Improved resourcetunable nearterm quantum algorithms for transition probabilities, with applications in physics and variational quantum linear algebra
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 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 shortdepth algorithm, allowing for the two input quantum states to be nonorthogonal. The extension comes at the cost of one ancilla qubit and at most only a constant four additional twoqubit gates. Building on this first procedure, we then derive a higherdepth approach based on Trotterization and Richardson extrapolation that requires fewer circuit evaluations. Third, we introduce a tunable approach that in effect interpolates between the lowdepth 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 proofofprinciple 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 nonorthogonal states may now be used with negligible increases in quantum resources, (b) we entirely avoid subroutines such as the Hadamard test that may require threequbit gates to be decomposed, and (c) in some cases fewer quantum circuit evaluations are required as compared to the previous stateoftheart in NISQ algorithms for transition probabilities.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.14213
 Bibcode:
 2022arXiv220614213S
 Keywords:

 Quantum Physics;
 Condensed Matter  Other Condensed Matter;
 Computer Science  Data Structures and Algorithms
 EPrint:
 12 pages, 5 figures