Improved resource-tunable near-term quantum algorithms for transition probabilities, with applications in physics and variational quantum linear algebra

TL;DR

This paper introduces resource-tunable, near-term quantum algorithms for calculating transition probabilities, applicable in physics and linear algebra, improving efficiency and hardware adaptability over previous methods.

Contribution

It extends existing algorithms to handle non-orthogonal states, reduces circuit evaluations with Trotterization and extrapolation, and offers a tunable approach balancing depth and measurement complexity.

Findings

Allows use of non-orthogonal states with minimal resource increase

Avoids complex subroutines like Hadamard test, simplifying implementation

Reduces quantum circuit evaluations compared to prior NISQ algorithms

Abstract

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. Here we present three related algorithms for calculating transition probabilities. First, we extend a previously published short-depth algorithm, allowing for the two input states to be non-orthogonal. Building on this first procedure, we then derive a higher-depth algorithm based on Trotterization and Richardson extrapolation that requires fewer circuit evaluations. Third, we introduce a tunable algorithm that allows for trading off circuit depth and measurement complexity, yielding an algorithm that can be tailored to specific hardware characteristics. Finally, we implement proof-of-principle numerics for models in physics and chemistry and for 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 small increases in quantum resources, (b) we (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.