We develop new approximate compilation schemes that significantly reduce the expense of compiling the Quantum Approximate Optimization Algorithm (QAOA) for solving the Max-Cut problem. Our main focus is on compilation with trapped-ion simulators using Pauli-$X$ operations and all-to-all Ising Hamiltonian $H_\text{Ising}$ evolution generated by Molmer-Sorensen or optical dipole force interactions, though some of our results also apply to standard gate-based compilations. Our results are based on principles of graph sparsification and decomposition; the former reduces the number of edges in a graph while maintaining its cut structure, while the latter breaks a weighted graph into a small number of unweighted graphs. Though these techniques have been used as heuristics in various hybrid quantum algorithms, there have been no guarantees on their performance, to the best of our knowledge. This work provides the first provable guarantees using sparsification and decomposition to improve quantum noise resilience and reduce quantum circuit complexity. For quantum hardware that uses edge-by-edge QAOA compilations, sparsification leads to a direct reduction in circuit complexity. For trapped-ion quantum simulators implementing all-to-all $H_\text{Ising}$ pulses, we show that for a $(1-\epsilon)$ factor loss in the Max-Cut approximation ($\epsilon>0)$, our compilations improve the (worst-case) number of $H_\text{Ising}$ pulses from $O(n^2)$ to $O(n\log(n/\epsilon))$ and the (worst-case) number of Pauli-$X$ bit flips from $O(n^2)$ to $O\left(\frac{n\log(n/\epsilon)}{\epsilon^2}\right)$ for $n$-node graphs. We demonstrate significant reductions in noise are obtained in our new compilation approaches using theory and numerical calculations for trapped-ion hardware. We anticipate these approximate compilation techniques will be useful tools in a variety of future quantum computing experiments.

翻译：我们开发了新的近似编译方案，可显著降低用于求解最大割问题的量子近似优化算法（QAOA）的编译开销。我们的研究主要集中于利用囚禁离子模拟器进行编译，该模拟器采用泡利-$X$操作以及由莫尔-索伦森相互作用或光偶极力相互作用生成的全连接伊辛哈密顿量$H_\text{Ising}$演化，尽管我们的部分结果也适用于标准的基于量子门的编译方案。我们的研究成果基于图稀疏化与分解原理：前者在保持图的割结构的同时减少其边数，后者将加权图分解为少量无权图。尽管这些技术已在多种混合量子算法中作为启发式方法使用，但据我们所知，其性能尚未得到理论保证。本研究首次提供了可证明的保证，利用稀疏化与分解来提升量子噪声鲁棒性并降低量子电路复杂度。对于采用逐边QAOA编译的量子硬件，稀疏化可直接降低电路复杂度。对于实现全连接$H_\text{Ising}$脉冲的囚禁离子量子模拟器，我们证明在最大割近似损失因子为$(1-\epsilon)$（$\epsilon>0$）的条件下，我们的编译方案将（最坏情况下）$H_\text{Ising}$脉冲数从$O(n^2)$改进至$O(n\log(n/\epsilon))$，并将（最坏情况下）泡利-$X$比特翻转数从$O(n^2)$改进至$O\left(\frac{n\log(n/\epsilon)}{\epsilon^2}\right)$（针对$n$节点图）。通过理论分析与针对囚禁离子硬件的数值计算，我们证明了新编译方案能实现显著的噪声抑制。我们预期这些近似编译技术将成为未来各类量子计算实验中的重要工具。