We study the task of agnostic tomography: given copies of an unknown $n$-qubit state $\rho$ which has fidelity $\tau$ with some state in a given class $C$, find a state which has fidelity $\ge \tau - \epsilon$ with $\rho$. We give a new framework, stabilizer bootstrapping, for designing computationally efficient protocols for this task, and use this to get new agnostic tomography protocols for the following classes: Stabilizer states: We give a protocol that runs in time $\mathrm{poly}(n,1/\epsilon)\cdot (1/\tau)^{O(\log(1/\tau))}$, answering an open question posed by Grewal, Iyer, Kretschmer, Liang [43] and Anshu and Arunachalam [6]. Previous protocols ran in time $\mathrm{exp}(\Theta(n))$ or required $\tau>\cos^2(\pi/8)$. States with stabilizer dimension $n - t$: We give a protocol that runs in time $n^3\cdot(2^t/\tau)^{O(\log(1/\epsilon))}$, extending recent work on learning quantum states prepared by circuits with few non-Clifford gates, which only applied in the realizable setting where $\tau = 1$ [33, 40, 49, 66]. Discrete product states: If $C = K^{\otimes n}$ for some $\mu$-separated discrete set $K$ of single-qubit states, we give a protocol that runs in time $(n/\mu)^{O((1 + \log (1/\tau))/\mu)}/\epsilon^2$. This strictly generalizes a prior guarantee which applied to stabilizer product states [42]. For stabilizer product states, we give a further improved protocol that runs in time $(n^2/\epsilon^2)\cdot (1/\tau)^{O(\log(1/\tau))}$. As a corollary, we give the first protocol for estimating stabilizer fidelity, a standard measure of magic for quantum states, to error $\epsilon$ in $n^3 \mathrm{quasipoly}(1/\epsilon)$ time.

翻译：我们研究不可知层析任务：给定未知 $n$ 量子比特态 $\rho$ 的若干副本，该态与给定类别 $C$ 中某态具有保真度 $\tau$，需找到一个与 $\rho$ 保真度 $\ge \tau - \epsilon$ 的态。我们提出一种新框架——稳定子自举，用于设计该任务的计算高效协议，并借此为以下类别获得新的不可知层析协议：稳定子态：我们给出运行时间为 $\mathrm{poly}(n,1/\epsilon)\cdot (1/\tau)^{O(\log(1/\tau))}$ 的协议，回答了 Grewal、Iyer、Kretschmer、Liang [43] 以及 Anshu 和 Arunachalam [6] 提出的公开问题。先前协议需要运行时间 $\mathrm{exp}(\Theta(n))$ 或要求 $\tau>\cos^2(\pi/8)$。具有稳定子维数 $n - t$ 的态：我们给出运行时间为 $n^3\cdot(2^t/\tau)^{O(\log(1/\epsilon))}$ 的协议，扩展了近期关于学习由含少量非克利福德门电路制备量子态的工作，该工作仅适用于 $\tau = 1$ 的可实现场景 [33, 40, 49, 66]。离散乘积态：若 $C = K^{\otimes n}$ 其中 $K$ 为某个 $\mu$ 分离的单量子比特态离散集，我们给出运行时间为 $(n/\mu)^{O((1 + \log (1/\tau))/\mu)}/\epsilon^2$ 的协议。这严格推广了先前仅适用于稳定子乘积态的保证 [42]。对于稳定子乘积态，我们进一步提出改进协议，其运行时间为 $(n^2/\epsilon^2)\cdot (1/\tau)^{O(\log(1/\tau))}$。作为推论，我们首次提出在 $n^3 \mathrm{quasipoly}(1/\epsilon)$ 时间内以误差 $\epsilon$ 估计稳定子保真度（量子态魔法的标准度量）的协议。