Ragone Michael, Bakalov Bojko N, Sauvage Frédéric, Kemper Alexander F, Ortiz Marrero Carlos, Larocca Martín, Cerezo M
Department of Mathematics, University of California Davis, Davis, USA.
Department of Mathematics, North Carolina State University, Raleigh, USA.
Nat Commun. 2024 Aug 22;15(1):7172. doi: 10.1038/s41467-024-49909-3.
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.
变分量子计算方案通过将初始状态送入参数化量子电路并测量某个算符的期望值来训练损失函数。尽管这些方案前景广阔，但由于电路的表现力、输入数据的纠缠、可观测量的局部性或噪声的存在所导致的贫瘠高原（BPs），阻碍了这些算法的可训练性。到目前为止，这些导致BPs的因素一直被视为相互独立的。在这项工作中，我们提出了一种通用的李代数理论，该理论为足够深的参数化量子电路的损失函数方差提供了精确表达式，即使在存在某些噪声模型的情况下也是如此。我们的结果使我们能够在一个框架下理解所有上述导致BPs的因素。这一理论飞跃解决了一个长期存在的关于损失集中度与电路生成元李代数维度之间联系的猜想。
