There has been much recent interest in near-term applications of quantum computers, i.e., using quantum circuits that have short decoherence times due to hardware limitations. Variational quantum algorithms (VQA), wherein an optimization algorithm implemented on a classical computer evaluates a parametrized quantum circuit as an objective function, are a leading framework in this space. An enormous breadth of algorithms in this framework have been proposed for solving a range of problems in machine learning, forecasting, applied physics, and combinatorial optimization, among others.

In this paper, we analyze the iteration complexity of VQA, that is, the number of steps that VQA requires until its iterates satisfy a surrogate measure of optimality. We argue that although VQA procedures incorporate algorithms that can, in the idealized case, be modeled as classic procedures in the optimization literature, the particular nature of noise in near-term devices invalidates the claim of applicability of off-the-shelf analyses of these algorithms. Specifically, noise makes the evaluations of the objective function via quantum circuits $biased$. Commonly used optimization procedures, such as SPSA and the parameter shift rule, can thus be seen as derivative-free optimization algorithms with biased function evaluations, for which there are currently no iteration complexity guarantees in the literature. We derive the missing guarantees and find that the rate of convergence is unaffected. However, the level of bias contributes unfavorably to both the constant therein, and the asymptotic distance to stationarity, i.e., the more bias, the farther one is guaranteed, at best, to reach a stationary point of the VQA objective.

中文翻译：

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变分量子算法的迭代复杂性


最近人们对量子计算机的近期应用非常感兴趣，即使用由于硬件限制而具有较短退相干时间的量子电路。变分量子算法 （VQA） 是在经典计算机上实现的优化算法将参数化量子电路评估为目标函数，是该领域的领先框架。该框架中已经提出了大量算法，用于解决机器学习、预测、应用物理学和组合优化等一系列问题。


在本文中，我们分析了 VQA 的迭代复杂性，即 VQA 在迭代满足最优性的代理度量之前所需的步骤数。我们认为，尽管 VQA 程序包含的算法在理想化情况下可以建模为优化文献中的经典程序，但近期设备中噪声的特殊性质使这些算法的现成分析的适用性声明无效。具体来说，噪声使通过量子电路对目标函数的评估$biased$。因此，常用的优化程序，如 SPSA 和参数偏移规则，可以看作是具有偏置函数评估的无导数优化算法，目前文献中没有迭代复杂性保证。我们推导出缺失的保证，发现收敛速率不受影响。然而，偏置水平对其中的常数和到平稳性的渐近距离都有不利的影响，即偏置越大，保证到达 VQA 物镜的静止点的距离就越远。