There is a folkloric belief that a depth-$\Theta(m)$ quantum circuit is needed to estimate the trace of the product of $m$ density matrices (i.e., a multivariate trace), a subroutine crucial to applications in condensed matter and quantum information science. We prove that this belief is overly conservative by constructing a constant quantum-depth circuit for the task, inspired by the method of Shor error correction. Furthermore, our circuit demands only local gates in a two dimensional circuit -- we show how to implement it in a highly parallelized way on an architecture similar to that of Google's Sycamore processor. With these features, our algorithm brings the central task of multivariate trace estimation closer to the capabilities of near-term quantum processors. We instantiate the latter application with a theorem on estimating nonlinear functions of quantum states with "well-behaved" polynomial approximations.

翻译：存在一种普遍认知：估计$m$个密度矩阵乘积的迹（即多元迹）需要深度为$\Theta(m)$的量子电路，该子程序对凝聚态物理和量子信息科学的应用至关重要。我们证明这一认知过于保守——受Shor纠错方法启发，我们构建了用于该任务的恒定量子深度电路。此外，我们的电路仅需二维电路中的局域门：我们展示了如何在类似谷歌Sycamore处理器的架构上以高度并行化方式实现该电路。这些特性使多元迹估计这一核心任务更贴近近中期量子处理器的能力。我们通过一个关于利用"良态"多项式近似估计量子态非线性函数的定理，对后一应用进行了实例化论证。