QAOA 期望值的评估可能与最优解计数一样困难

中文摘要 AI 辅助

评估期望值是变分量子特征求解器（VQE）以及参数化量子电路和其他量子算法更普遍的关键任务。我们研究了已被充分研究的用于 MaxCut 问题的量子近似优化算法（QAOA）案例。Wang 等人[arXiv:2511.20212]的近期工作表明，对于任意 QAOA 深度 p≥2，该任务在一般情况下是 NP 难的，补充了过去关于 p=1 时对任意问题图都存在可高效计算的公式的结果。我们强化了这种二分性，表明对于 p≥2，在确定性多项式时间图灵归约下，精确或指数精度的代价期望值评估是 #P 难的。即使对于评估单个两两关联函数⟨Z⊗Z⟩以及高度受限的算法参数集，p≥2 时的难解性仍然成立。我们的证明改进了 Wang 等人的 NP 难解性构造，该构造可从 QAOA 劳伦特多项式的最大指数中恢复最大割值，我们利用一种不同且更简单的构造，除了提取最优割值外，还提取与最大割总数成比例的值。因此，我们表明 QAOA 从 p=1 到 p=2 的期望值难解性转变不仅是从可处理性到优化难解性，而且到计数最优解的难解性。作为应用，我们的结果意味着对于计算 QAOA 电路的梯度和 Hessian 也有类似的难解性结果。

英文摘要

Evaluating expectation values is a critical task for variational quantum eigensolvers, parameterized quantum circuits, and many other quantum algorithms. We consider the well-studied case of the Quantum Approximate Optimization Algorithm (QAOA) for the MaxCut problem. Recent work of Wang et al. [arXiv:2511.20212] showed this task to be NP-hard in general for any QAOA depth $p\geq 2$, complementing past results showing efficiently computable formulas for $p=1$ with arbitrary problem graphs. We sharpen this dichotomy showing that for $p\geq 2$ exact or exponentially precise cost expectation value evaluation is #P-hard under deterministic polynomial-time Turing reductions. Hardness at $p\geq 2$ is shown to remain even for evaluating single pairwise correlators $\langle Z\otimes Z\rangle $, as well as for highly restricted sets of algorithm parameters. Our proof refines the NP-hardness construction of Wang et al. that recovers the maximum cut value from the largest exponent of a QAOA Laurent polynomial, utilizing a distinct and simpler construction that extracts a value proportional to the total number of maximum cuts, in addition to the optimal cut value. Thus we show that the QAOA expectation value hardness transition from $p=1$ to $p=2$ is not only from tractability to optimization hardness, but to that of counting optimal solutions. As an application we show our results imply analogous hardness results for computing gradients and Hessians of QAOA circuits.

发表机构

• Quantum Artificial Intelligence Laboratory (QuAIL), NASA Ames Research Center（量子人工智能实验室（QuAIL），NASA艾姆斯研究中心）
• USRA Research Institute for Advanced Computer Science（美国大学空间研究协会高级计算机科学研究院）

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