Collaborative Research: FET: Small: Minimum Quantum Circuit Size Problems, Variants, and Applications

现代计算的一个核心追求是找到一种尽可能有效的方法来解决计算问题。抽象地说，给出一个函数的真值表，可以确定正确计算该函数的布尔电路的最小大小吗？对它的研究称为最小电路尺寸问题(MCSP)，从一开始就伴随着理论计算机科学的发展。它表现出各种各样的，有时甚至是神秘的属性，这些属性在研究计算基础方面被证明是卓有成效的。例如，MCSP的高效算法意味着重要任务的高效学习算法以及公钥密码学的破解；另一方面，MCSP有助于证明不可行的结果，例如解决问题所需的计算资源的下界。本项目的目标是通过MCSP的透镜来推进量子信息处理。本项目中研究的工具和对象在量子信息处理和量子物理中具有重要的应用价值。例如，我们可能能够构建新的量子密码协议，证明基本问题的量子资源下界，以及通过研究MCSP的量子版本来表明估计虫洞体积的难度。此外，该项目还可以促进计算机科学家和物理学家之间的进一步合作。研究部分将辅之以协调一致的教育和外联计划。这将包括开发量子计算课程和升级计算机科学的理论课程，在参与的大学建立量子计算小组，向广大受众传播研究结果，通过成功的项目(例如，为波特兰高中生提供实习机会的星期六学院)吸引当地社区，以及在两所大学举办量子编码竞赛。它们构成了这个项目的重要组成部分，以促进人们对量子计算的更大兴趣和熟练程度。该项目旨在研究沿三个推力计算各种物体的最小量子电路大小。1)研究经典和量子对象上的最小量子电路尺寸问题的框架，包括函数、量子态和酉算符。将开发新的工具来确定它们的硬度和与量子信息处理中的基本问题的联系。量子复杂性理论的新图景将被识别出来。2)在文[1]的框架下，进一步探讨了模拟量子系统和制备基态所需的最小电路尺寸的确定问题。这两个问题代表了量子计算机最可行的一些应用，这里的新发现将描绘这些应用的算法限制，并将它们与量子计算中的其他基本原语联系起来。3)将探索更多的输入模型，包括简洁的经典描述和纯量子输入(例如，寄存器中的量子态)，形式处理将伴随着新的应用，包括用于经典验证量子资源的新协议以及新的量子伪随机原语。所有这些方面的工作都将扩大量子信息处理的范围和最小电路尺寸问题。此外，它还将提供研究基本量子原语的新方法，并将计算机科学和量子物理联系起来。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A central pursuit in modern computing is to find a method as efficient as possible to solve a computational problem. Abstractly, given the truth table of a function, can one decide the minimum size of a boolean circuit that correctly computes the function? The investigation into it, termed the minimum circuit size problem (MCSP), has accompanied the development of theoretical computer science since the beginning. It manifests diverse and sometimes mysterious properties that prove to be fruitful in investigating the foundations of computing. For instance, efficient algorithms for MCSP imply efficient learning algorithms for important tasks as well as breaking public-key cryptography; and on the other hand, MCSP is useful to demonstrate no-go results such as lower bounds on the computational resources necessary to solve a problem. The goal of this project is to advance quantum information processing through the lens of MCSP. The tools and objects studied in this project can have significant applications in quantum information processing and quantum physics. For instance, we might be able to build novel quantum cryptographic protocols, prove quantum resources lower bounds for basic problems, and show the hardness of estimating the wormhole volume by studying quantum versions of MCSP. Furthermore, this project could stimulate further collaboration between computer scientists and physicists. The research component will be supplemented by a concerted education and outreach plan. This will include developing courses in quantum computing and upgrading the theory curriculum in computer science, establishing groups in quantum computing at participating universities, disseminating the findings to a broad audience, and engaging local communities via successful programs (e.g., Saturday Academy that provides internships to Portland high school students), and hosting quantum coding contests across the two universities. They form a vital part of this project to promote greater interest and proficiency in quantum computing. This project aims to investigate the minimum quantum circuit size for computing various objects along three thrusts. 1) A framework for studying minimum quantum circuit size problems on classical and quantum objects, including functions, quantum states, and unitary operators. New tools will be developed to establish their hardness and connections to fundamental problems in quantum information processing. A new landscape of quantum complexity theory will be identified. 2) Under the framework developed in thrust 1, the problems of deciding the minimum size circuit for simulating a quantum system andfor preparing ground states will be further explored. These two problems represent some of the most viable applications of quantum computers, and the new findings here will depict the algorithmic limits of these applications as well as connect them to other basic primitives in quantum computing. 3) More input models will be explored, including succinct classical descriptions and purely quantum inputs (e.g., a quantum state in a register), and the formal treatment will be accompanied by novel applications, including new protocols for classically verifying quantum resources as well as novel quantum pseudorandom primitives. The proposed work in all these thrusts will expand the scope of quantum information processing and the minimum circuit size problem. Moreover, it will provide new approaches to studying basic quantum primitives and bridging computer science and quantum physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.