We revisit the problem of Pauli shadow tomography: given copies of an unknown $n$-qubit quantum state $\rho$, estimate $\text{tr}(P\rho)$ for some set of Pauli operators $P$ to within additive error $\epsilon$. This has been a popular testbed for exploring the advantage of protocols with quantum memory over those without: with enough memory to measure two copies at a time, one can use Bell sampling to estimate $|\text{tr}(P\rho)|$ for all $P$ using $O(n/\epsilon^4)$ copies, but with $k\le n$ qubits of memory, $\Omega(2^{(n-k)/3})$ copies are needed. These results leave open several natural questions. How does this picture change in the physically relevant setting where one only needs to estimate a certain subset of Paulis? What is the optimal dependence on $\epsilon$? What is the optimal tradeoff between quantum memory and sample complexity? We answer all of these questions. For any subset $A$ of Paulis and any family of measurement strategies, we completely characterize the optimal sample complexity, up to $\log |A|$ factors. We show any protocol that makes $\text{poly}(n)$-copy measurements must make $\Omega(1/\epsilon^4)$ measurements. For any protocol that makes $\text{poly}(n)$-copy measurements and only has $k < n$ qubits of memory, we show that $\widetilde{\Theta}(\min\{2^n/\epsilon^2, 2^{n-k}/\epsilon^4\})$ copies are necessary and sufficient. The protocols we propose can also estimate the actual values $\text{tr}(P\rho)$, rather than just their absolute values as in prior work. Additionally, as a byproduct of our techniques, we establish tight bounds for the task of purity testing and show that it exhibits an intriguing phase transition not present in the memory-sample tradeoff for Pauli shadow tomography.

翻译：我们重新审视泡利影子层析成像问题：给定未知$n$量子比特量子态$\rho$的副本，估计一组泡利算符$P$的迹$\text{tr}(P\rho)$，要求加法误差不超过$\epsilon$。该问题已成为探索含量子存储器协议相较于无存储器协议优势的经典试验平台：当单次可测量两个副本的存储容量足够时，可通过贝尔采样以$O(n/\epsilon^4)$个副本估计所有$P$的$|\text{tr}(P\rho)|$；但当存储容量仅为$k\le n$量子比特时，所需副本数呈$\Omega(2^{(n-k)/3})$增长。这些结果遗留了若干自然问题：在仅需估计特定泡利子集的物理相关场景下，上述结论如何变化？对$\epsilon$的最优依赖关系是什么？量子存储容量与样本复杂度之间的最优权衡为何？我们完整回答了这些问题。对于任意泡利子集$A$及任意测量策略族，我们完全刻画了最优样本复杂度（至多相差$\log |A|$因子）。我们证明：任何需进行$\text{poly}(n)$次副本测量的协议，必须实施$\Omega(1/\epsilon^4)$次测量。对于仅含$k < n$量子比特存储器、且需进行$\text{poly}(n)$次副本测量的协议，我们证明$\widetilde{\Theta}(\min\{2^n/\epsilon^2, 2^{n-k}/\epsilon^4\})$个副本既是必要的也是充分的。我们提出的协议还可估计$\text{tr}(P\rho)$的真实值，而不仅限于先前工作中的绝对值。此外，作为技术副产品，我们建立了纯度测试任务的紧界，并揭示其中存在泡利影子层析成像存储-样本权衡中未曾出现的奇特相变现象。