自适应在保真度量子态测试中的威力
On the Power of Adaptivity in Testing Quantum States in Fidelity
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中文总结 AI 辅助
本研究探讨量子态认证、等价性和独立性测试在保真度度量下的样本复杂度，证明认证无需自适应，而等价性测试中自适应可显著降低样本需求，并给出相应算法与下界。
中文摘要 AI 辅助
我们研究了量子态认证、等价性测试和独立性测试问题。在认证问题中，给定未知量子态$\rho$的样本以及态$\sigma$的描述，目标是测试是否$\rho=\sigma$，或者$\rho$和$\sigma$在给定的距离度量下是否相距很远。在等价性测试中，$\sigma$也是未知的，只能通过样本访问。独立性测试决定$\rho_{AC}=\rho_A\otimes\rho_C$是否成立，或者是否远离乘积态。这些问题在迹距离下对于决策间隙$\varepsilon$的样本复杂度现已得到充分理解：在单拷贝测量设置下，对于$d$维态，所有三个任务都可以使用相同的非自适应方法解决，该方法使用$\Theta(d^{3/2}/\varepsilon^2)$个样本，并且即使没有自适应，通常也是最优的。在这项工作中，我们考虑以保真度表示的决策间隙，并研究这些问题之间可能的分离以及自适应如何提供帮助。我们证明，对于秩为$r$的态$\sigma$，关于保真度的认证问题不会从自适应中受益，并且需要$\widetilde{\Theta}(r^{3/2}/\varepsilon)$个样本。对于等价性测试和独立性测试，我们提供了自适应算法，分别使用$\widetilde{O}(\min\{d^{3/2}/\varepsilon^2,d^{9/4}/\varepsilon\})$和$\widetilde{O}(\min\{(d_Ad_C)^{3/2}/\varepsilon^2,d_A^{9/4}d_C^{3/4}/\varepsilon\})$个样本，其中$d_A\geq d_C$。我们的主要技术是一个框架，该框架使用部分学习并归约到$\ell_2$距离下的测试，改编自分布测试文献。我们通过证明在非自适应情况下即使对于量子比特也需要$\widetilde{\Omega}(1/\varepsilon^2)$个样本，表明自适应在保真度等价性测试中很重要，从而显示出与认证问题的分离。
英文摘要
We study the problems of quantum state certification, equivalence testing and independence testing. In certification, given samples of an unknown quantum state $ρ$ and the description of a state $σ$, the goal is to test whether $ρ=σ$, or whether $ρ$ and $σ$ are far in a given distance measure. In equivalence testing, $σ$ is also unknown and only accessible via samples. Independence testing decides whether $ρ_{AC}=ρ_A\otimesρ_C$, or is far from being a product. The sample complexities of these problems are now well-understood for a decision gap $\varepsilon$ in trace distance: in the single-copy measurement setting with $d$-dimensional states, all three tasks can be solved using the same non-adaptive approach, which uses $Θ(d^{3/2}/\varepsilon^2)$ samples and is optimal in general, even without adaptivity. In this work, we consider decision gaps expressed in fidelity and study possible separations between these problems and how adaptivity can help. We prove that certification with respect to fidelity for a state $σ$ of rank $r$ does not benefit from adaptivity and requires $\widetildeΘ(r^{3/2}/\varepsilon)$ samples. For equivalence testing and independence testing, we provide adaptive algorithms using $\widetilde{O}(\min\{d^{3/2}/\varepsilon^2,d^{9/4}/\varepsilon\})$ and $\widetilde{O}(\min\{(d_Ad_C)^{3/2}/\varepsilon^2,d_A^{9/4}d_C^{3/4}/\varepsilon\})$ samples, for $d_A\geq d_C$, respectively. Our main technique is a framework that uses partial learning and a reduction to testing in $\ell_2$-distance, adapted from the distribution testing literature. We show that adaptivity matters for equivalence testing in fidelity by proving that $\widetildeΩ(1/\varepsilon^2)$ samples are necessary in the non-adaptive case even for qubits, showing a separation from certification.
发表机构
- National University of Singapore（新加坡国立大学）
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