Differentiation of Linear Optical Circuits

TL;DR

This paper introduces classical and quantum algorithms for efficiently computing gradients in linear optical circuits, enabling advanced optimization in quantum applications like chemistry and machine learning.

Contribution

It develops a general framework for gradient evaluation in linear optical circuits and proposes algorithms that could offer quantum speed-ups over classical methods.

Findings

Gradient evaluation algorithms for linear optical circuits

Comparison of classical and quantum sampling complexities

Potential quantum speed-ups identified in specific cases

Abstract

Linear optical circuits with single-photon sources offer a promising platform for quantum chemistry and machine learning. However, current applications are all based on support vector machines or gradient-free optimization methods. This paper develops classical and quantum algorithms for evaluating the analytic gradients of linear optical circuits with respect to their phase parameters. First, we set up a general framework by characterising the class of observables whose expectation values can be estimated efficiently by sampling from a passive linear optical circuit with finitely many photons. We then show how to compute the gradients of the expectation values of a special class of ``non-interacting'' observables arising in full-counting-statistics. Our differentiation algorithm uses the Halmos dilation and requires evaluating two circuits of twice the size, using one additional photon. Building on the methods of full-counting-statistics, we show how to recover the gradients of arbitrary observables from the gradient of a non-interacting characteristic function. Throughout the paper, we compare the performance of classical and quantum algorithms on the same estimation problems, analysing the sampling complexity of the algorithms and suggesting different cases for which quantum speed-ups could be obtained.