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：我们提出了一种编译量子模拟电路的新方法，针对出现在Trotter、qDRIFT及多乘积公式中的电路，将其编译为Clifford与非Clifford操作，可使非Clifford操作的数量减少至多4倍。事实上，在许多情况下，总门数也会减少。我们证明，实现一个指数化对易泡利算符和至多需要m个（受控）旋转门，其中m是不同非零特征值的数量（忽略符号）。因此，我们可以将相互对易的哈密顿项收集为若干组，这些组满足本工作中确定的若干对称性之一，从而实现对整组项的低成本模拟。我们进一步表明，在某些情况下，通过将哈密顿项部分分配到多个组中可降低成本，并提供一种多项式时间经典算法，能够贪婪地将项分配到合适的组中。我们还专门讨论了这些优化在费米子动力学情况下的应用，并对我们的分组策略在qDRIFT中针对6量子比特和4量子比特海森堡模型、LiH、H₂进行了大量数值模拟，观察到非Clifford门数量减少了1.8至3.2倍。这表明基于Trotter的二次量子化化学模拟可能比此前认为的更加实用。