Ye Qi

Sitan Chen, Weiyuan Gong, Qi Ye et al.

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.

Weiyuan Gong, Jonas Haferkamp, Qi Ye et al.

We study the sample complexity of the prototypical tasks quantum purity estimation and quantum inner product estimation. In purity estimation, we are to estimate $tr(ρ^2)$ of an unknown quantum state $ρ$ to additive error $ε$. Meanwhile, for quantum inner product estimation, Alice and Bob are to estimate $tr(ρσ)$ to additive error $ε$ given copies of unknown quantum state $ρ$ and $σ$ using classical communication and restricted quantum communication. In this paper, we show a strong connection between the sample complexity of purity estimation with bounded quantum memory and inner product estimation with bounded quantum communication and unentangled measurements. We propose a protocol that solves quantum inner product estimation with $k$-qubit one-way quantum communication and unentangled local measurements using $O(median\{1/ε^2,2^{n/2}/ε,2^{n-k}/ε^2\})$ copies of $ρ$ and $σ$. Our protocol can be modified to estimate the purity of an unknown quantum state $ρ$ using $k$-qubit quantum memory with the same complexity. We prove that arbitrary protocols with $k$-qubit quantum memory that estimate purity to error $ε$ require $Ω(median\{1/ε^2,2^{n/2}/\sqrtε,2^{n-k}/ε^2\})$ copies of $ρ$. This indicates the same lower bound for quantum inner product estimation with one-way $k$-qubit quantum communication and classical communication, and unentangled local measurements. For purity estimation, we further improve the lower bound to $Ω(\max\{1/ε^2,2^{n/2}/ε\})$ for any protocols using an identical single-copy projection-valued measurement. Additionally, we investigate a decisional variant of quantum distributed inner product estimation without quantum communication for mixed state and provide a lower bound on the sample complexity.