超越矩匹配的随机量子电路
Random Quantum Circuits Beyond Moment Matching
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中文总结 AI 辅助
本文研究随机量子电路，证明强近似酉k-设计在Kolmogorov距离下以最优界限逼近Porter-Thomas分布，并发现局部不变性可指数改进精度，应用于熵估计。
中文摘要 AI 辅助
随机量子电路旨在高效地复现理想随机量子演化的统计性质。一种方法是构造近似酉设计，其以受控误差匹配Haar随机酉矩阵直至指定阶数的矩。在本工作中，我们建立了这些设计在多大程度上准确复现单个输出概率的完整分布的定量保证。我们证明，对于n量子比特上的每个强ε-近似酉k-设计，每个单个输出概率的分布与有限维Porter-Thomas分布（参数为1和2^n-1的beta分布）之间的Kolmogorov距离为O(√(2^n/(k(k+2^n)))+ε)。该界限在常数因子内是最优的，这意味着要获得实质上更好的均匀界限需要额外的结构。我们进一步证明，局部不变性导致对k的依赖呈指数改进，并且在更强的总变差度量下也有保证：每个在作用于log k+O(1)个量子比特的酉平移下不变的强ε-近似酉k-设计，在Kolmogorov距离上达到误差2^{-Ω(k)}+O(ε)，在总变差距离上达到误差2^{-Ω(k)}+O(ε log(2/ε))。这些结果既识别了矩匹配所保证的分布精度，也识别了能实质改进它的额外结构。作为应用，我们证明如果强近似设计的每个输出概率的边际分布与其Haar对应物的Kolmogorov距离在η以内，那么输出分布的期望Shannon熵与其Haar值的差异为O(√η)。
英文摘要
Random quantum circuits aim to efficiently reproduce the statistical properties of ideal random quantum evolution. One approach is to construct approximate unitary designs, which match the moments of Haar-random unitaries up to a prescribed order with controlled error. In this work, we establish quantitative guarantees for how accurately these designs reproduce the full distributions of individual output probabilities. We show that, for every strong $\varepsilon$-approximate unitary $k$-design on $n$ qubits, the distribution of each individual output probability is within $O\left(\sqrt{\frac{2^n}{k(k+2^n)}}+\varepsilon\right)$ in Kolmogorov distance of the finite-dimensional Porter-Thomas distribution, a beta distribution with parameters $1$ and $2^n-1$. This bound is optimal up to constant factors, implying that a substantially better uniform bound requires additional structure. We further show that local invariance yields an exponential improvement in the dependence on $k$, with guarantees also in the stronger metric of total variation: every strong $\varepsilon$-approximate unitary $k$-design invariant under unitary translations acting on $\log k+O(1)$ qubits achieves error $2^{-Ω(k)}+O(\varepsilon)$ in Kolmogorov distance and $2^{-Ω(k)}+O(\varepsilon\log(2/\varepsilon))$ in total variation distance. These results identify both the distributional accuracy guaranteed by moment matching and the additional structure that substantially improves it. As an application, we show that if the marginal distribution of each output probability of a strong approximate design is within Kolmogorov distance $η$ of its Haar counterpart, then the expected Shannon entropy of the output distribution differs from its Haar value by $O(\sqrtη)$.
发表机构
- National Taiwan University（国立台湾大学）
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