#### Date on Senior Honors Thesis

5-2018

#### Document Type

Senior Honors Thesis

#### Degree Name

B.S.

#### Department

Physics and Astronomy

#### Author's Keywords

quantum algebra; quantum approximation; quantum computing; numerical experiment; physics; computer science

#### Abstract

Quantum computation is of current ubiquitous interest in physics, computer science, and the public interest. In the not-so-distant future, quantum computers will be relatively common pieces of research equipment. Eventually, one can expect an actively quantum computer to be a common feature of life. In this work, I study the approximation efficiency of several common universal quantum gate sets at short sequence lengths using an implementation of the Solovay-Kitaev algorithm. I begin by developing from almost nothing the relevant formal mathematics to rigorously describe what one means by the terms universal gate set and covering efficiency. I then describe some interesting results on the asymptotic covering properties of certain classes of universal gate sets and discuss the theorem which the Solovay-Kitaev algorithm is based on.

Moving from mathematical introduction to experimental method, I then describe how sets will be compared. I use the commonly studied sets H+T, Pauli+V, V, and Clifford+T to determine which is the most efficient at approximating randomly generated unitaries. By doing so, we get an understanding of how well each set would perform in the context of a general quantum computer processor. This was accomplished by using the same implementation of the Solovay-Kitaev algorithm throughout, with roughly equal-sized preprocessed libraries formed from each gate set, over approximations for 10,000 randomly generated unitary matrices at algorithm depth n=5. Ultimately, the Pauli+V and V sets were the most efficient and had similar performance qualities. On average the Pauli+V set produced approximations of length 15,491 and accuracy 0.0002686. The V basis produced approximations of average sequence length 16,403 and accuracy 0.0001465. This performance is about equal given this particular implementation of the Solovay-Kitaev algorithm.

We conclude that this result is somewhat surprising as the general behavior and efficiency of these particular choices of gate set are expected to be similar. It is possible though that the asymptotic efficiencies of these gate sets vary by a relatively wide margin and this has effected the experiment. It is also possible that some aspect of a naive implementation of the Solovay-Kitaev algorithm resulted in the Hadamard gate based sets performing more poorly than the V basis sets overall. Due to constraints on computational power, this result could also be limited to this particular accuracy regime and could even out as tolerance π is taken to be arbitrarily small. Further possibilities of this result as well as further work are then discussed.

#### Recommended Citation

Mode, Brent A., "Efficient quantum approximation : examining the efficiency of select universal gate sets in approximating 1-qubit quantum gates." (2018). *College of Arts & Sciences Senior Honors Theses.* Paper 170.

Retrieved from https://ir.library.louisville.edu/honors/170

#### Lay Summary

Quantum computing is an interesting application of quantum mechanics that stands to change how we think about computers. Underlying them are quantum logic gates, similar to the logic gates that make up the CPU in a conventional computer. There are an infinite number of these quantum logic gates, so it is necessary to approximate them. By utilizing a well-known algorithm, it is possible to examine different ways of approximating quantum logic gates and determine which is best. This paper uses commonly discussed universal quantum gate sets to explore these approximations.