Variational Quantum Algorithms (VQAs), such as the Quantum Approximate Optimization Algorithm (QAOA) of [Farhi, Goldstone, Gutmann, 2014], have seen intense study towards near-term applications on quantum hardware. A crucial parameter for VQAs is the \emph{depth} of the variational ``ansatz'' used -- the smaller the depth, the more amenable the ansatz is to near-term quantum hardware in that it gives the circuit a chance to be fully executed before the system decoheres. In this work, we show that approximating the optimal depth for a given VQA ansatz is intractable. Formally, we show that for any constant $\epsilon>0$, it is QCMA-hard to approximate the optimal depth of a VQA ansatz within multiplicative factor $N^{1-\epsilon}$, for $N$ denoting the encoding size of the VQA instance. (Here, Quantum Classical Merlin-Arthur (QCMA) is a quantum generalization of NP.) We then show that this hardness persists in the even ``simpler'' QAOA-type settings. To our knowledge, this yields the first natural QCMA-hard-to-approximate problems.

翻译：变分量子算法（VQAs），如[Farhi, Goldstone, Gutmann, 2014]提出的量子近似优化算法（QAOA），在近期量子硬件的应用中备受关注。VQA的关键参数是所使用的变分"拟设"的深度——深度越小，拟设越易于在近期量子硬件上实现，因为这样电路就有可能在系统退相干前完全执行。本研究表明，近似给定VQA拟设的最优深度是棘手的。形式上，我们证明对于任意常数ε>0，在VQA实例编码大小N的N^{1-ε}倍乘因子范围内近似最优深度是QCMA难的（这里，量子经典梅林-亚瑟（QCMA）是NP的量子推广）。我们进一步证明，这种难度在更"简单"的QAOA型设置中依然存在。据我们所知，这给出了首个自然的QCMA难近似问题。