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$C$中某个态的保真度为$\tau$，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，并据此获得以下类别的新不可知层析协议：稳定子态：我们提出运行时间为$\mathrm{poly}(n,1/\epsilon)\cdot (1/\tau)^{O(\log(1/\tau))}$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$\mathrm{exp}(\Theta(n))$运行时间或要求$\tau>\cos^2(\pi/8)$。稳定子维度为$n - t$的态：我们提出运行时间为$n^3\cdot(2^t/\tau)^{O(\log(1/\epsilon))}$的协议，扩展了近期关于通过含少量非克利福德门电路制备量子态的学习研究，该研究仅适用于$\tau = 1$的可实现场景[33, 40, 49, 66]。离散乘积态：若$C = K^{\otimes n}$（其中$K$为$\mu$分离的单量子比特态离散集），我们提出运行时间为$(n/\mu)^{O((1 + \log (1/\tau))/\mu)}/\epsilon^2$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（量子态魔力量度的标准度量）的协议。