Improved Resource-Tunable Near-Term Quantum Algorithms for Transition Probabilities, with Applications in Physics and Variational Quantum Linear Algebra

Transition amplitudes and transition probabilities are relevant to many areas of physics simulation, including the calculation of response properties and correlation functions. These quantities can also be related to solving linear systems of equations. This study presents three related algorithms for calculating transition probabilities. First, a previously published short-depth algorithm is extended, allowing for the two input states to be non-orthogonal. Building on this first procedure, a higher-depth algorithm based on Trotterization and Richardson extrapolation is derived that requires fewer circuit evaluations. Third, a tunable algorithm that allows for trading off circuit depth and measurement complexity is introduced, yielding an algorithm that can be tailored to specific hardware characteristics. Finally, proof-of-principle numerics are presented for models in physics and chemistry and for a subroutine in variational quantum linear solving (VQLS). The primary benefits of these approaches are that a) arbitrary non-orthogonal states may now be used with small increases in quantum resources, b) they (like another recently proposed method) entirely avoid subroutines such as the Hadamard test that may require three-qubit gates to be decomposed, and c) in some cases fewer quantum circuit evaluations are required as compared to the previous state-of-the-art in NISQ algorithms for transition probabilities.