AF: Small: Approximation algorithms for quantum mechanical problems

计算机科学中的一个基本问题是约束满足问题Max-SAT，它的问题是：给定一组关于n比特的布尔约束，通过对所有比特的赋值可以同时满足的最大约束数是多少？尽管MAX-SAT有许多应用，但遗憾的是，人们认为它不可能有效地求解。幸运的是，这类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，存在一个物理激励的MAX-SAT推广，称为1H，它涉及到极低温度下量子系统的重要性质的计算。不幸的是，与MAX-SAT一样，人们也认为黄体生成素难以治愈。因此，这个项目提出了这样一个问题：我们能否通过近似算法的框架来计算k-Lh的近似解？这个问题的解决将使我们深入了解我们近似计算自然界中量子系统的性质的能力。所取得的成果将通过各种途径传播，包括会议、高中讲习班和旨在让普通公众接触研究前沿的工程学公共讲座系列。在更高的层次上，这个项目的目标是为各种类型的局部哈密顿问题设计多项式时间近似算法，从物理激励的特殊情况到更一般的设置。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可有效表示的量子态如何很好地逼近涉及局部哈密顿空间的真正量子问题的解。

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.