Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic estimation
Sitan Chen, Weiyuan Gong, Qi Ye, Zhihan Zhang
TL;DR
The paper introduces stabilizer bootstrapping, a general framework for agnostic tomography that yields efficient protocols across several structured state classes. By iteratively collecting high-correlation stabilizers via Bell difference sampling and amplifying fidelity with low-correlation projections, the authors derive polynomial-to-quasi-polynomial time algorithms for agnostic tomography of stabilizer states, high-stabilizer-dimension states, discrete product states, and stabilizer-product states, with corresponding list-decoding guarantees. A key corollary is an efficient estimator for stabilizer fidelity (magic) in quasipolynomial time, enabling practical magic quantification in $n$-qubit systems. The paper also provides information-theoretic lower bounds and discusses limitations and open questions, including whether the 1/τ dependence can be tightened and how to extend the framework to broader state classes. Overall, stabilizer bootstrapping offers a unified, scalable approach to agnostic quantum state learning beyond realizable models, with impactful implications for quantum device characterization and quantum magic estimation.
Abstract
We study the task of agnostic tomography: given copies of an unknown $n$-qubit state $ρ$ which has fidelity $τ$ with some state in a given class $C$, find a state which has fidelity $\ge τ- ε$ with $ρ$. 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 $\mathrm{poly}(n,1/ε)\cdot (1/τ)^{O(\log(1/τ))}$, answering an open question posed by Grewal, Iyer, Kretschmer, Liang [43] and Anshu and Arunachalam [6]. Previous protocols ran in time $\mathrm{exp}(Θ(n))$ or required $τ>\cos^2(π/8)$. States with stabilizer dimension $n - t$: We give a protocol that runs in time $n^3\cdot(2^t/τ)^{O(\log(1/ε))}$, extending recent work on learning quantum states prepared by circuits with few non-Clifford gates, which only applied in the realizable setting where $τ= 1$ [33, 40, 49, 66]. Discrete product states: If $C = K^{\otimes n}$ for some $μ$-separated discrete set $K$ of single-qubit states, we give a protocol that runs in time $(n/μ)^{O((1 + \log (1/τ))/μ)}/ε^2$. 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 $(n^2/ε^2)\cdot (1/τ)^{O(\log(1/τ))}$. As a corollary, we give the first protocol for estimating stabilizer fidelity, a standard measure of magic for quantum states, to error $ε$ in $n^3 \mathrm{quasipoly}(1/ε)$ time.