量子近似优化中态制备的复杂性障碍

中文摘要 AI 辅助

对于许多重要的优化问题，由于计算复杂性，我们在实践中通常只能求助于近似解。与精确优化设置不同，近似优化允许采用超越是否找到最优解的性能度量，并具有不同的权衡和复杂性。对于MaxCut，接近1的（普通）近似比可以与相对于随机割的接近零的改进（增益）共存。对于标准编码，无条件的经典MaxCut增益硬度差距意味着，任何在每次输入上以至少逆多项式成功概率恢复最优经典增益的固定正比例的统一高效量子或混合过程，都将把NP置于BQP中。因此，在标准假设下，这样的过程被认为是不可能的。我们广泛探讨了在最坏情况障碍在量子算法全景中适用或不适用的位置。我们证明了该障碍在量子随机访问优化（QRAO）压缩下仍然存在，并适用于经典最优值与松弛最优值之间。对于每个输入，乘积态都能达到经典最优值。因此，达到经典阈值的障碍并非源于对纠缠的需求。对于每量子比特$d\in\{2,3\}$个变量，已知解码器将编码能量增益按精确因子$1/d^2$转移到解码后的平均增益。将此恒等式与MaxCut增益硬度相结合，为QRAO提供了操作性制备障碍。我们还构造了具有相对量子松弛超额$\Theta(1/n)$的困难$n$量子比特族，而最大混合态的能量近似比为$1-\Theta(1/n)$，编码能量增益为零，因此解码后的平均增益也为零。我们的结果将松弛紧致性和能量近似的影响与操作可访问性分离开来，促使在基准测试和性能评估中进行更全面的核算。

英文摘要

For many important optimization problems we are restricted to approximate solutions in practice due to computational complexity. Distinct from the exact optimization setting, approximate optimization admits performance measures beyond whether the optimum is found, with different tradeoffs and complexity. For MaxCut, a near-unity (ordinary) approximation ratio can coexist with near-zero improvement (gain) over a random cut. For the standard encoding, the unconditional classical MaxCut-Gain hardness gap implies that \emph{any uniformly efficient quantum or hybrid procedure recovering a fixed positive fraction of the optimal classical gain on every input, with at least inverse-polynomial success probability, would place} NP \emph{in} BQP. Such a procedure is therefore believed impossible under standard assumptions. We broadly address where our worst-case barriers do or do not apply across the quantum algorithm landscape. We prove that the barrier survives quantum random access optimization (QRAO) compression and applies between the classical and relaxed optimal values. For every input, a product state attains the classical optimum. Thus the barrier to reaching the classical threshold does not arise from a need for entanglement. For $d\in\{2,3\}$ variables per qubit, the known decoder transfers encoded energy gain to decoded mean gain by the exact factor $1/d^2$. Combining this identity with MaxCut-Gain hardness gives an operational preparation barrier for QRAO. We also construct hard $n$-qubit families with relative quantum relaxation excess $Θ(1/n)$, while the maximally mixed state has energy approximation ratio $1-Θ(1/n)$, zero encoded energy gain, and hence zero decoded mean gain. Our results separate the effects of relaxation tightness and energy approximation from operational accessibility, motivating more comprehensive accounting in benchmarking and performance assessment.

发表机构

• 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所）

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