Improved Adaptive Estimation of Quantum Partition Functions with Heisenberg Scaling

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(sqrt n) always exists and can be generated from overlap estimates with O(sqrt n) further preparations. Given such a schedule and classically chosen temperatures, we estimate the log partition function with O(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 O(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.