We provide a new approach for compiling quantum simulation circuits that appear in Trotter, qDRIFT and multi-product formulas to Clifford and non-Clifford operations that can reduce the number of non-Clifford operations by a factor of up to $4$. In fact, the total number of gates reduce in many cases. We show that it is possible to implement an exponentiated sum of commuting Paulis with at most $m$ (controlled)-rotation gates, where $m$ is the number of distinct non-zero eigenvalues (ignoring sign). Thus we can collect mutually commuting Hamiltonian terms into groups that satisfy one of several symmetries identified in this work which allow an inexpensive simulation of the entire group of terms. We further show that the cost can in some cases be reduced by partially allocating Hamiltonian terms to several groups and provide a polynomial time classical algorithm that can greedily allocate the terms to appropriate groupings. We further specifically discuss these optimizations for the case of fermionic dynamics and provide extensive numerical simulations for qDRIFT of our grouping strategy to 6 and 4-qubit Heisenberg models, $LiH$, $H_2$ and observe a factor of 1.8-3.2 reduction in the number of non-Clifford gates. This suggests Trotter-based simulation of chemistry in second quantization may be even more practical than previously believed.

翻译：我们提出了一种新方法，用于编译特罗特、qDRIFT及多乘积公式中的量子模拟电路，将其转化为克利福德和非克利福德操作，从而将非克利福德操作的数量减少至原来的$1/4$。事实上，在许多情况下，总门数也会减少。我们证明，对于一组对易的泡利算符的指数和，其实现最多只需$m$个（受控）旋转门，其中$m$是不同非零特征值（忽略符号）的个数。因此，我们可以将相互对易的哈密顿项收集到满足本文识别的若干对称性的组中，从而以较低成本模拟整个项组。我们进一步表明，在某些情况下，可以通过将哈密顿项部分分配到多个组来降低成本，并提出了一个多项式时间的经典算法，该算法能贪婪地将项分配到合适的组中。我们特别针对费米子动力学情形讨论了这些优化，并对qDRIFT中的分组策略在6量子比特和4量子比特的海森堡模型、$LiH$、$H_2$上进行了广泛的数值模拟，观察到非克利福德门数量减少了1.8-3.2倍。这表明基于特罗特的二次量子化化学模拟可能比先前认为的更为实用。