information and then map the resulting expressions to a polymer model from statistical mechanics. In this map, a polymer is a connected set of qubits that produces a trivial syndrome. They then use a cluster expansion technique to control the partition functions that appear and to show convergence when rotation angles are small enough.
Why this matters: earlier threshold proofs focused on stochastic error models and numerical work hinted that coherent errors are also suppressed, especially for the surface code. This paper closes a gap by providing a rigorous, analytic proof that coherent Z-rotation errors admit a nonzero threshold for a wide class of topological codes in the code-capacity setting. The proof also gives insight into how interference is preserved and exploited by error correction, and it borrows tools (Fourier methods and cluster expansions) used earlier in classical coding and statistical mechanics.
Important caveats: the proof works in the code-capacity setting, meaning the syndrome measurements are assumed to be noiseless and the errors act only on data qubits. It also assumes the use of maximum-likelihood Pauli recovery and that the number of logical qubits is bounded in the code family. The paper does not prove a threshold for the more realistic circuit-level setting where syndrome extraction itself can be faulty or where measurements introduce coherent errors. Also, while the rotation-angle threshold is independent of code size, the argument gives a constant bound rather than a precise numerical threshold value, and the prefactor linear in the number of physical qubits means one needs code distance to grow sufficiently quickly (superlogarithmically) to make the logical error vanish in large codes.
Overall, the work gives the first rigorous demonstration that coherent single-qubit Z rotations can be suppressed by topological QLDPC codes under ideal syndrome measurement. It is a step toward extending threshold theorems to more realistic coherent noise and noisy measurements, and it frames coherent-error suppression in a statistical-mechanical language that may guide future analysis.