Solving the electronic structure problem using the Variational Quantum Eigensolver (VQE) technique involves the measurement of the Hamiltonian expectation value. The current hardware can perform only projective single-qubit measurements, and thus, the Hamiltonian expectation value is obtained by measuring parts of the Hamiltonian rather than the full Hamiltonian. This restriction makes the measurement process inefficient because the number of terms in the Hamiltonian grows as O(N⁴) with the size of the system, N. To optimize the VQE measurement, one can try to group as many Hamiltonian terms as possible for their simultaneous measurement. Single-qubit measurements allow one to group only the terms commuting within the corresponding single-qubit subspaces or qubit-wise commuting. We found that the qubit-wise commutativity between the Hamiltonian terms can be expressed as a graph and the problem of the optimal grouping is equivalent to finding a minimum clique cover (MCC) for the Hamiltonian graph. The MCC problem is NP-hard, but there exist several polynomial heuristic algorithms to solve it approximately. Several of these heuristics were tested in this work for a set of molecular electronic Hamiltonians. On average, grouping qubit-wise commuting terms reduced the number of operators to measure three times less compared to the total number of terms in the considered Hamiltonians.

中文翻译：

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使用最小集团覆盖率的变分量子本征求解器中的测量优化

使用变分本征求解器（VQE）技术解决电子结构问题涉及哈密顿期望值的测量。当前的硬件只能执行投影单量子位测量，因此，汉密尔顿期望值是通过测量部分汉密尔顿而不是全部汉密尔顿获得的。这种限制使得测量过程效率低下，因为在哈密顿术语的数目的增长作为ø（Ñ ⁴）与系统的大小，Ñ。为了优化VQE测量，可以尝试对尽可能多的汉密尔顿项进行同时测量。单量子位测量仅允许将在相应的单量子位子空间内进行换向或逐位换向的项进行分组。我们发现，哈密顿项之间的按位交换性可以表示为图，并且最佳分组的问题等效于为哈密顿图找到最小派系覆盖度（MCC）。MCC问题是NP难的，但存在几种多项式启发式算法可以对其进行近似求解。在这项工作中，针对一组分子电子哈密顿量对这些启发式方法中的几种进行了测试。一般，