A Lie Algebraic Theory of Barren Plateaus for Deep Parameterized Quantum Circuits

TL;DR

This paper develops a Lie algebraic framework to analyze and unify the understanding of barren plateaus in deep variational quantum circuits, accounting for expressiveness, entanglement, locality, and noise.

Contribution

It introduces a comprehensive Lie algebraic theory providing exact variance expressions for loss functions, unifying multiple sources of barren plateaus in quantum circuits.

Findings

Provides an exact variance formula for deep quantum circuits

Unifies different sources of barren plateaus under a single framework

Confirms the connection between loss concentration and Lie algebra dimension

Abstract

Variational quantum computing schemes train a loss function by sending an initial state through a parametrized quantum circuit, and measuring the expectation value of some operator. Despite their promise, the trainability of these algorithms is hindered by barren plateaus (BPs) induced by the expressiveness of the circuit, the entanglement of the input data, the locality of the observable, or the presence of noise. Up to this point, these sources of BPs have been regarded as independent. In this work, we present a general Lie algebraic theory that provides an exact expression for the variance of the loss function of sufficiently deep parametrized quantum circuits, even in the presence of certain noise models. Our results allow us to understand under one framework all aforementioned sources of BPs. This theoretical leap resolves a standing conjecture about a connection between loss concentration and the dimension of the Lie algebra of the circuit's generators.