[论文解读] Physics-Constrained Hardware-Efficient Ansatz on Quantum Computers that is Universal, Systematically Improvable, and Size-consistent

Xiaoxiao Xiao, Hewang Zhao+3 more|arXiv (Cornell University)|Jul 7, 2023

Quantum Computing Algorithms and Architecture被引用 4

一句话总结

本文提出了一种物理约束的、硬件高效变分量子算法构造（XYZ2F），该构造具有普适性、可系统性改进且尺寸一致性良好，确保了更优的可扩展性和精度。通过施加基本物理约束并采用线性最近邻量子比特架构，该方法在比启发式构造更少层数的情况下实现了化学精度，尤其在强关联体系（如海森堡模型及H₂O、N₂等分子）中表现卓越。

ABSTRACT

Variational wavefunction ansätze are at the heart of solving quantum many-body problems in physics and chemistry. Previous designs of hardware-efficient ansatz (HEA) on quantum computers are largely based on heuristics and lack rigorous theoretical foundations. In this work, we introduce a physics-constrained approach for designing HEA with rigorous theoretical guarantees by imposing a few fundamental constraints. Specifically, we require that the target HEA to be universal, systematically improvable, and size-consistent, which is an important concept in quantum many-body theories for scalability, but has been overlooked in previous designs of HEA. We extend the notion of size-consistency to HEA, and present a concrete realization of HEA that satisfies all these fundamental constraints while only requiring linear qubit connectivity. The developed physics-constrained HEA is superior to other heuristically designed HEA in terms of both accuracy and scalability, as demonstrated numerically for the Heisenberg model and some typical molecules. In particular, we find that restoring size-consistency can significantly reduce the number of layers needed to reach certain accuracy. In contrast, the failure of other HEA to satisfy these constraints severely limits their scalability to larger systems with more than ten qubits. Our work highlights the importance of incorporating physical constraints into the design of HEA for efficiently solving many-body problems on quantum computers.

研究动机与目标

• 为解决现有硬件高效构造（HEA）设计中缺乏理论基础的问题，这些设计大多基于启发式方法，且在可扩展性和精度方面缺乏保证。
• 借鉴密度泛函理论中的成功方法，建立严格的物理约束——普适性、系统性改进和尺寸一致性——作为HEA的设计原则。
• 开发一种具体的HEA实现（XYZ2F），满足全部三项约束，同时仅需线性量子比特连接和原生门。
• 证明满足这些约束可显著提升收敛速度并减少实现化学精度所需的电路深度。
• 实现在近场量子硬件上对强关联体系（包括分子和自旋模型）的可扩展且精确的变分量子模拟。

提出的方法

• 该构造由具有固定重复结构的参数化量子线路（PQC）的乘积构成，可通过增加层数实现系统性改进。
• 通过确保构造可乘法地分解为子系统，电路架构强制实现尺寸一致性，从而在大系统中保持正确的缩放行为。
• 通过允许每个线路模块表示任意泡利算符的指数，实现普适性，确保构造可逼近任意量子态。
• 该构造仅使用单量子比特旋转和最近邻两量子比特纠缠门，可在线性连接的量子硬件上实现。
• 采用逐层优化策略，以缓解稀释梯度问题并提升可训练性。
• 在海森堡模型和多种分子（包括H₄、H₆、H₈、LiH、H₂O和N₂）上对构造进行了数值测试，以精确对角化和UCCSD为基准。

Figure 1: Heuristically designed HEA and physics-constrained HEA. (a) $R_{y}R_{z}$ full (EfficientSU2) ansatz. The $R_{y}$ full ansatz can be obtained by removing the columns of $R_{z}$ gates, while the $R_{y}$ linear ansatz can be derived from the $R_{y}$ full ansatz by further replacing the all-to

实验结果

研究问题

• RQ1 能否设计出具有严格理论保证（包括普适性、系统性改进和尺寸一致性）的硬件高效构造？
• RQ2 强制实现尺寸一致性如何影响在量子化学模拟中达到化学精度所需的收敛速度和电路深度？
• RQ3 物理约束的构造能否在强关联体系中实现比启发式设计的HEA更优的精度和可扩展性？
• RQ4 逐层优化策略是否能有效缓解大规模变分量子算法中的稀释梯度问题？
• RQ5 所提出的构造能否扩展至二维体系，同时保持其理论和实际优势？

主要发现

• 物理约束的构造XYZ2F在H₄、H₆、H₈、LiH、H₂O和N₂上实现了化学精度（1 mH），且所需层数显著少于其他HEA，展现出更优的收敛性能。
• 在1.5 Å间距的H₃–H₃梯形体系中，XYZ2F仅需约37层即可达到化学精度，而其他HEA如XYZ1F和ASWAP收敛速度慢得多。
• 其他HEA在解离区域（3.0 Å）和中间距离完全失效，凸显了尺寸一致性在可扩展性中的关键作用。
• 非尺寸一致HEA在超过十个量子比特后无法继续扩展，凸显了在大体系模拟中强制实施该约束的必要性。
• 逐层优化策略有效缓解了稀释梯度问题，实现了稳定训练并随着层数增加实现系统性改进。
• 该构造在具有挑战性的强关联区域（如接近解离极限）仍保持高精度，而UCCSD在此区域性能显著下降。

Figure 2: Examples for Theorem 1 with six qubits. (a), (b) quantum circuits appeared in UCC arising from the double excitation $a_{0}^{\dagger}a_{2}^{\dagger}a_{4}a_{5}$ and the single excitation $a_{1}^{\dagger}a_{4}$ , respectively; (c) the XYZ1F circuit. Both (a) and (b) can be represented by (c)