Barren Plateaus in Variational Quantum Computing

TL;DR

The paper provides a comprehensive synthesis of barren plateaus in variational quantum computing, identifying the BP as a consequence of the curse of dimensionality in exponentially large Hilbert spaces. It lays out the theoretical foundations for probabilistic and deterministic concentration, and links BP to circuit expressiveness, input states, measurements, and noise. The review analyzes architectures prone to BP (hardware-efficient and problem-inspired ansatzes) and surveys mitigation strategies, including shallow circuits, small dynamical Lie algebras, adaptive/variable-structure ansatzes, and smarter initialization, while acknowledging limits and the possibility of BP persisting in worst cases. It also situates BP in a broader context, drawing connections to other quantum learning paradigms and classical vanishing-gradient problems, and discusses open questions and implications for achieving practical quantum advantage.

Abstract

Variational quantum computing offers a flexible computational paradigm with applications in diverse areas. However, a key obstacle to realizing their potential is the Barren Plateau (BP) phenomenon. When a model exhibits a BP, its parameter optimization landscape becomes exponentially flat and featureless as the problem size increases. Importantly, all the moving pieces of an algorithm -- choices of ansatz, initial state, observable, loss function and hardware noise -- can lead to BPs when ill-suited. Due to the significant impact of BPs on trainability, researchers have dedicated considerable effort to develop theoretical and heuristic methods to understand and mitigate their effects. As a result, the study of BPs has become a thriving area of research, influencing and cross-fertilizing other fields such as quantum optimal control, tensor networks, and learning theory. This article provides a comprehensive review of the current understanding of the BP phenomenon.

Figures (1)

• Figure 1: Variational quantum computing. An $n$-qubit quantum computer is initialized to a state $\rho$ that is evolved through a Parametrized Quantum Circuit (PQC) $\mathcal{U}_{\boldsymbol{\theta}}$. At the output of the circuit we perform a finite set of measurements which are used to estimate a loss function $\ell_{\boldsymbol{\theta}}(\rho,O)={\rm Tr}[\mathcal{U}_{\boldsymbol{\theta}}(\rho)O]$. A classical computer takes as input the finite sample estimate of $\ell_{\boldsymbol{\theta}}(\rho,O)$ (or its partial derivatives) and attempts to find new sets of parameters which train the PQC and minimize the loss. As such by writing the loss in the vectorized form $\ell_{\boldsymbol{\theta}}(\rho,O) = \langle \rho(\boldsymbol{\theta}),O\rangle$ one can see that at the core of variational quantum computing lies the manipulation and comparison---via polynomially many samples---of the exponentially large operators $\rho(\boldsymbol{\theta})=\mathcal{U}_{\boldsymbol{\theta}}(\rho)$ and $O$.