Fault tolerance with budgeted correlated noise

Threshold theorems cover local stochastic noise and several non-Markovian models with quantitative locality or norm bounds. Unrestricted adversarial noise is not correctable. The open programme is to identify which bounded correlations remain tolerable while keeping geometry, decoding and total overhead implementable.

August 2026 preprint: Breuckmann–Golowich–Vazirani allow a globally correlated adversary to corrupt up to $N^{1-o(1)}$ physical qudits per time step. Their construction uses polynomial space, subpolynomial multiplicative depth overhead and, after recursive composition, constant local dimension. Reducing an unbounded alphabet is therefore not the remaining generic problem; small-qubit architectures, quantitative constants and space overhead are.

Bharti–Haug–Tanggara's preprint gives a logarithmic reliability contribution to memory space-time cost: fixed small logical registers and sufficiently wide amortized computations have different relative-overhead conclusions. Gong–Hu's self-calibration theorem applies to their structured control-error and local-convexity setting, not all correlated noise.

Decoding milestones also need model labels. Bazzi–Khater prove additive inapproximability for minimum-weight Pauli decoding of surface/toric codes under depolarizing noise, and separate X/Z decoding for their colour-code setting; this does not make the usual independent-error surface-code matching problem NP-hard. Krishnamoorthy et al. express CSS degenerate maximum-likelihood decoding through partition functions and give statistical certificates, or exact certificates when suitable estimators are available. Neither supplies an unconditional efficient optimal decoder for every CSS code. Related: A6, A7, O5.

Source disclosures

Breuckmann et al. report that proofs and writing are their own, with ChatGPT used for final-draft typographical and presentation checks. Source

Bharti et al. disclose generative-AI assistance with ideation, editing, organization and preparation of parts of the manuscript. Source

Gong–Hu acknowledge ChatGPT/Claude for proof-idea discussions and presentation, retaining responsibility for the proofs and results. Source

Bazzi–Khater report reviewed ChatGPT assistance with editing, notation and literature search. Source

Krishnamoorthy et al. disclose large-language-model use for text optimization. Source

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

preprint: Adversarial fault tolerance with per-step corruption budget, polynomial space and constant alphabet after recursion. Evidence

preprint: Memory reliability lower bound; fixed-width overhead and wide-register amortization must be distinguished. Evidence

preprint: Efficient syndrome-based calibration in the stated control-error model and local-convexity regime. Evidence

preprint: Minimum-weight decoding inapproximability for specific code/noise models; not a hardness result for independent-X/Z surface-code matching. Evidence

preprint: Statistical decoding certification and estimator-dependent exact certificates; no universal fast optimal-decoding theorem. Evidence