Recent advances in the field of adiabatic quantum computing and the closely related field of quantum annealers has 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 than other common practices. 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 to generate 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 constraints as NP-hard.

中文翻译：

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为几个线性约束构造司机哈密顿量

绝热量子计算领域和密切相关的量子退火器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，与其他常见做法相比，该方法能够使用更稀疏的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——一个允许我们将为任意一组约束寻找驱动者哈密顿量的复杂性分类为 NP-hard。