Quantum gates


Yard, J. (2016). Quantum gates. Perimeter Institute. https://pirsa.org/16060049


Yard, Jon. Quantum gates. Perimeter Institute, Jun. 08, 2016, https://pirsa.org/16060049


          @misc{ pirsa_16060049,
            doi = {},
            url = {https://pirsa.org/16060049},
            author = {Yard, Jon},
            keywords = {Other},
            language = {en},
            title = {Quantum gates},
            publisher = {Perimeter Institute},
            year = {2016},
            month = {jun},
            note = {PIRSA:16060049 see, \url{https://pirsa.org}}


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.