非负无纠缠量子证明的近最优间隙放大
Near-Optimal Gap Amplification for Nonnegative Unentangled Quantum Proofs
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中文总结 AI 辅助
本文针对非负无纠缠量子证明类QMA⁺(2)，证明了近最优间隙放大结果，揭示其复杂度相变，结合对称子空间投影与德芬蒂定理完成证明，为量子复杂度理论提供关键进展。
中文摘要 AI 辅助
我们研究由计算基下振幅为非负的无纠缠量子证明所刻画的类QMA⁺(2)的间隙放大问题。该类由Jeronimo和Wu于STOC 2023年提出，其行为对完备性-可靠性间隙高度敏感：尽管在某些小常数间隙下它能捕获NEXP的能力，但在较大常数间隙下它等价于QMA(2)，这与QMA(2)形成鲜明对比——QMA(2)因Harrow和Montanaro的乘积测试（FOCS 2010、JACM 2013）而具有强间隙放大特性。本文中，我们对所有满足c-s=1/poly(n)的完备性c和可靠性s，证明NEXP = QMA⁺(2,c,s) = QMA⁺(2,1-1/poly(n),1/4+1/poly(n))。该结果为QMA⁺(2)提供了清晰的复杂度相变，因为QMA^ℝ(2)（表示证明限制为实振幅的QMA(2)）满足QMA^ℝ(2)=QMA⁺(2,1-1/poly(n),1/4-1/poly(n))。我们的放大是最优的：若对可靠性稍作改进，将导致QMA^ℝ(2)=NEXP的坍缩。我们的证明结合了对称子空间投影与Bassirian、Fefferman和Marwaha（ITCS 2024）提出的关系QMA⁺(1)=NEXP，主要技术要素是希尔伯特-施密特范数下与维度无关的德芬蒂定理，适用于所考虑寄存器数量呈对数增长的情况。
英文摘要
We study gap amplification of the class $\mathsf{QMA}^{+}(2)$ characterized by unentangled quantum proofs whose amplitudes are nonnegative in the computational basis. This class was recently introduced by Jeronimo and Wu (STOC 2023), and its behavior depends sharply on the completeness-soundness gap: although it captures the power of $\mathsf{NEXP}$ for some small constant gap, it is equal to $\mathsf{QMA}(2)$ for larger constant gap. This is in stark contrast to $\mathsf{QMA}(2)$ where strong gap amplification is known due to the product test by Harrow and Montanaro (FOCS 2010, JACM 2013). In this paper, we prove for every completeness $c$ and soundness $s$ with $c-s=1/\mathrm{poly}(n)$, \[ \mathsf{NEXP} = \mathsf{QMA}^{+}(2,c,s) = \mathsf{QMA}^{+}\left(2,1-\frac1{\mathrm{poly}(n)},\frac14+\frac1{\mathrm{poly}(n)}\right). \] Our result gives a clean complexity phase transition for $\mathsf{QMA}^{+}(2)$ since we have \[ \mathsf{QMA}^{\mathbb R}(2) = \mathsf{QMA}^{+}\left(2,1-\frac1{\mathrm{poly}(n)},\frac14-\frac1{\mathrm{poly}(n)}\right), \] where $\mathsf{QMA}^{\mathbb R}(2)$ denotes $\mathsf{QMA}(2)$ with witnesses restricted to real amplitudes. Our amplification is thus optimal in the sense that a slight improvement of our soundness would have the collapse \[ \mathsf{QMA}^{\mathbb R}(2)=\mathsf{NEXP}. \] Our proof combines symmetric-subspace projections with the relation $\mathsf{QMA}^{+}(1)=\mathsf{NEXP}$ of Bassirian, Fefferman, and Marwaha (ITCS 2024). The main technical ingredient is a dimension-independent de Finetti theorem in Hilbert-Schmidt norm that applies when the number of registers under consideration grows logarithmically.
