Many applications of quantum simulation require one to prepare and then characterize quantum states by efficiently estimating $k$-body reduced density matrices ($k$-RDMs), from which observables of interest may be obtained. For instance, the fermionic 2-RDM contains the energy, charge density, and energy gradients of an electronic system, while the qubit 2-RDM contains the spatial correlation functions of magnetic systems. Naive estimation of such RDMs requires repeated state preparations for each matrix element, which makes for prohibitively large computation times. However, commuting matrix elements may be measured simultaneously, allowing for a significant cost reduction. In this work, we design schemes for such a parallelization with near-optimal complexity in the system size $N$. We first describe a scheme to sample all elements of a qubit $k$-RDM using only $\mathcal{O}(3^k{\log }^{k-1}N)$ unique measurement circuits, an exponential improvement over prior art. We then describe a scheme for sampling all elements of the fermionic 2-RDM using only $\mathcal{O}(N^2)$ unique measurement circuits, each of which requires only a local $\mathcal{O}(N)$-depth measurement circuit. We prove a lower bound of $\mathrm{\Omega }({\epsilon }^{-2}{N}^k)$ on the number of state preparations, Clifford circuits, and measurement in the computational basis required to estimate all elements of a fermionic $k$-RDM, making our scheme for sampling the fermionic 2-RDM asymptotically optimal. We finally construct circuits to sample the expectation value of a linear combination of $\omega$ anticommuting two-body fermionic operators with only $\mathcal{O}(\omega )$ gates on a linear array. These circuits allows for sampling any linear combination of fermionic 2-RDM elements in $\mathcal{O}(N^4/\omega )$ time, with a significantly lower measurement circuit complexity than prior art. Our results improve the viability of near-term quantum simulation of molecules and strongly correlated material systems.

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量子态部分层析成像的近最佳测量调度

量子仿真的许多应用都需要准备并通过有效估计来表征量子态 $ķ$体密度矩阵$ķ$-RDM），可以从中获得感兴趣的可观测值。例如，铁离子2-RDM包含电子系统的能量，电荷密度和能量梯度，而量子位2-RDM包含磁性系统的空间相关函数。这种RDM的幼稚估计需要为每个矩阵元素重复进行状态准备，这会导致计算时间过长。但是，可以同时测量通勤矩阵元素，从而可以显着降低成本。在这项工作中，我们设计了这样的并行化方案，并且系统大小具有接近最佳的复杂度$ñ$。我们首先描述一种采样量子位的所有元素的方案$ķ$-RDM仅使用 $\mathcal{Ø}（3^{ķ}{\mathrm{日志}}^{ķ-\mathrm{1个}}ñ）$独特的测量电路，是对现有技术的指数级改进。然后，我们描述了仅使用一个采样费米电子2-RDM的所有元素的方案$\mathcal{Ø}（{ñ}^2）$ 独特的测量电路，每个电路仅需要一个本地 $\mathcal{Ø}（ñ）$深度测量电路。我们证明了$\mathrm{\Omega }（{\epsilon }^{-2}{ñ}^{ķ}）$ 关于状态准备，Clifford回路的数量，以及在估算费米离子所有元素所需的计算基础上的测量 $ķ$-RDM，这使我们的渐近最优的铁离子2-RDM采样方案成为可能。最后，我们构造电路以采样以下项的线性组合的期望值：$\omega$ 反换位两体费米子算子 $\mathcal{Ø}（\omega ）$线性阵列上的门。这些电路允许采样铁氧体2-RDM元素的任何线性组合$\mathcal{Ø}（{ñ}^4/\omega ）$与现有技术相比，测量时间复杂度大大降低。我们的结果提高了分子和强相关材料系统的近期量子模拟的可行性。