有限量子记忆下的最优稳定子测试与学习
Optimal Stabilizer Testing and Learning with Limited Quantum Memory
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中文总结 AI 辅助
研究在有限相干量子记忆下稳定子态的测试与学习，发现测试与学习之间的样本复杂度差距在记忆约束下消失，并给出了精确的样本复杂度界限。
中文摘要 AI 辅助
我们研究在有限相干量子记忆下的稳定子态测试与学习。在此设置中，算法依次接收未知$n$量子比特态的副本，但在测量之间只能保留$k$量子比特的相干量子记忆。在无记忆限制下，Gross、Nezami和Walter的开创性工作表明，使用$6$个副本即可测试$n$量子比特稳定子态，这与维度无关，而学习的复杂度为$\Theta(n)$。我们证明，这种测试与学习之间的分离在记忆约束下消失。更具体地，我们证明：(1) 在$k$量子比特记忆框架下，测试稳定子态的样本复杂度为$\Theta(n-k)$。我们的上界通过一个与隐藏移位问题的新联系得到，下界则通过随机正交群的组合学对似然比进行平均情况界的新方法证明。(2) 在非自适应框架下，使用$k$量子比特记忆学习稳定子态的样本复杂度为$\Theta(n^2/k)$。作为我们技术的进一步应用，我们证明了纯度测试的指数下界，即使在整个协议中记忆保持相干。我们的主要结果表明，相干量子记忆是导致稳定子测试与学习之间通常分离的资源。特别地，即使有$k=0.99n$量子比特的记忆，也不存在常数副本的稳定子测试器；此外，对于$k=cn$量子比特的记忆（$0<c<1$），稳定子测试与学习一样困难，两者都需要$\Theta(n)$个副本。
英文摘要
We study stabilizer state testing and learning with limited coherent quantum memory. Here an algorithm sequentially receives copies of an unknown $n$-qubit state, but may keep only $k$ qubits of coherent quantum memory between measurements. With unrestricted memory, seminal work of Gross, Nezami and Walter showed how to test $n$-qubit stabilizer states using $6$ copies, which is dimension independent, unlike the learning complexity of $Θ(n)$. We show that this testing-vs-learning separation is lost under memory constraints. More concretely we show that (1) The sample complexity of testing stabilizer states in the $k$-qubit memory framework is $Θ(n-k)$. Our upper bound goes via a novel connection to the hidden shift problem and the lower bound is proven using a novel approach to average case bounds on likelihood ratios via combinatorics of the stochastic orthogonal group. (2) The sample complexity of learning stabilizer states with $k$ qubits of memory, in the non-adaptive framework, is $Θ(n^2/k)$. As a further application of our techniques, we prove an exponential lower bound for purity testing even when the memory may be left coherent throughout the protocol. Our main results identify coherent quantum memory as the resource enabling the usual separation between stabilizer testing and learning. In particular, even with $k=0.99n$ qubits of memory, there is no constant-copy stabilizer tester; furthermore for $k=cn$ qubits of memory (for $0< c < 1$), stabilizer testing is as hard as learning, with both requiring $Θ(n)$ copies.
发表机构
- IBM Research（IBM研究院）
- Dahlem Center for Complex Quantum Systems（达赫姆复杂量子系统中心）
- Silicon Valley（硅谷）
机构由 AI 辅助整理，请以论文原文为准。
