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.