DQI sampling below the replica-symmetry-breaking transition

Determine whether Decoded Quantum Interferometry can efficiently perform Gibbs sampling at inverse temperatures below the replica-symmetry-breaking transition \(\beta_{RSB}(k,D)\), by using polynomial-size quantum circuits that approach the Holevo information-theoretic decoding limit sufficiently closely.

Background

The paper establishes that DQI+USD can sample in part of the shattered phase, but its performance does not reach the replica-symmetry-breaking transition. The Holevo bound supplies an information-theoretic upper bound on the inverse temperature accessible through reliable quantum decoding, and the authors show that this bound lies above the static transition for the Erdős–Rényi ensemble. The unresolved issue is whether efficient quantum circuits can approach that bound closely enough to cross the replica-symmetry-breaking barrier.

References

Can DQI efficiently perform Gibbs sampling at temperatures below the replica symmetry breaking (RSB) transition? The RSB transition occurs at a colder temperature than the shattering/dynamical transition (\beta_{RSB}\ge \beta_{dyn}), and is thought to be more challenging to reach classically.

— Gibbs Sampling in the Shattered Phase by Decoded Quantum Interferometry  (2609.40345 - Zhou et al., 30 Sep 2026) in Section 2, Main Results, concluding list of open questions

Is there an ensemble of Hamiltonians where DQI efficiently samples at temperatures beyond those reached by all known classical algorithms?

— Gibbs Sampling in the Shattered Phase by Decoded Quantum Interferometry  (2609.40345 - Zhou et al., 30 Sep 2026) in Section 2, Main Results, concluding list of open questions

We leave an ensemble-dependent characterization of \beta_{\mathrm{Holevo}} and a more precise comparison to \beta_{RSB} as future work.

— Gibbs Sampling in the Shattered Phase by Decoded Quantum Interferometry  (2609.40345 - Zhou et al., 30 Sep 2026) in Appendix G, subsection Comparing beta_Holevo with beta_RSB