We study the complexity of learning quantum states in various models with respect to the stabilizer formalism and obtain the following results: - We prove that $\Omega(n)$ $T$-gates are necessary for any Clifford+$T$ circuit to prepare computationally pseudorandom quantum states, an exponential improvement over the previously known bound. This bound is asymptotically tight if linear-time quantum-secure pseudorandom functions exist. - Given an $n$-qubit pure quantum state $|\psi\rangle$ that has fidelity at least $\tau$ with some stabilizer state, we give an algorithm that outputs a succinct description of a stabilizer state that witnesses fidelity at least $\tau - \varepsilon$. The algorithm uses $O(n/(\varepsilon^2\tau^4))$ samples and $\exp\left(O(n/\tau^4)\right) / \varepsilon^2$ time. In the regime of $\tau$ constant, this algorithm estimates stabilizer fidelity substantially faster than the na\"ive $\exp(O(n^2))$-time brute-force algorithm over all stabilizer states. - In the special case of $\tau > \cos^2(\pi/8)$, we show that a modification of the above algorithm runs in polynomial time. - We improve the soundness analysis of the stabilizer state property testing algorithm due to Gross, Nezami, and Walter [Comms. Math. Phys. 385 (2021)]. As an application, we exhibit a tolerant property testing algorithm for stabilizer states. The underlying algorithmic primitive in all of our results is Bell difference sampling. To prove our results, we establish and/or strengthen connections between Bell difference sampling, symplectic Fourier analysis, and graph theory.

翻译：我们研究了在稳定子形式框架下量子态学习的复杂性，并获得了以下结果：- 我们证明对于任何Clifford+$T$电路而言，制备计算伪随机量子态至少需要$\Omega(n)$个$T$门，这一结果相较此前已知的界呈指数级改进。若线性时间量子安全伪随机函数存在，该下界是渐进紧的。- 给定一个与某个稳定子态保真度至少为$\tau$的$n$量子比特纯量子态$|\psi\rangle$，我们提出了一种算法，可输出能够见证保真度至少为$\tau - \varepsilon$的稳定子态的简洁描述。该算法使用$O(n/(\varepsilon^2\tau^4))$个样本和$\exp\left(O(n/\tau^4)\right) / \varepsilon^2$时间。在$\tau$为常数的情况下，该算法估算稳定子保真度的速度显著快于对所有稳定子态进行暴力搜索的朴素$\exp(O(n^2))$时间算法。- 在$\tau > \cos^2(\pi/8)$的特殊情形下，我们证明上述算法的改进版本可在多项式时间内运行。- 我们改进了Gross、Nezami和Walter [Comms. Math. Phys. 385 (2021)]提出的稳定子态性质测试算法的合理性分析。作为应用，我们展示了一种容忍性稳定子态性质测试算法。我们所有结果的核心算法原语是贝尔差分采样。为证明我们的结果，我们建立并/或强化了贝尔差分采样、辛傅里叶分析与图论之间的关联。