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. - 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体系下多种模型中学习量子态的复杂性，并得到以下结果：- 我们证明，任何Clifford+$T$电路制备计算伪随机量子态都需要$\Omega(n)$个$T$门，较先前已知的界实现了指数级改进。若线性时间量子安全伪随机函数存在，该界是渐近紧的。- 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$|\psi\rangle$，我们提出一种算法，可输出一个见证保真度至少为$\tau - \varepsilon$的稳定子态的简洁描述。该算法使用$O(n/(\varepsilon^2\tau^4))$个样本，运行时间为$\exp\left(O(n/\tau^4)\right) / \varepsilon^2$。在$\tau$为常数区域中，该算法估计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。- 我们改进了Gross、Nezami与Walter [Comms. Math. Phys. 385 (2021)]提出的稳定子态性质测试算法的可靠性分析。作为应用，我们展示了一种容错性稳定子态性质测试算法。我们所有结果中的核心算法原语是贝尔差采样。为证明我们的结果，我们建立并/或加强了贝尔差采样、辛傅里叶分析以及图论之间的联系。