Variational quantum simulation: a case study for understanding warm starts

TL;DR

This paper investigates the effectiveness of warm starts in variational quantum algorithms, analyzing their ability to mitigate barren plateaus and improve trainability in quantum simulation tasks.

Contribution

It provides a theoretical analysis of warm starts, proving conditions for trainability and highlighting limitations related to minima outside trainable regions.

Findings

Gradients remain substantial near initializations in small regions.

Convexity guarantees support trainability for polynomial time-steps.

Good minima may lie outside regions with guaranteed trainability.

Abstract

The barren plateau phenomenon, characterized by loss gradients that vanish exponentially with system size, poses a challenge to scaling variational quantum algorithms. Here we explore the potential of warm starts, whereby one initializes closer to a solution in the hope of enjoying larger loss variances. Focusing on an iterative variational method for learning shorter-depth circuits for quantum real time evolution we conduct a case study to elucidate the potential and limitations of warm starts. We start by proving that the iterative variational algorithm will exhibit substantial (at worst vanishing polynomially in system size) gradients in a small region around the initializations at each time-step. Convexity guarantees for these regions are then established, suggesting trainability for polynomial size time-steps. However, our study highlights scenarios where a good minimum shifts outside the region with trainability guarantees. Our analysis leaves open the question whether such minima jumps necessitate optimization across barren plateau landscapes or whether there exist gradient flows, i.e., fertile valleys away from the plateau with substantial gradients, that allow for training. While our main focus is on this case study of variational quantum simulation, we end by discussing how our results work in other iterative settings.