Predicting properties of quantum thermal states from a single trajectory

TL;DR

The paper tackles the challenge of efficiently estimating thermal expectation values for quantum Gibbs states by introducing a single-trajectory Gibbs-sampling framework that exploits autocorrelation times shorter than the mixing time. It combines burn-in with interleaved, detailed-balance measurements implemented via Gaussian-filtered quantum phase estimation (GQPE) to preserve the Gibbs ensemble, and shows that for observables commuting with the Hamiltonian, energy estimates can be obtained with logarithmic overhead in the precision. A key result is that detailed-balanced measurements do not degrade the spectral gap, and the autocorrelation time is bounded by the reciprocal of the spectral gap, enabling substantial reductions in sampling cost compared with multi-trajectory schemes. The authors also extend the measurement toolkit to non-commuting observables using the weighted operator Fourier transform (WOFT), which reduces measurement disturbance, and provide a thorough resource analysis comparing HPQPE, local measurements, and GQPE. Overall, the work offers a practical, scalable approach to quantum thermodynamics that could accelerate simulations of molecular and materials properties at finite temperature.

Abstract

Estimating thermal expectation values of observables is a fundamental task in quantum physics, quantum chemistry, and materials science. While recent quantum algorithms have enabled efficient quantum preparation of thermal states, observable estimation via sampling remains costly: a straightforward implementation separates successive measurements by a full mixing time in order to ensure samples are approximately independent. In this work, we show that the sampling cost can be substantially reduced by using a single Gibbs-sampling trajectory. After a single burn-in period, we interleave coherent measurements that satisfy detailed balance with respect to the target Gibbs state. The efficiency of this approach rests on the fact that, in many settings, the autocorrelation time can be significantly shorter than the mixing time. For energy estimation (and more generally for observables commuting with the Hamiltonian), we implement the required measurements using Gaussian-filtered quantum phase estimation with only logarithmic overhead. We also introduce a weighted operator Fourier transform technique to mitigate measurement-induced disturbance for general observables.

Figures (2)

• Figure 1: Two approaches for estimating $\operatorname{tr}(H \rho_{\beta})$ using a quantum Gibbs sampling algorithm (Gibbs sampling channel). (a) Multiple-trajectory approach: Run $N$ independent trajectories, each applying the Gibbs sampling channel for the mixing time $t_{mix}$ followed by a measurement of $H$. Conventionally, $t_{mix}$ is used since quantum measurement collapses the state and may effectively reinitialize the Gibbs sampling dynamics. Estimating $\operatorname{tr}(H\rho_\beta)$ to precision $\epsilon$ requires $N = \mathcal{O}(\operatorname{var}_H / \epsilon^2)$. (b) Single-trajectory approach (this work): First, run a burn-in stage by applying the Gibbs sampling channel for the mixing time $t_{mix}$. Then, enter the sampling stage, where measurements are performed after each fixed evolution time $\Delta t$ (a tunable parameter, set to $\Delta t = 1$ for simplicity). Measurements are chosen to satisfy detailed balance, implemented via Gaussian-filtered QPE, thus preserving the Gibbs state ensemble and avoiding extra burn-in periods. In this approach, effectively independent samples are obtained approximately every autocorrelation time $t_{aut}$. The autocorrelation time leverages a warm start from the previous measurement and depends on the observable of interest, and can therefore be much shorter than the mixing time. As a result, the total Gibbs sampling evolution time needed is $t_{mix} + N t_{aut}$, which is significantly smaller than the $N t_{mix}$ required in the multiple-trajectory approach.
• Figure 2: Three examples illustrating that the autocorrelation time of the energy observable can be much smaller than the mixing time. Let $\beta$ be a constant inverse temperature. (a)(b) correspond to the three-qubit Ising model $H = - \alpha Z_1 Z_2 - h (Z_1 + Z_2) - \gamma Z_3$ under Glauber dynamics: (a) Asymmetric double-well ($\alpha \gg h \gg \gamma > 0$) with negligible Gibbs weight in the smaller well; the mixing time is $\Omega(e^{\beta \alpha})$ since transitions between the two wells require overcoming an energy barrier of order $\alpha$, while the autocorrelation time is $\mathcal{O}(1)$ because typical stationary samples lie in the dominant well. (b) Symmetric double-well ($h=0$ and $\gamma>0$); the mixing time is $\Omega(e^{\beta \alpha})$, while the autocorrelation time is $\mathcal{O}(1)$ because the energy does not distinguish the two wells and its dominant fluctuations come from the decoupled spin $Z_3$. (c) A classical birth--death chain on $\left\vert 0 \right\rangle,\dots,\left\vert m \right\rangle$; the mixing time scales as $\Omega(m)$, since starting from $\left\vert m \right\rangle$ the chain requires $\Omega(m)$ time to reach the low-energy states, whereas the autocorrelation time is $\mathcal{O}(1)$ since the Gibbs distribution is heavily concentrated near the low-energy states.

Theorems & Definitions (36)

• Theorem 1: Informal
• Lemma 1: Informal
• Lemma 2
• Lemma 3
• Lemma 4: Contractivity of a detailed-balance channel under the weighted norm
• Lemma 5
• Definition 6: Spectral covariance weight
• Lemma 7
• Lemma 8
• Lemma 9