费米子和玻色子高斯态的最优测试
Optimal testing of fermionic and bosonic Gaussian states
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中文总结 AI 辅助
本文针对五类高斯量子态提出常数样本复杂度的性质测试器，证明紧的 $\Theta(1/\varepsilon^2)$ 样本复杂度，并通过谱间隙下界和匹配下界确立最优性，且扩展到容忍设置。
中文摘要 AI 辅助
我们研究量子态性质测试的问题：给定一个纯态 $|\psi\rangle$，它要么（A）属于某类纯态 $C$，要么（B）在迹距离上与 $C$ 中所有态相距 $\varepsilon$，确定哪种情况以高概率发生。已知存在某些类 $C$，其性质测试的样本复杂度必然随系统规模缩放。在本工作中，我们确定了仅需与规模无关的样本数即可进行测试的类。我们证明了五个重要高斯量子态类的性质测试具有紧的样本复杂度 $\Theta(1/\varepsilon^2)$：1）费米子高斯态，2）斯莱特行列式，3）玻色子高斯态，4）零均值玻色子高斯态，5）玻色子相干态。虽然其中一些类的性质测试器此前已被研究，但那些分析仅给出了随模式数多项式缩放的上界。在每种情况下，我们的测试器是两或三个副本的“顶部”不可约表示的投影。我们通过下界化拒绝概率随与类距离增加的速度来获得常数样本复杂度；主要技术部分涉及界定测试投影子对称扩展的谱间隙。然后，我们通过为每个类证明基于二进制态区分的匹配下界来确立最优性。我们的测试器和最优性声明也扩展到容忍设置，其中在情况（A）中 $\psi$ 可能与 $C$ 相距 $O(\varepsilon)$，而非完全位于其中。
英文摘要
We study the problem of property testing quantum states: given a pure state $|ψ\rangle$ that either (A) belongs to some class $C$ of pure states, or (B) is $\varepsilon$-far in trace distance from all states in $C$, determine which is the case with high probability. It is known that there exists classes $C$ for which the sample complexity of property testing necessarily scales with the size of the system. In this work, we identify classes for which testing only requires a size-independent number of samples. We prove a tight sample complexity of $Θ(1/\varepsilon^2)$ for the property testing of five important classes of Gaussian quantum states: 1) Fermionic Gaussian states, 2) Slater determinants, 3) bosonic Gaussian states, 4) zero-mean bosonic Gaussian states, and 5) bosonic coherent states. While property testers for some of these classes have been studied previously, those analyses only gave upper bounds scaling polynomially in the number of modes. In each case, our tester is the projection onto the ``top'' irrep of two or three copies. We get to constant sample complexity by lower bounding how quickly the rejection probability increases with distance from the class; the main technical component involves bounding the spectral gap of symmetric extension of the test projector. We then establish optimality by proving matching lower bounds based on binary state discrimination for each class. Our testers and optimality claims also extend to the tolerant setting, wherein $ψ$ may be $O(\varepsilon)$-close to $C$ in case (A), rather than lying exactly inside.
发表机构
- University of Texas at Austin（德克萨斯大学奥斯汀分校）
- IBM Research（IBM研究院）
- QuICS, University of Maryland（马里兰大学量子信息与计算科学中心）
- Sandia National Laboratories（桑迪亚国家实验室）
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