Single-shot state preparation thresholds in quantum LDPC codes: soundness, confinement, and statistical mechanics

Abstract

Single-shot error correction does not by itself establish equally direct and robust logical-state initialization. We study preparation of CSS quantum low-density parity-check code states using one noisy round of the code's checks. We prove that linear confinement over a weight range growing at least logarithmically, together with at least logarithmic distance, suffices for a nonzero threshold under independent local-stochastic readout and data noise in the relevant CSS sector. The threshold is certified by ideal recovery of a decoder-defined logical channel. The key is a correction rule that preserves the logical class under local syndrome changes, allowing the proof to proceed without a local-stochastic bound on the preparation residual. This gives a single-block guarantee for quantum expander codes without dimension jumps or batching. Under the stronger linear-soundness hypothesis, we establish local-stochastic residual noise, enabling composition with compatible fault-tolerant gadgets. For a retained-record formulation, we also derive an exact statistical-mechanical representation of optimal final recovery at fixed preparation decoder. A decoder-dependent logical-sector order parameter characterizes asymptotically successful block decoding. Monte Carlo results for the three-dimensional toric code provide finite-size evidence for a threshold beyond both linear sufficient conditions. These results distinguish logical robustness from local-stochastic residual-noise control and identify decoder consistency as a route to single-shot initialization.