A leading proposal for verifying near-term quantum supremacy experiments on noisy random quantum circuits is linear cross-entropy benchmarking. For a quantum circuit $C$ on $n$ qubits and a sample $z \in \{0,1\}^n$, the benchmark involves computing $|\langle z|C|0^n \rangle|^2$, i.e. the probability of measuring $z$ from the output distribution of $C$ on the all zeros input. Under a strong conjecture about the classical hardness of estimating output probabilities of quantum circuits, no polynomial-time classical algorithm given $C$ can output a string $z$ such that $|\langle z|C|0^n\rangle|^2$ is substantially larger than $\frac{1}{2^n}$ (Aaronson and Gunn, 2019). On the other hand, for a random quantum circuit $C$, sampling $z$ from the output distribution of $C$ achieves $|\langle z|C|0^n\rangle|^2 \approx \frac{2}{2^n}$ on average (Arute et al., 2019).
In analogy with the Tsirelson inequality from quantum nonlocal correlations, we ask: can a polynomial-time quantum algorithm do substantially better than $\frac{2}{2^n}$? We study this question in the query (or black box) model, where the quantum algorithm is given oracle access to $C$. We show that, for any $\varepsilon \ge \frac{1}{\mathrm{poly}(n)}$, outputting a sample $z$ such that $|\langle z|C|0^n\rangle|^2 \ge \frac{2 + \varepsilon}{2^n}$ on average requires at least $\Omega\left(\frac{2^{n/4}}{\mathrm{poly}(n)}\right)$ queries to $C$, but not more than $O\left(2^{n/3}\right)$ queries to $C$, if $C$ is either a Haar-random $n$-qubit unitary, or a canonical state preparation oracle for a Haar-random $n$-qubit state. We also show that when $C$ samples from the Fourier distribution of a random Boolean function, the naive algorithm that samples from $C$ is the optimal 1-query algorithm for maximizing $|\langle z|C|0^n\rangle|^2$ on average.

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量子霸权 Tsirelson 不等式

在嘈杂的随机量子电路上验证近期量子霸权实验的一个主要建议是线性交叉熵基准测试。对于 $n$ 量子位上的量子电路 $C$ 和样本 $z \in \{0,1\}^n$，基准涉及计算 $|\langle z|C|0^n \rangle|^2 $，即在全零输入上从 $C$ 的输出分布测量 $z$ 的概率。在关于估计量子电路输出概率的经典难度的强烈猜想下，给定 $C$ 的多项式时间经典算法都不能输出字符串 $z$ 使得 $|\langle z|C|0^n\rangle|^ 2$ 远大于 $\frac{1}{2^n}$（Aaronson 和 Gunn，2019 年）。另一方面，对于随机量子电路 $C$，从 $C$ 的输出分布中采样 $z$ 实现 $|\langle z|C|0^n\rangle|^2 \approx \frac{2} {2^n}$ 平均（Arute 等人，
与量子非局域相关性中的 Tsirelson 不等式类似，我们问：多项式时间量子算法能比 $\frac{2}{2^n}$ 做得更好吗？我们在查询（或黑盒）模型中研究这个问题，其中量子算法被授予 oracle 访问 $C$ 的权限。我们证明，对于任何 $\varepsilon \ge \frac{1}{\mathrm{poly}(n)}$，输出一个样本 $z$ 使得 $|\langle z|C|0^n\rangle| ^2 \ge \frac{2 + \varepsilon}{2^n}$ 平均至少需要 $\Omega\left(\frac{2^{n/4}}{\mathrm{poly}(n)} \right)$ 查询 $C$，但不超过 $O\left(2^{n/3}\right)$ 查询 $C$，如果 $C$ 是 Haar-random $n$- qubit unitary，或 Haar-random $n$-qubit 状态的规范状态准备预言机。我们还表明，当 $C$ 从随机布尔函数的傅立叶分布中采样时，