Recent advances in the field of adiabatic quantum computing and the closely related field of quantum annealing have centered around using more advanced and novel Hamiltonian representations to solve optimization problems. One of these advances has centered around the development of driver Hamiltonians that commute with the constraints of an optimization problem—allowing for another avenue to satisfying those constraints instead of imposing penalty terms for each of them. In particular, the approach is able to use sparser connectivity to embed several practical problems on quantum devices in comparison to the standard approach of using penalty terms. However, designing the driver Hamiltonians that successfully commute with several constraints has largely been based on strong intuition for specific problems and with no simple general algorithm for generating them for arbitrary constraints. In this work, we develop a simple and intuitive algebraic framework for reasoning about the commutation of Hamiltonians with linear constraints—one that allows us to classify the complexity of finding a driver Hamiltonian for an arbitrary set of linear constraints as NP-complete. Because unitary operators are exponentials of Hermitian operators, these results can also be applied to the construction of mixers in the quantum alternating operator ansatz framework.

绝热量子计算领域和量子退火密切相关领域的最新进展集中在使用更先进和新颖的哈密顿表示来解决优化问题。这些进步之一集中在司机哈密顿量的发展上，这些哈密顿量与优化问题的约束相适应——允许另一种途径来满足这些约束，而不是对每个约束强加惩罚项。特别是，与使用惩罚项的标准方法相比，该方法能够使用稀疏连接将几个实际问题嵌入到量子设备上。然而，设计能够成功处理多个约束的驱动程序哈密顿量在很大程度上是基于对特定问题的强烈直觉，并且没有简单的通用算法来为任意约束生成它们。在这项工作中，我们开发了一个简单直观的代数框架，用于推理具有线性约束的哈密顿量的对易——该框架允许我们将为任意一组线性约束寻找驱动程序哈密顿量的复杂性分类为 NP 完全。因为酉算符是厄米算符的指数，所以这些结果也可以应用于量子交替算符 ansatz 框架中的混频器的构造。我们开发了一个简单直观的代数框架，用于推理具有线性约束的哈密顿量的对易——该框架允许我们将为任意一组线性约束寻找驱动者哈密顿量的复杂性分类为 NP 完全。因为酉算符是厄米算符的指数，所以这些结果也可以应用于量子交替算符 ansatz 框架中的混频器的构造。我们开发了一个简单直观的代数框架，用于推理具有线性约束的哈密顿量的对易——该框架允许我们将为任意一组线性约束寻找驱动者哈密顿量的复杂性分类为 NP 完全。因为酉算符是厄米算符的指数，所以这些结果也可以应用于量子交替算符 ansatz 框架中的混频器的构造。