这篇论文揭示了相干量子记忆是实现稳定子状态测试与学习分离的关键资源，给出了精确的样本复杂度上下界，对量子计算理论研究者有启发。
该论文研究在最多k量子比特相干量子记忆限制下，n量子比特稳定子状态的测试与学习。结果表明测试的样本复杂度为Θ(n−k)，而学习的样本复杂度非自适应时为Θ(n^2/k)。与无记忆限制时测试仅需6个副本不同，即使k=0.99n，测试也无法常数副本完成；当k=cn(0<c<1)时，测试与学习同样困难，均需Θ(n)个副本。此外，论文还证明纯化测试在允许连贯记忆时也有指数下界。
Optimal Stabilizer Testing and Learning with Limited Quantum Memory
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.