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Decoded Quantum Interferometry
Gibbs Sampling
Topological Barrier
Spin Glasses
Quantum Decoding
Gibbs Sampling in the Shattered Phase by Decoded Quantum Interferometry
🤖 AI Summary
This study addresses the limitation that classical Ising spin glasses in the broken phase are constrained by topological barriers, rendering stable algorithms such as Glauber dynamics ineffective for Gibbs sampling at low temperatures. To overcome this, we propose the Decoded Quantum Interferometry (DQI) technique, which reformulates sampling as a quantum decoding problem. Theoretical analysis is conducted on k-spin glass models defined over random Erdős–Rényi hypergraphs using the Prange method. We provide the first proof that DQI can surmount both “shattering” and “disorder chaos” topological barriers. Specifically, when the average degree satisfies D = αk, the algorithm achieves successful sampling at inverse temperatures β < tanh⁻¹(1/α), far beyond the dynamical threshold. This performance significantly surpasses the limits of classical stable algorithms, substantially expanding the accessible temperature range for efficient sampling.
📝 Abstract
We apply Decoded Quantum Interferometry (DQI) to sample from the Gibbs measures of classical Ising spin Hamiltonians. We show that this Gibbs sampling problem reduces to a quantum decoding problem, and the temperature achievable by DQI is determined by the performance of decoding algorithms. We then focus on the task of Gibbs sampling for classical Ising $k$-spin glasses (or Max-$k$-XORSAT) on random Erdős-Rényi hypergraphs with average degree $D\ge k$. In a temperature range beginning asymptotically at the predicted dynamical phase transition, $β_{\rm dyn}(k,D) = \sqrt{(2\ln k)/D}\times [1+o_{k\to\infty}(1)]$, we show that shattering and disorder chaos form a topological barrier that obstructs many algorithms, including Glauber dynamics and any algorithm whose output distribution is "stable" under perturbations of the input. In contrast, we prove that this barrier can be broken both by a classical algorithm based on Prange's method, and by DQI equipped with a quantum decoder. For example, when $D=αk$ with fixed $α>1$, both Prange's algorithm and DQI can sample at any inverse temperature $β< \tanh^{-1}(1/α)$ for sufficiently large $k$, well beyond the dynamical threshold $β_{\rm dyn} \sim \sqrt{2\ln k / (αk)}$. Therefore, our results show that DQI can overcome topological barriers that obstruct stable algorithms.
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