# 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 are also closely related to solving linear systems of equations in quantum linear algebra. Here we present three related algorithms for calculating transition probabilities with respect to arbitrary operators and states. First, we extend a previously published short-depth algorithm, allowing for the two input quantum states to be non-orthogonal. The extension comes at the cost of one ancilla qubit and at most only a constant four additional two-qubit gates. Building on this first procedure, we then derive a higher-depth approach based on Trotterization and Richardson extrapolation that requires fewer circuit evaluations. Third, we introduce a tunable approach that in effect interpolates between the low-depth method and the method of fewer circuit evaluations. This tunability between circuit depth and measurement complexity allows the algorithm to be tailored to specific hardware characteristics. Finally, we implement proof-of-principle numerics for toy models in physics and chemistry and for use a subroutine in variational quantum linear solving (VQLS). The primary benefits of our approaches are that (a) arbitrary non-orthogonal states may now be used with negligible increases in quantum resources, (b) we 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.

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