Algebraic and number theoretic methods for quantum circuits

这项研究项目的目的是开发技术，以减少执行实际量子计算所需的资源量。量子计算是一种基于量子物理定律而不是经典物理的计算形式。它有可能比目前存在的任何一台计算机都强大得多。对于某些计算问题，存在有效的量子算法，尽管还没有已知的有效的经典算法。最著名的例子是肖尔1994年提出的将整数分解为素数的量子算法。 20年前，当量子计算学科首次进入计算机科学的主流时，它最初主要是理论上的兴趣，因为实用量子计算机的开发实际上还需要几十年的时间。然而，这种情况似乎正在改变。现在有许多政府、公共和私人组织，包括加拿大的几个组织，正在积极尝试建造可扩展的量子计算机，并正在取得切实的进展。与此同时，越来越明显的是，即使可以建造一台可伸缩的量子计算机，使其容错所需的实际资源也将是惊人的--据一些人估计，一项需要几百个逻辑量子比特的计算可能需要数十万个物理量子比特，以及在被认为在短期内可行的硬件上花费数百万年的计算时间。 带着这一点，现实的考虑突然被推到了量子计算的前景中。使量子计算可行所需的资源削减可能来自各种方面，例如，更好的纠错方案，改进的组件协议，如状态蒸馏，改进的算法，以及改进的编译技术。我的研究主要集中在后一个领域，特别是改进的么正逼近方法和逻辑量子电路的优化。

The purpose of this research project is to develop techniques for reducing the amount of resources required for performing practical quantum computing. Quantum computing is a form of computing that is based on the laws of quantum physics, rather than classical physics. It has the potential to be vastly more powerful than any of the computers existing today. There are certain computational problems for which there exist efficient quantum algorithms, although no efficient classical algorithm is known. The best-known example is Shor's 1994 quantum algorithm for factoring integers into primes. When the subject of quantum computing first emerged into the mainstream of computer science two decades ago, it was initially mostly of theoretical interest, as the development of practical quantum computers was literally decades away. However, this seems to be changing. There are now a number of government, public, and private organizations, including several in Canada, that are actively trying to build a scalable quantum computer and are making tangible progress. At the same time, it is becoming clear that, even if a scalable quantum computer can be built, the actual resources required to make it fault tolerant will be staggering - by some estimates, a computation that requires, say, a few hundred logical qubits could require hundreds of thousands of physical qubits and millions of years of computing time on the sort of hardware that is considered realistic in the near term. With this, practical considerations are suddenly thrust into the foreground of quantum computing. The resource reductions required to make quantum computing feasible can potentially come from a variety of places, for example, better error correction schemes, improved protocols for components such as state distillation, improved algorithms, and improved compilation techniques. My research is primarily focused in the latter area, and specifically on improved methods for unitary approximation and the optimization of logical quantum circuits.