CIF: Small: Quantum LDPC codes: structure and logical operations

量子计算机算法在解决缺乏有效经典解的基本搜索问题方面有着重要的前景。因此，构建功能齐全的量子计算机的普遍挑战目前正被世界各地的学术界和工业界的工程师们所追求，它也得到了物理学和计算机科学界的理论研究的支持。由于量子态，无论具体的体系结构，不可避免地暴露在固有的噪声中，量子计算的可靠性取决于将噪声控制程序直接纳入计算过程。这一挑战引发了有关量子纠错码的构造、分析和实现的各种问题。这个项目专门研究一类被称为低密度奇偶校验码的量子码，这种码被认为是促进容错计算的有前途的竞争者。该项目的预期成果包括构建和分析计算过程中噪音缓解的新方法，以及开发和教授涵盖量子信息科学的本科课程。量子编码理论的最新进展集中在量子稳定器代码的构建和解码上，其生成器作用于少量的量子比特。这种被称为低密度奇偶校验码的码已经成为支持量子计算机容错操作的主要候选码。过去几年设计的几个相关代码族获得了以前认为这类代码不可能实现的参数。借助于同调代数、平衡乘积码，特别是提升乘积码的方法，证明了新的设计具有与码块长度成正比的距离，从而产生了渐近良好的码族。这个项目将分析的一个问题领域是平衡乘积码的逻辑运算符的构造，这些逻辑运算符在本地作用于例如单个物理量子比特或一小群这样的量子比特。这种类型的逻辑运算符是实现代码的首选，并且构成了一个活跃的、正在发展的研究领域。本项目还将研究两类重要的量子低密度码，即Lift乘积码和量子Tanner码的共同底层结构。这项研究的一个主要目标是推进量子局部可测码的构造，这在量子复杂性理论中是重要的。该项目还将解决基于特定图形和群体动作的提升产品代码的构建，旨在构建具有改进参数的小长度或中等长度的新稳定器代码。该奖项反映了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。

Quantum computer algorithms hold significant promise in tackling fundamental search problems that lack efficient classical solutions. The general challenge of constructing functioning quantum computers is thus currently pursued by engineers in academia and industry worldwide, and it is also supported by theoretical research in the physics and computer science communities. Since quantum states, irrespective of the specific architecture, are inevitably exposed to inherent noise, the reliability of quantum computations hinges on incorporating noise-controlling procedures directly into computational processes. This challenge gives rise to various issues concerning the construction, analysis, and implementation of quantum error-correcting codes. This project specifically delves into the study of a class of quantum codes known as low-density parity-check codes, which are considered promising contenders for facilitating fault-tolerant computations. The anticipated outcomes of this project include constructions and analysis of new methods of noise mitigation in the computation process, as well as development and teaching of undergraduate courses covering quantum information science.Recent advances in quantum coding theory have focussed on the construction and decoding of quantum stabilizer codes, whose generators act on a small number of qubits. Such codes, termed low-density parity-check codes, have been the primary candidate for supporting fault-tolerant operation of quantum computers. Several related code families, designed in the last few years, attain parameters that were previously thought impossible for this class of codes. Relying on methods of homological algebra, balanced product codes and, in particular, lifted product codes, new designs were shown to have distance proportional to the block length of the code, resulting in asymptotically good code families. One problem area that this project will analyze is constructions of logical operators for balanced product codes that act locally, for instance, on a single physical qubit or a small group of such qubits. Logical operators of this type are preferable for implementing the codes, and constitute an active, developing research area. This project will also study common underlying structures in the two prominent families of quantum low-density codes, namely, lifted product codes and quantum Tanner codes. A key goal of this study is to advance constructions of quantum locally testable codes, which are important in quantum complexity theory. This project also will address constructions of lifted product codes based on specific graphs and group actions, aiming at constructing new stabilizer codes of small or moderate length with improved parameters.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.