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arXiv 2010.09720quant-phcond-mat.dis-nn
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变分量子线路训练中的突变

Abrupt Transitions in Variational Quantum Circuit Training

Ernesto Campos, Aly Nasrallah, Jacob Biamonte

备注：
11 pages, 8 figures, RevTeX
发表信息：
Phys. Rev. A 103, 032607 (2021)
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摘要：

变分量子算法主导了现代量子处理器的基于门的应用。所谓的{ \it逐层可训练性猜想}出现在贯穿变分量子计算文献的各种著作中。该猜想断言量子线路可以分段训练，例如~可以按顺序训练几层以最小化目标函数。这里我们证明这个猜想是错误的。通过考虑指数接近（量子比特数）单位矩阵的目标函数来找到反例。在有限的设置中，我们发现量子线路的训练能力突然转变，以最小化这些目标函数。具体来说，我们发现，低于一个关键的（目标门依赖）阈值，电路训练终止接近身份，并保持接近身份的后续添加块训练分段。临界层深度将突然训练任意接近目标，从而使目标函数最小化。这些发现为变分量子线路的分而治之的可训练性提供了新的线索，并适用于当代文献的广泛收集。

英文摘要：

Variational quantum algorithms dominate gate-based applications of modern quantum processors. The so called, {\it layer-wise trainability conjecture} appears in various works throughout the variational quantum computing literature. The conjecture asserts that a quantum circuit can be trained piece-wise, e.g.~that a few layers can be trained in sequence to minimize an objective function. Here we prove this conjecture false. Counterexamples are found by considering objective functions that are exponentially close (in the number of qubits) to the identity matrix. In the finite setting, we found abrupt transitions in the ability of quantum circuits to be trained to minimize these objective functions. Specifically, we found that below a critical (target gate dependent) threshold, circuit training terminates close to the identity and remains near to the identity for subsequently added blocks trained piece-wise. A critical layer depth will abruptly train arbitrarily close to the target, thereby minimizing the objective function. These findings shed new light on the divide-and-conquer trainability of variational quantum circuits and apply to a wide collection of contemporary literature.

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