[Paper Review] A Lie Algebraic Theory of Barren Plateaus for Deep Parameterized Quantum Circuits

Michael Ragone, Bojko Bakalov+5 more|arXiv (Cornell University)|Sep 17, 2023

Quantum Computing Algorithms and Architecture20 citations

TL;DR

This paper introduces a general Lie algebraic framework to exactly compute the variance of loss in deep parametrized quantum circuits, unifying expressiveness, data entanglement, locality, and noise as sources of barren plateaus.

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.

Motivation & Objective

• Motivate and quantify barren plateaus (BPs) in variational quantum algorithms.
• Unify disparate BP sources under a single Lie algebra framework based on the dynamical Lie algebra (DLA).
• Provide exact expressions for loss variance under broad conditions and noisy settings.
• Link loss concentration to the dimension of the DLA and to generalized entanglement and locality notions.

Proposed method

• Model the parametrized circuit as U(θ)=∏l e^{iHl θl} with generators Hl in i𝔲(2^n).
• Define the dynamical Lie algebra 𝔤 as the Lie closure of {iHl}, and decompose it as 𝔤=𝔤1⊕…⊕𝔤k, with 𝔤k abelian.
• Use, when deep, an ε-approximate 2-design over G=e^{𝔤} to compute parameter-averaged quantities with Weingarten calculus.
• Introduce 𝔤-purity 𝒫𝔤(H)=Tr[H𝔤^2] via projection H𝔤 of H into 𝔤ℂ, linking variance to generalized entanglement and locality.
• Prove Theorem 1 giving exact mean and variance of the loss: E[ℓθ]=Tr[ρ𝔤k O𝔤k] and Var[ℓθ]=∑j=1^{k-1} 𝒫𝔤j(ρ)𝒫𝔤j(O)/dim(𝔤j).
• Address noise via SPAM and coherent errors, showing algebraic decoherence effects on 𝔤-purity and locality.

Experimental results

Research questions

• RQ1 Under what conditions can the loss variance of deep parametrized quantum circuits be computed exactly?
• RQ2 How do expressiveness (DLA dimension), generalized entanglement (𝒫𝔤), and generalized locality of measurements impact barren plateaus?
• RQ3 How do SPAM and coherent noise alter loss variance within the same Lie-algebraic framework?
• RQ4 Can a single framework unify known BP sources (expressiveness, locality, entanglement, noise) for general architectures?
• RQ5 What are the implications for when a circuit forms a 2-design and how many layers are needed?

Key findings

• The loss mean is determined by the center 𝔤k of the DLA and can vanish if 𝔤 has no center.
• The loss variance is a sum over noncentral simple components 𝔤j, scaled by dim(𝔤j) and the 𝔤-purities of ρ and O.
• Barren plateaus arise if the DLA dimension is exponentially large or if 1/𝒫𝔤(ρ) or 1/𝒫𝔤(O) grows as Ω(b^n).
• Deep circuits with polynomially scaling DLAs do not necessarily have BP from expressiveness alone; initial state entanglement and locality play roles.
• Noise can induce or suppress concentration via algebraic decoherence affecting 𝒫𝔤(ρ) and 𝒫𝔤(O), and coherent errors can increase expressiveness by enlarging the DLA.
• The work provides exact variance expressions rather than bounds and shows how to quantify BP origins within a single framework.