Collaborative Research: CIF: Medium: QODED: Quantum codes Optimized for the Dynamics between Encoded Computation and Decoding using Classical Coding Techniques

利用自然界中的量子现象来处理信息，有望在计算和通信等几个任务中获得令人兴奋的收益和优势。然而，在发生这种现象的微小尺度上，信息载体，如原子，是脆弱的，极易受到噪声的影响。为了构建可扩展且可靠的量子系统，人们需要通过纠错技术来减轻噪声。量子纠错码（QECC）在更大的数学空间中对数据进行编码，以便利用冗余来检测和纠正错误。对于量子计算，人们需要定期进行这样的纠错，它试图保持数据的原样，同时还需要执行主动计算来处理数据，这会不断改变数据。为了容错，这样的量子计算机需要找到一种方法，在保持资源需求最小化的同时，仔细平衡计算和解码（纠错）。这是一项非常具有挑战性的任务，本项目开发了研究现代qecc的方法，这些方法利用经典纠错理论的技术优化了编码计算和解码之间的动态。该团队还将在这些领域培养年轻人才，并激励代表性不足的群体加入不断增长的量子劳动力队伍。著名的QECC阈值定理表明，只要每个硬件组件满足特定QECC函数的保真度阈值，量子信息就可以得到无限期的保护。该项目将提供对理想代码结构的具体理解，由实际约束驱动，并因此处理阈值和计算开销作为该结构的功能。具体而言，本研究计划的智力贡献可概括如下：(1)研究者最近的研究成果产生了系统的方法来综合稳定器码的逻辑运算。研究小组将首先将这种方法应用于量子低密度奇偶校验（QLDPC）代码，以了解它们在逻辑计算中的效用。(2)该团队将探索诸如串联和提升代数原型等策略，以结合QLDPC代码和量子Reed-Muller代码等代数代码的最佳方面。(3)对于QEC，已经观察到具有对称消息更新的迭代解码器由于与错误退化相关的循环和“量子”捕获集而在QLDPC码上失败。该团队将利用他们的经典专业知识开发基于带有噪声综合征的消息传递的单次算法，了解非线性消息更新的影响，并分析错误层。(4)研究人员还将确定这些方法的理想图结构如何与在这些混合qecc上实现逻辑操作相互作用。最后，该团队将把这些见解与其他有前途的方法结合起来，如基于测量的量子计算，这将使结果应用于广泛的技术领域。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Leveraging quantum phenomena in nature for processing information promises exciting gains and advantages in several tasks such as computing and communications. At the minute scale where such phenomena occur, however, the information carriers, such as atoms, are fragile and highly susceptible to noise. In order to build scalable and reliable quantum systems, one needs to mitigate noise through error-correction techniques. A quantum error correcting code (QECC) encodes data in a larger mathematical space so that the redundancy can be use to detect and correct errors. For quantum computation, one needs to regularly do such error correction, which tries to preserve data as it is, while also performing active computation to process the data, which keeps altering it. In order to be fault-tolerant, such a quantum computer needs to find ways of carefully balancing computation and decoding (error correction) while keeping resource requirements at a minimum. This is a very challenging task, and this project develops methods to study modern QECCs that are optimized for the dynamics between encoded computation and decoding using techniques from classical error-correction theory. The team will also nurture young talent in these areas and inspire underrepresented groups to join the growing quantum workforce.The celebrated threshold theorem of QEC established that quantum information can be protected indefinitely as long as each hardware component meets a fidelity threshold that is a function of the specific QECC. This project will provide a concrete understanding of desirable code structure, motivated by practical constraints, and hence address both thresholds and computational overhead as a function of this structure. Specifically, the intellectual contributions of the proposed research plan can be summarized as follows: (1) Recent results of investigators have produced systematic methods to synthesize logical operations on stabilizer codes. The team of researchers will begin by applying such methods to quantum low-density parity-check (QLDPC) codes in order to understand their utility for logical computation. (2) The team will explore strategies such as concatenation and lifting algebraic protographs to combine the best aspects of QLDPC codes and algebraic codes such as quantum Reed-Muller codes. (3) For QEC, it has been observed that iterative decoders with symmetric message updates fail on QLDPC codes due to cycles and "quantum" trapping sets related to error degeneracy. The team will leverage their classical expertise to develop single-shot algorithms based on message-passing with noisy syndromes, understand the effect of non-linear message updates, and analyze error floors. (4) The investigators will also determine how the desirable graph structure for such methods interplay with realizing logical operations on these hybrid QECCs. Finally, the team will combine these insights with other promising approaches such as measurement-based quantum computation, which will make the results apply across a wide array of technologies.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.