Algebraic and number theoretic methods for quantum circuits

该研究项目的目的是开发减少执行实际量子计算所需资源量的技术。量子计算是一种基于量子物理定律的计算形式，而不是经典物理。它有可能比今天存在的任何计算机都强大得多。对于某些计算问题，存在有效的量子算法，尽管没有已知的有效的经典算法。最著名的例子是Shor在1994年提出的将整数分解为素数的量子算法。 二十年前，当量子计算的主题首次成为计算机科学的主流时，它最初主要是理论上的兴趣，因为实际量子计算机的发展实际上还有几十年的时间。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，正在积极尝试构建可扩展的量子计算机，并取得了切实的进展。 与此同时，越来越清楚的是，即使可以构建可扩展的量子计算机，使其容错所需的实际资源也将是惊人的-根据一些估计，需要几百个逻辑量子位的计算可能需要数十万个物理量子位和数百万年的计算时间，这在短期内被认为是现实的。 有了这个，实际的考虑突然被推到量子计算的前台。使量子计算可行所需的资源减少可能来自各种地方，例如，更好的纠错方案，改进的组件协议，如状态蒸馏，改进的算法和改进的编译技术。我的研究主要集中在后一个领域，特别是对酉近似和优化逻辑量子电路的改进方法。

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.