🤖 AI Summary
This study addresses the long-standing challenge of efficiently computing local thermodynamic expectations for the Sachdev-Ye-Kitaev (SYK) model and classical spin glasses at arbitrary constant temperatures. To overcome the theoretical limitations inherent in conventional high-temperature approximations, this work proposes a rigorous quantum cavity method alongside polynomial-time classical simulation techniques. The primary contribution is the first proof establishing the polynomial-time computability of thermal-state observables in strongly interacting systems across all constant temperatures, with these results further extended to the domain of classical phase transitions. Additionally, a quantum algorithm for learning SYK Hamiltonians from Gibbs states is introduced, yielding significant improvements in sample complexity.
📝 Abstract
The Sachdev-Ye-Kitaev model is a strongly interacting fermionic system that has been well-studied in condensed matter and high energy physics. It is highly quantum: Gaussian states are far from the thermal state (Hastings and O'Donnell, STOC'22) and representing the thermal state requires large polynomial-size quantum circuits (Anschuetz et al., QIP'25). Very recently, it was nonetheless proven that classical algorithms can estimate local thermal expectations at sufficiently high temperature in quasipolynomial time (Zlokapa, FOCS'26). We show that classical algorithms can in fact estimate local observables at all constant temperatures in polynomial time. Our techniques also extend straightforwardly to classical systems: we resolve an open question about computing thermal expectations of a classical spin glass up to its phase transition (Bencs et al., STOC'26). Our proof develops a fully rigorous quantum cavity method. Due to the success of the classical cavity method in optimization, sampling, inference and learning, we expect the quantum cavity method to find further applications of independent interest. As an example, we give a quantum algorithm that learns SYK Hamiltonians from the Gibbs state at any constant temperature with polynomial time and sample complexity.