Algebraic and number theoretic methods for quantum circuits

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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.