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). 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 task of multivariate trace estimation, crucial to applications in condensed matter and estimating nonlinear functions of quantum states, 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：有一种民间信念认为，估计m个密度矩阵乘积的迹（即多元迹）需要深度为Θ(m)的量子电路。我们通过构造一个用于该任务的恒定量子深度电路，证明这一信念过于保守，其灵感来自Shor纠错方法。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的方式实现它。凭借这些特性，我们的算法使多元迹估计任务（对凝聚态物理和量子态非线性函数估计等应用至关重要）更接近近期量子处理器的能力边界。我们通过一个关于利用“良好行为”多项式逼近估计量子态非线性函数的定理，实例化了后一个应用。