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 ε 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⪯ H⪯ hI with h=O(n) and block-encoding normalization α=Θ(h). For constant inverse temperature, we show that such a schedule of length O(√n) always exists and can be generated from overlap estimates with Õ(√n) further preparations. Given such a schedule and classically chosen temperatures, we estimate the log partition function with Õ(n5/4/ε) expected calls to state preparation and to the block encoding, improving by n1/4 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 Õ(n/ε). 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.
Related
- AI estimation of partition functions with the PULSE method for the thermodynamic properties of disordered compounds
- Finding CFTs from spatial positivity of partition functions
- Trace Relations and Partition Functions
- From Z to all: cyclic sum rules for thermal CFT partition functions
- Some universalities in the partition functions of low-dimensional gravity models
BibTeX
@misc{wang2026improved,
title = {{Improved Adaptive Estimation of Quantum Partition Functions with Heisenberg Scaling}},
author = {Yufei Wang and Daniel Stilck França and Samuel Slezak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06197},
eprint = {2610.06197},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06197},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Improved Adaptive Estimation of Quantum Partition Functions with Heisenberg Scaling
AU - Yufei Wang
AU - Daniel Stilck França
AU - Samuel Slezak
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06197
UR - https://arxiv.org/abs/2610.06197
AB - We give quantum algorithms that estimate the log partition function of an n-qubit quantum Hamiltonian to additive error ε 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⪯ H⪯ hI with h=O(n) and block-encoding normalization α=Θ(h). For constant inverse temperature, we show that such a schedule of length O(√n) always exists and can be generated from overlap estimates with Õ(√n) further preparations. Given such a schedule and classically chosen temperatures, we estimate the log partition function with Õ(n^5/4/ε) 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 Õ(n/ε). 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.
ER -