Decoded Quantum Interferometry with Polar Codes: Capacity-Approaching Decoders Meet a Minimum-Distance Ceiling

Abstract Decoded Quantum Interferometry (DQI) [Jordan et al., Nature 646, 831 (2025)] turns efficient decoders for a code C⊥ into samplers for high-quality solutions of a dual optimization problem, with performance governed by the largest error weight ℓ at which uniformly random errors are decodable from their syndrome. We give the first instantiation and evaluation of DQI with polar codes, whose successive-cancellation (SC) decoders are capacity-achieving on average-case errors despite poor minimum distance. This is precisely the regime in which DQI’s average-case requirement diverges most sharply from worst-case decoding guarantees. We characterize the dual instance family, a dense hierarchical max-XORSAT whose variable degrees are powers of two, and evaluate DQI’s guarantee via the semicircle law with empirically measured, layer-resolved decoder failure rates weighted by the DQI tridiagonal eigenvector. Three findings emerge. (i) At practical block lengths (m ≤ 2048) the achievable weight fraction plateaus at ℓ∗/m ≈ 0.049 at rate 1/2, far below the ceiling h−1(1/2) ≈ 0.110, reflecting slow polarization. (ii) At the operating point, all observed list-decoding (SCL-8) failures are maximum-likelihood ambiguities, that is, consistent solutions of equal or lesser weight induced by minimum-weight dual codewords (dmin=8 at m=1024, rate 1/2). The binding constraint on ℓ is therefore information-theoretic and cannot be lifted by any decoder. (iii) The resulting net guarantee, ⟨s⟩/m ∈ [0.691, 0.708] after fidelity accounting, is matched or exceeded by simulated annealing at modest compute (best 0.724, mean 0.706, at a 10°ø budget of ∼100 s; mean best 0.698 across five instances). Polar codes at practical block lengths are thus a structurally explained negative example for DQI: capacity-achieving average-case decoding is necessary but not sufficient, because the dual code’s distance spectrum independently caps the usable error weight. This motivates DQI with distance-enhanced (CRC-aided or pre-transformed) polar variants, whose dual optimization problems we identify.