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arXiv:2603.21595

Thermal expectation estimation via single-trajectory Gibbs sampling with non-destructive measurements

Hongrui Chen, Jiaqing Jiang, Bowen Li, Lexing Ying

quant-phmath-phSubmitted Mar 23, 2026• Announced Mar 24, 2026• 0 citations• Revised May 29, 2026 (v2)
v1 · Mar 23, 2026v2 · May 29, 2026
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Abstract

Estimating thermal expectation values of quantum many-body systems is a central challenge in physics, chemistry, and materials science. Standard quantum Gibbs sampling protocols address this task by preparing the Gibbs state from scratch after every measurement, incurring a full mixing time cost at each step. Recent advances in single-trajectory Gibbs sampling [Jiang et al. 2026] substantially reduce this overhead: once stationarity is reached, measurements can be collected along a single trajectory without re-thermalizing, provided the measurement channel preserves the Gibbs ensemble. However, explicit constructions of such non-destructive measurements have been limited primarily to observables that commute with the Hamiltonian. In this work, we fundamentally extend the single-trajectory framework to arbitrary, non-commuting observables. We provide two measurement constructions that extract measurement information without fully destroying the Gibbs state, thereby eliminating the need for full re-mixing between samples. First, we construct a measurement that satisfies exact detailed balance. This ensures the system remains in equilibrium throughout the trajectory, allowing measurement outcomes to decorrelate in an autocorrelation time that could be significantly shorter than the global mixing time. Second, assuming the underlying quantum Gibbs sampler has a positive spectral gap, we design a simplified measurement scheme that ensures the post-selected state serves as a warm start for rapid re-mixing. This approach successfully decouples the resampling cost from the global mixing time. Both measurement schemes admit efficient quantum circuit implementations, requiring only polylogarithmic Hamiltonian simulation time.

Comments
41 pages; typos corrected

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Estimating thermal expectation values of quantum many-body systems is a central challenge in physics, chemistry, and materials science. Standard quantum Gibbs sampling protocols address this task by preparing the Gibbs state from scratch after every measurement, incurring a full mixing time cost at each step. Recent advances in single-trajectory Gibbs sampling [Jiang et al. 2026] substantially reduce this overhead: once stationarity is reached, measurements can be collected along a single trajectory without re-thermalizing, provided the measurement channel preserves the Gibbs ensemble. However, explicit constructions of such non-destructive measurements have been limited primarily to observables that commute with the Hamiltonian. In this work, we fundamentally extend the single-trajectory framework to arbitrary, non-commuting observables. We provide two measurement constructions that extract measurement information without fully destroying the Gibbs state, thereby eliminating the need for full re-mixing between samples.
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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.
We present an efficient protocol for estimating MMM observables to precision ε\varepsilonε from only O(log⁡(M)/ε2)\mathcal{O}(\log (M)/\varepsilon^2)O(log(M)/ε2) copies of a Gibbs state ρ∝exp⁡(−βH)ρ\propto\exp(-βH)ρ∝exp(−βH), given access to its Hamiltonian HHH. The protocol uses single-copy, nonadaptive measurements and a total Hamiltonian simulation time of O~(βM/ε2)\widetilde{\mathcal{O}}(βM/\varepsilon^2)O(βM/ε2); we show that the sample complexity is optimal in a black-box setting. The key idea of our protocol is a new interpretation of quantum Gibbs samplers as detailed-balanced measurement channels: measurements that preserve the Gibbs state when outcomes are marginalized, and thus enable Gibbs samples to be reused for measuring subsequent observables. Consequently, shadow tomography of thermal states admits a general efficient algorithm when controlled Hamiltonian simulation is available, substantially boosting the data acquisition rate in quantum thermal simulation workflows.
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Preparing thermal and ground states is an essential quantum algorithmic task for quantum simulation. In this work, we construct the first efficiently implementable and exactly detailed-balanced Lindbladian for Gibbs states of arbitrary noncommutative Hamiltonians. Our construction can also be regarded as a continuous-time quantum analog of the Metropolis-Hastings algorithm. To prepare the quantum Gibbs state, our algorithm invokes Hamiltonian simulation for a time proportional to the mixing time and the inverse temperature βββ, up to polylogarithmic factors. Moreover, the gate complexity reduces significantly for lattice Hamiltonians as the corresponding Lindblad operators are (quasi-) local (with radius ∼β\simβ∼β) and only depend on local Hamiltonian patches. Meanwhile, purifying our Lindbladians yields a temperature-dependent family of frustration-free "parent Hamiltonians", prescribing an adiabatic path for the canonical purified Gibbs state (i.e., the Thermal Field Double state). These favorable features suggest that our construction serves as a quantum algorithmic counterpart to classical Markov chain Monte Carlo sampling.
Providing evidence that quantum computers can efficiently prepare low-energy or thermal states of physically relevant interacting quantum systems is a major challenge in quantum information science. A newly developed quantum Gibbs sampling algorithm [Chen et al., 2023] provides an efficient simulation of the detailed-balanced dissipative dynamics of non-commutative quantum systems. The running time of this algorithm depends on the mixing time of the corresponding quantum Markov chain, which has not been rigorously bounded except in the high-temperature regime. In this work, we establish a polylog(n) upper bound on its mixing time for various families of random n × n sparse Hamiltonians at any constant temperature. We further analyze how the choice of the jump operators for the algorithm and the spectral properties of these sparse Hamiltonians influence the mixing time. Our result places this method for Gibbs sampling on par with other efficient algorithms for preparing low-energy states of quantumly easy Hamiltonians.
Providing evidence that quantum computers can efficiently prepare low-energy or thermal states of physically relevant interacting quantum systems is a major challenge in quantum information science. A newly developed quantum Gibbs sampling algorithm [Chen et al., 2023] provides an efficient simulation of the detailed-balanced dissipative dynamics of non-commutative quantum systems. The running time of this algorithm depends on the mixing time of the corresponding quantum Markov chain, which has not been rigorously bounded except in the high-temperature regime. In this work, we establish a polylog(n) upper bound on its mixing time for various families of random n × n sparse Hamiltonians at any constant temperature. We further analyze how the choice of the jump operators for the algorithm and the spectral properties of these sparse Hamiltonians influence the mixing time. Our result places this method for Gibbs sampling on par with other efficient algorithms for preparing low-energy states of quantumly easy Hamiltonians.