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:我们提供了一种新的方法,用于汇编在Trotter、qDRIFT和多产品公式中出现的量子模拟电路,这些电路出现在克里福德和非克利福德操作中,可以将非克利福德操作的数量减少高达4美元的系数。事实上,在许多情况下,门的总数会减少许多。我们表明,可以实施一种推算式总和,即用最多(受控)美元对保罗斯进行通勤,甚至以美元(调控)旋转门为单位,在那里,美元是不同的非零电子价值的数量(点火标志)。因此,我们可以将汉密尔顿术语相互通缩入能够满足这项工作中确定的若干对称的组合之一,这样可以使整个术语组的模拟费用便宜。我们进一步表明,在某些情况下,可以将汉密尔顿术语部分分配给几个组,提供一种可以贪婪地将术语分配给适当集团的多时制算法。我们还具体讨论了这些优化的超度动态案例,并在以前为4DRIF-H 和4 美元的递减系数组中提供了广泛的数字模拟。