There has been significant interest in understanding how practical constraints on contemporary quantum devices impact the complexity of quantum learning. For the classic question of tomography, recent work tightly characterized the copy complexity for any protocol that can only measure one copy of the unknown state at a time, showing it is polynomially worse than if one can make fully-entangled measurements. While we now have a fairly complete picture of the rates for such tasks in the near-term and fault-tolerant regimes, it remains poorly understood what the landscape in between looks like. In this work, we study tomography in the natural setting where one can make measurements of $t$ copies at a time. For sufficiently small $\epsilon$, we show that for any $t \le d^2$, $\widetilde{\Theta}(\frac{d^3}{\sqrt{t}\epsilon^2})$ copies are necessary and sufficient to learn an unknown $d$-dimensional state $\rho$ to trace distance $\epsilon$. This gives a smooth and optimal interpolation between the known rates for single-copy and fully-entangled measurements. To our knowledge, this is the first smooth entanglement-copy tradeoff known for any quantum learning task, and for tomography, no intermediate point on this curve was known, even at $t = 2$. An important obstacle is that unlike the optimal single-copy protocol, the optimal fully-entangled protocol is inherently biased and thus precludes naive batching approaches. Instead, we devise a novel two-stage procedure that uses Keyl's algorithm to refine a crude estimate for $\rho$ based on single-copy measurements. A key insight is to use Schur-Weyl sampling not to estimate the spectrum of $\rho$, but to estimate the deviation of $\rho$ from the maximally mixed state. When $\rho$ is far from the maximally mixed state, we devise a novel quantum splitting procedure that reduces to the case where $\rho$ is close to maximally mixed.

翻译：当前量子设备在实际操作中的约束如何影响量子学习的复杂性，已成为研究热点。针对经典的状态层析问题，最新工作严格刻画了每次仅能测量单个未知拷贝的协议所需的拷贝复杂度，结果表明其复杂度比允许全纠缠测量时呈多项式增长。尽管我们已对近中期及容错量子计算场景下的速率有较完整认知，但两者之间的过渡区域仍缺乏清晰理解。本研究关注每次可对$t$个拷贝进行测量的自然场景，在$\epsilon$足够小的条件下，我们证明：对于任意$t \le d^2$，学习未知$d$维量子态$\rho$至迹距离$\epsilon$所需拷贝数量的下界与上界均为$\widetilde{\Theta}(\frac{d^3}{\sqrt{t}\epsilon^2})$。该结果实现了单拷贝测量与全纠缠测量两种已知速率之间的光滑最优插值。据我们所知，这是首个在任何量子学习任务中发现的平滑纠缠-拷贝折衷关系，而此前状态层析在该曲线上（即便在$t=2$时）均无任何中间点存在。核心难点在于：最优单拷贝协议相反，最优全纠缠协议具有固有偏差性，无法直接采用朴素批处理方法。为此，我们设计了一种新型两阶段方案：首先利用Keyl算法，基于单拷贝测量获得对$\rho$的粗略估计。关键洞见在于采用Schur-Weyl采样并非用于估计$\rho$的谱分布，而是估计$\rho$与最大混合态的偏差。当$\rho$远离最大混合态时，我们提出新型量子拆分过程，将其约化至$\rho$接近最大混合态的情形。