Improved Adaptive Estimation of Quantum Partition Functions with Heisenberg Scaling

Abstract:We give quantum algorithms that estimate the log partition function of an $n$-qubit quantum Hamiltonian to additive error $\epsilon$ with Heisenberg scaling, while minimizing calls to thermal-state purification preparation. The input is a block encoding of $H$, unitaries preparing thermofield-double states at chosen inverse temperatures, with their inverses and controlled versions, and an adaptive slowly-varying cooling schedule. The system-size bounds below assume $0\preceq H\preceq hI$ with $h=O(n)$ and block-encoding normalization $\alpha=\Theta(h)$. For constant inverse temperature, we show that such a schedule of length $O(\sqrt n)$ always exists and can be generated from overlap estimates with $\widetilde O(\sqrt n)$ further preparations. Given such a schedule and classically chosen temperatures, we estimate the log partition function with $\widetilde O(n^{5/4}/\epsilon)$ expected calls to state preparation and to the block encoding, improving by $n^{1/4}$ on non-adaptive strategies. To achieve our result, we use a recursive-doubling identity that expresses each schedule increment as a weighted sum of logarithms of thermofield-double overlaps and of short imaginary-time steps implemented by quantum singular value transformation combined with previously developed variance reduction techniques. If thermofield doubles can be prepared controlled on a temperature register, a quantum inverse-binomial log estimator combined with quantum mean estimation reduces both query counts to $\widetilde O(n/\epsilon)$. We show that the latter scaling is optimal in block-encoding queries up to polylogarithmic factors and discuss end-to-end complexities for one-dimensional partition-function estimation.

arXiv:2610.06197 [quant-ph]

(or arXiv:2610.06197v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.06197 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.