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 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量子态学习的复杂性，并获得以下结果： - 我们证明，任何Clifford+$T$电路制备计算伪随机量子态至少需要$\Omega(n)$个$T$门，这相比先前已知的界实现了指数级改进。若存在线性时间量子安全伪随机函数，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，能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$\tau$为常数的情形下，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$\exp(O(n^2))$时间暴力搜索的朴素算法。 - 对于$\tau > \cos^2(\pi/8)$的特殊情形，我们证明上述算法的改进版本可在多项式时间内运行。 - 我们展示了一种针对稳定子态的容错性质检验算法。 所有结果的核心算法基元是贝尔差分采样。为证明结论，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联系。