AF: Small: Approximation algorithms for quantum mechanical problems

在计算机科学中的一个基本问题是约束满足问题MAX-SAT，它问：给定一组布尔约束的n位，什么是最大数量的约束，可以同时满足分配给所有位？尽管MAX-SAT有许多应用，但不幸的是，它被认为是不可能有效解决的。幸运的是，这些问题通常可以通过所谓的近似算法近似解决。在量子环境中，存在MAX-SAT的物理动机推广，称为LH，它涉及在非常低的温度下计算量子系统的重要属性。不幸的是，像MAX-SAT，LH也被认为是棘手的。因此，该项目提出了一个问题：我们可以通过近似算法的框架计算近似解k-LH？这个问题的解决将使我们深入了解我们近似计算自然界量子系统性质的能力。所获得的成果将通过各种途径传播，包括会议，高中研讨会和旨在向公众展示研究前沿的工程公开讲座系列。在高层次上，该项目的目的是设计多项式时间近似算法的各种类的局部哈密顿问题（LH），从物理动机的特殊情况下，更一般的设置。所使用的技术的灵感主要来自近似算法和量子信息理论领域的想法。在其他结果中，该项目的一个关键目标是深入了解经典有效表示的量子态如何近似解决涉及局部哈密顿基空间的真正量子问题。

A fundamental problem in computer science is the constraint satisfaction problem MAX-SAT, which asks: Given a set of Boolean constraints on n bits, what is the maximum number of constraints which can be simultaneously satisfied by an assignment to all the bits? Despite its many applications, MAX-SAT is unfortunately believed to be impossible to solve efficiently. Fortunately, such problems can often be solved approximately via so-called approximation algorithms.In the quantum setting, a physically motivated generalization of MAX-SAT exists, known as LH, which concerns the computation of important properties of quantum systems at very low temperatures. Unfortunately, like MAX-SAT, LH is also believed intractable. This project hence asks the question: Can we compute approximate solutions to k-LH via the framework of approximation algorithms? The resolution of this question will yield deep insight into our ability to approximately compute properties of quantum systems in nature. The results obtained will be disseminated through a variety of avenues, including conferences, high school workshops, and engineering public lecture series aimed at exposing the general public to the frontiers of research. At a high level, this project aims to design polynomial-time approximation algorithms for a variety of classes of the local Hamiltonian problem (LH), from physically motivated special cases to more general settings. The techniques used are inspired primarily by ideas from the fields of approximation algorithms and quantum information theory. Among other results, a key aim of the project is to obtain insight into how well classically efficiently representable quantum states can approximate solutions to genuinely quantum problems involving ground spaces of local Hamiltonians.