SYK thermal expectations are classically easy at any temperature

Abstract

Estimating thermal expectations of local observables is a natural target for quantum advantage. We give a simple classical algorithm that approximates thermal expectations, and we show it has quasi-polynomial cost $n^{O(\log n/ε)}$ for all temperatures above a phase transition in the free energy. For many natural models, this coincides with the entire fast-mixing, quantumly easy phase. Our results apply to the Sachdev-Ye-Kitaev (SYK) model at any constant temperature -- including when the thermal state is highly entangled and satisfies polynomial quantum circuit lower bounds, a sign problem, and nontrivial instance-to-instance fluctuations. Our analysis of the SYK model relies on the replica trick to control the complex zeros of the partition function.

Figures (9)

• Figure 1: Common obstructions to classical algorithms. The SYK model satisfies all of the above properties liu2018quantumanschuetz2025stronglyhastings2023fieldramkumar2025high.
• Figure 2: (a) Cartoon of a phase transition identified by the complex (Fisher) zeros of $Z(\beta)$ asymptotically intersecting the real axis in the thermodynamic limit. (b) Illustration of Barvinok's algorithm to estimate $Z(\beta)$ via an analytic continuation of a Taylor expansion from the origin. Because the high-temperature phase is characterized by Fisher zeros bounded away from the real axis, a zero-free strip of constant height is always available for Barvinok's algorithm to analytically continue along. For constant $\beta$, this yields a quasi-polynomial time to estimate (up to inverse polynomial additive error) $\log Z(\beta)$ and, if the zero-free region is stable upon perturbation by local observables, thermal expectations. For many natural systems (including SYK), the high-temperature phase corresponds to the non-glassy, fast-mixing phase.
• Figure 3: Complex zeros of $Z(\beta)$ obtained from exact diagonalization of a single $n=30, q=4$ SYK instance ($J=1$). Brightness indicates magnitude and hue indicates phase; zeros are indicated in white. The region $|\Re \beta| \gtrsim 0.01$ is zero-free, as well as $|\Im \beta| \lesssim 3.4$, exceeding the zero-free region identified analytically in \ref{['eq:sykregion']}. In Fig. \ref{['fig:sykzeros300']} of \ref{['app:syk']}, we verify that the zero-free region persists up to at least $|\Re \beta|, |\Im \beta| \leq 300$ and when $H$ is perturbed by a local observable.
• Figure 4: Illustrative schematic of the mapping $\phi$ used to avoid zeros of the partition function $Z_H(z)$ when applying Barvinok's interpolation. Red dots are example zeros that are avoided by the mapping.
• Figure 5: Red lines show the contour of $\Re(c_L - \beta \mathcal{J} \cos c_L/2) = 0$; blue lines show the contour of $\Im(c_L - \beta \mathcal{J} \cos c_L/2) = 0$; we set $\mathcal{J} = 1$. A black dot indicates the intersection at the $c_*$ branch identified as the leading saddle in khramtsov2021spectral. At $\beta \approx 1.32i$, the leading saddle has multiplicity 2, causing a complex zero in the partition function (as derived in \ref{['eq:132']}). At $|\Im \beta| \gtrsim 1.32$, the two saddles can cancel each other in the action, leading to additional complex zeros.

Theorems & Definitions (65)

• Remark : Universal high-temperature phase
• Remark : Estimating nontrivial fluctuations
• Lemma 1: QMV from QPF; implicit in Lemma 11 of bravyi2021complexity
• Definition A.1: Closed and open disk
• Lemma 2: Taylor truncation under a zero-free/analytic disk
• proof
• Theorem A.1: Barvinok interpolation under a zero-free disk
• proof
• Remark A.1: Fully polynomial time approximation scheme (FPTAS)
• Lemma 3: Disk to strip; Lemma 2.2.3 of barvinok2016combinatorics