A differentiable quantum phase estimation algorithm

The simulation of electronic properties is a pivotal issue in modern electronic structure theory, driving significant efforts over the past decades to develop protocols for computing energy derivatives. In this work, we address this problem by developing a strategy to integrate the quantum phase estimation algorithm within a fully differentiable framework. This is accomplished by devising a smooth estimator able to tackle arbitrary initial states. We provide analytical expressions to characterize the statistics and algorithmic cost of this estimator. Furthermore, we provide numerical evidence that the estimation accuracy is retained when an arbitrary state is considered and that it exceeds the one of standard majority rule. We explicitly use this procedure to estimate chemically relevant quantities, demonstrating our approach through ground-state and triplet excited state geometry optimization with simulations involving up to 19 qubits. This work paves the way for new quantum algorithms that combine interference methods and quantum differentiable programming.