训练变分量子算法是NP难的，即使在局部意义上也是如此
Training Variational Quantum Algorithms Is NP-Hard, Even Locally
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中文总结 AI 辅助
该研究证明，即使仅寻找变分量子算法训练目标的局部最小值或其附近参数，也是强NP难的，揭示了变分量子训练的基本计算障碍。
中文摘要 AI 辅助
变分量子算法（VQAs）通常依赖经典优化来训练参数化量子电路。这种训练旨在最小化一个目标函数，其效率对于这些算法的实际成功至关重要。然而，已知在电路参数上全局最小化此类训练目标是$\mathsf{NP}$难的，这限制了高效训练的一般保证前景。在本快报中，我们证明，即使是寻找此类VQA训练目标的局部最小值这一较弱任务，也是强$\mathsf{NP}$难的，包括当目标允许高效经典评估时。此外，我们表明，即使对于寻找与某个局部最小值的$\ell_p$距离严格小于$\pi/2$的参数向量这一任务，对于每个$p\geq 1$，这种难度仍然存在。我们的核心技术结果是，逼近厄米三角多项式的局部最小值是强$\mathsf{NP}$难的。通过显式构造其训练目标重现这些困难实例的量子电路，我们获得了到VQA训练的多项式时间归约。我们的结果确立了变分量子训练的基本计算障碍：即使在最坏情况下，达到局部最小值附近仍然是困难的。
英文摘要
Variational quantum algorithms (VQAs) generally rely on classical optimization to train parameterized quantum circuits. This training seeks to minimize an objective function, and its efficiency is central to the practical success of these algorithms. However, globally minimizing such training objectives over the circuit parameters is known to be $\mathsf{NP}$-hard, limiting the prospect of general guarantees for efficient training. In this Letter, we prove that even the weaker task of finding a local minimum of such VQA training objectives is strongly $\mathsf{NP}$-hard, including when the objective admits efficient classical evaluation. Moreover, we show that this hardness persists even for the task of finding a parameter vector within $\ell_p$-distance strictly less than $π/2$ of some local minimizer, for every $p\geq 1$. Our central technical result is that approximating a local minimizer of a Hermitian trigonometric polynomial is strongly $\mathsf{NP}$-hard. By explicitly constructing quantum circuits whose training objectives reproduce these hard instances, we obtain a polynomial-time reduction to VQA training. Our results establish a fundamental computational barrier to variational quantum training: even reaching the vicinity of a local minimum remains hard in the worst case.
发表机构
- Singapore Institute of Technology（新加坡理工大学）
- Singapore University of Technology and Design（新加坡科技设计大学）
- Quantum Innovation Centre (Q.InC), Agency for Science, Technology and Research (A*STAR)（A*STAR量子创新中心）
- Archimedes Research Unit on AI, Data Science and Algorithms, Athena RC（雅典娜研究中心人工智能、数据科学与算法阿尔基米德研究组）
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