Trainability of Parametrised Linear Combinations of Unitaries

TL;DR

This paper proves that linear combinations of trainable parametrised quantum circuits remain trainable, addressing barren plateaus and enabling more expressive quantum models with potential quantum speed-up.

Contribution

It analytically demonstrates the trainability of linear combinations of parametrised circuits, extending to incoherent superpositions, and supports this with numerical evidence.

Findings

LCU of trainable circuits is trainable despite barren plateaus

Analytical expression for variance of expectation values derived

Numerical results on fermionic Gaussian unitaries support theoretical claims

Abstract

A principal concern in the optimisation of parametrised quantum circuits is the presence of barren plateaus, which present fundamental challenges to the scalability of applications, such as variational algorithms and quantum machine learning models. Recent proposals for these methods have increasingly used the linear combination of unitaries (LCU) procedure as a core component. In this work, we prove that an LCU of trainable parametrised circuits is still trainable. We do so by analytically deriving the expression for the variance of the expectation when applying the LCU to a set of parametrised circuits, taking into account the postselection probability. These results extend to incoherent superpositions. We support our conclusions with numerical results on linear combinations of fermionic Gaussian unitaries (matchgate circuits). Our work shows that sums of trainable parametrised circuits are still trainable, and thus provides a method to construct new families of more expressive trainable circuits. We argue that there is a scope for a quantum speed-up when evaluating these trainable circuits on a quantum device.