Abstract:

Fault-tolerant quantum computers will compute by applying

a sequence of elementary unitary operations, or gates, to an

error-protected subspace. While algorithms are typically expressed

over arbitrary local gates, there is unfortunately no known theory

that can correct errors for a continuous set of quantum gates.

However, theory does support the fault-tolerant construction of

various finite gate sets, which in some cases generate circuits that

can approximate arbitrary gates to any desired precision. In this

talk, I will present a framework for approximating arbitrary qubit

unitaries over a very general but natural class of gate sets derived

from the theory of integral quaternions over number fields, where the

complexity of a unitary is algebraically encoded in the length of a

corresponding quaternion. Then I will explore the role played by

higher-dimensional generalizations of the Pauli gates in various

physical and mathematical settings, from classifying bulk-boundary

correspondences of abelian fractional quantum Hall states to

generating optimal symmetric quantum measurements with surprising

connections to Hilbert's 12th problem on explicit class field theory

for real quadratic number fields.