Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic estimation
This addresses the challenge of efficiently characterizing quantum states in noisy or imperfect settings, with applications in quantum computing and magic estimation, though it is incremental as it builds on prior work in tomography.
The paper tackles the problem of agnostic tomography for unknown quantum states, introducing a new framework called stabilizer bootstrapping that yields computationally efficient protocols for various state classes, achieving runtimes such as poly(n,1/ε)·(1/τ)^{O(log(1/τ))} for stabilizer states and n^3·(2^t/τ)^{O(log(1/ε))} for states with stabilizer dimension n-t.
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.