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arXiv:2603.16845

Efficiently estimating quantum thermal properties from exponentially fewer samples

Chi-Fang Chen, András Gilyén

quant-phSubmitted Mar 17, 2026• Announced Mar 18, 2026• 0 citations• Revised Sep 29, 2026 (v2)
v1 · Mar 17, 2026v2 · Sep 29, 2026
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Abstract

We present an efficient protocol for estimating $M$ observables to precision $\varepsilon$ from only $\mathcal{O}(\log (M)/\varepsilon^2)$ copies of a Gibbs state $ρ\propto\exp(-βH)$, given access to its Hamiltonian $H$. The protocol uses single-copy, nonadaptive measurements and a total Hamiltonian simulation time of $\widetilde{\mathcal{O}}(βM/\varepsilon^2)$; 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.

Comments
24 pages, 2 figures, small corrections, clarifications, and shaving off a log factor in the gate complexity

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Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown ddd-dimensional quantum state ρρρ and a known collection of observables E1,…,EME_1,\ldots,E_ME1​,…,EM​, the goal is to estimate all expectation values {Tr(ρEi)}i=1M\{\text{Tr}(ρE_i)\}_{i=1}^M{Tr(ρEi​)}i=1M​ to additive accuracy ε\varepsilonε with probability at least 1−δ1-δ1−δ. An elusive open question from the seminal shadow tomography work of Aaronson is whether this task admits a dimension-independent sample complexity with only polylogarithmic dependence on MMM, as suggested by the best-known lower bounds. In this work, we propose two different quantum protocols for shadow tomography with the best sample complexity O(log⁡(M)log⁡(M/δ)ε2), O\left( \frac{\log(M)\log(M/δ)}{\varepsilon^2} \right), O(ε2log(M)log(M/δ)​), which is polylogarithmic in the number of observables and independent of the dimension of the unknown state, thereby answering Aaronson's original question while also providing an exponential improvement in the prior best dimension independent sample complexity of shadow tomography from Sinha (STOC 2025) and, more recently, Chen, O'Donnell, Pelecanos, and Wright. Our approach first reduces the general shadow tomography problem to a finite-ensemble estimation problem via a minimax argument. We then develop an observable-independent protocol that repeatedly applies the pretty-good measurement while updating the prior distribution over the finite ensemble according to the measurement outcomes. A tail analysis of the resulting estimation error yields simultaneous accuracy guarantees for all observables and a cubic-logarithmic upper bound. We also introduce a refined recovery-label measurement for the same finite ensemble, which yields the bound in our main theorem.
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While recent advances have established efficient quantum algorithms for preparing Gibbs states of finite-dimensional systems, comparable complexity results for bosonic and other infinite-dimensional models remain unexplored. We introduce the first general rigorous Gibbs sampling framework for bosonic many-body systems, showing that physically relevant bosonic models admit gapped dissipative generators, enabling efficient preparation of thermal states, provided that the spectral gap scales favourably with the number of modes. Although our results hold for broad classes of models, we illustrate them using Bose-Hubbard Hamiltonians, both within and beyond the mean-field regime. In both cases, we show that the associated dissipative generators maintain a positive spectral gap, thereby implying exponential convergence to the thermal state. For the Gibbs sampler corresponding to the full Bose-Hubbard model, we obtain a spectral-gap lower bound Ce−cnCe^{-cn}Ce−cn, where nnn is the number of modes. We apply our results to provide a Gibbs-state preparation algorithm on qubit hardware, with runtime polynomial in the number of modes and the inverse spectral gap, and thereby obtain a quantum algorithm to compute thermal properties of the model. This provides the first mathematically controlled route to Gibbs sampling in infinite-dimensional systems, with implications for quantum simulation, thermalization, and many-body complexity, where quantum advantages may arise.
We investigate the problem of efficiently estimating expectation values of large sets of observables from copies of an unknown many-body quantum state. This task, known as shadow tomography, is crucial for quantum simulation and quantum chemistry, where the relevant observables often include k -body fermionic or k -local Pauli operators. A key goal is to achieve sample efficiency, computational efficiency, and few-copy measurements. We introduce the notion of triply efficient shadow tomography to formalize these requirements. Prior work has shown that single-copy measurements suffice for k -local Pauli operators with constant k . However, we prove that for k -body fermionic observables and the full set of n -qubit Pauli operators, single-copy protocols fail to achieve sample-efficient tomography. We then present new protocols based on measuring two copies of the state at a time that overcome these lower bounds, providing a triply efficient protocol.