Fontana Enrico, Herman Dylan, Chakrabarti Shouvanik, Kumar Niraj, Yalovetzky Romina, Heredge Jamie, Sureshbabu Shree Hari, Pistoia Marco
Global Technology Applied Research, JPMorganChase, New York, NY, 10017, USA.
Computer and Information Sciences, University of Strathclyde, Glasgow, G1 1XQ, UK.
Nat Commun. 2024 Aug 22;15(1):7171. doi: 10.1038/s41467-024-49910-w.
Variational quantum algorithms, a popular heuristic for near-term quantum computers, utilize parameterized quantum circuits which naturally express Lie groups. It has been postulated that many properties of variational quantum algorithms can be understood by studying their corresponding groups, chief among them the presence of vanishing gradients or barren plateaus, but a theoretical derivation has been lacking. Using tools from the representation theory of compact Lie groups, we formulate a theory of barren plateaus for parameterized quantum circuits whose observables lie in their dynamical Lie algebra, covering a large variety of commonly used ansätze such as the Hamiltonian Variational Ansatz, Quantum Alternating Operator Ansatz, and many equivariant quantum neural networks. Our theory provides, for the first time, the ability to compute the exact variance of the gradient of the cost function of the quantum compound ansatz, under mixing conditions that we prove are commonplace.
变分量子算法是近期量子计算机常用的一种启发式算法，它利用自然表示李群的参数化量子电路。据推测，变分量子算法的许多性质可以通过研究其相应的群来理解，其中最主要的是梯度消失或贫瘠高原的存在，但一直缺乏理论推导。利用紧致李群表示理论的工具，我们为可观测量位于其动力学李代数中的参数化量子电路建立了一种贫瘠高原理论，涵盖了多种常用的近似方法，如哈密顿变分近似、量子交替算子近似以及许多等变量子神经网络。我们的理论首次提供了在我们证明很常见的混合条件下计算量子复合近似代价函数梯度的精确方差的能力。
