Thermal-Drift Sampling: Generating Random Thermal Ensembles for Quantum Chaos Diagnostics

Abstract

Random thermal states of many-body Hamiltonians underpin studies of thermalization, chaos, and quantum phase transitions, yet their generation remains costly when each Hamiltonian must be prepared individually. We introduce the thermal-drift channel, a measurement-based operation that implements a tunable nonunitary drift along a chosen Pauli term. Based on this channel, we present a measurement-controlled sampling algorithm that generates thermal states together with their Hamiltonian "labels" for general physical models. We prove that the total gate count of our algorithm scales cubically with system size, quadratically with inverse temperature, and as the inverse error tolerance to the two-thirds power, with logarithmic dependence on the allowed failure probability. We also show that the induced label distribution approaches a normal distribution reweighted by the thermal partition function, which makes an explicit trade-off between accuracy and effective range. Numerical simulations for a 2D Heisenberg model validate the predicted scaling and distribution. As an application, we compute unfolding-free level-spacing ratio statistics from sampled thermal states of a 2D transverse-field Ising model and observe a crossover toward the Wigner--Dyson prediction, demonstrating a practical and scalable route to chaos diagnostics and random matrix universality studies on near-term quantum hardware.

Figures (4)

• Figure 1: Difference between classical sampling and measurement-controlled sampling. The task is to sample a thermal state together with its Hamiltonian label. (a) Sampling the label via classical computers. The thermal state is prepared using existing thermal-state preparation circuits. (b) Sampling the thermal state and label via one structured random circuit. The label is generated by a random walk based on outcomes of mid-circuit measurements.
• Figure 2: Numerical verification of the theoretical predictions for the thermal-drift sampling algorithm applied to a $3\times 3$ two-dimensional Heisenberg model. (a) Precision $\epsilon$ between the output state and the ideal thermal state as a function of the inverse temperature $\beta$, for different step scalings. (b) Empirical marginal distribution of a nearest-neighbor $YY$ interaction coefficient, compared with the theoretical prediction from Theorem \ref{['thm:distribution']}. Here $Y_{i,j}$ denotes the Pauli-$Y$ operator on the site at row $i$ and column $j$. (c) Trade-off between inverse precision and effective sample range at fixed $\beta$, as the number of steps $N$ increases from $C\beta$ to $C\beta^{3}$, where $C$ is a fixed constant. See the Supplementary Material for more experimental details.
• Figure 3: Level-spacing ratio statistics of the sampled thermal states for a $3 \times 3$ 2D transverse-field Ising model. Shown is the distribution of the adjacent level-spacing ratio $r$. The orange histogram in the background corresponds to the initial state, while the blue histogram in the foreground corresponds to the output state generated by the sampling algorithm. The blue dashed curve indicates the Wigner--Dyson prediction, and the orange dash-dotted curve indicates the Poisson prediction. See the Supplementary Material for additional experimental details.
• Figure S1: A circuit implementing the channel $\mathcal{N}_{\approx}$. Repeating $\mathcal{N}_{\approx}$ and measuring $AB$ in the computational basis until the outcome is not in $\Pi^\circlearrowleft$ implements $\mathcal{N}_{\sigma}$.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Lemma S1: Fróchet derivative of matrix exponential, Nicholas2008functions
• Lemma S2
• Lemma S3: Baker--Campbell--Hausdorff formula for $e^Xe^Y e^X$, hall2015lie
• Lemma S4: Freedman Inequality for Hermitian matrices, tropp2011freedman
• Lemma S5: Lattice-based random walk, spitzer2001principles
• Theorem S6
• Theorem S7
• Lemma S8