### Linear Algebra in Quantum Computation

**George Cybenko**

Dartmouth College

The quantum computing paradigm has attracted much attention recently and implementations
of simple but real quantum computers are happening faster than people expected.
While the principles and intuition underlying quantum mechanics are profound
and deep, quantum computation as a formal mathematical model is relatively straightforward
to understand in terms of ideas well-known to applied mathematicians. Unfortunately,
most developments of quantum computing are couched in the language and symbolism
of physics, not the traditional mathematics of functional analysis and matrix
algebra.

The first part of this talk will be an overview of quantum computing using
simple linear algebra and probability ideas leading to some of the important
quantum algorithms people have discovered. We will then present some results
about the reducibility of quantum computations to simple quantum gate operations
using Givens-like rotations applied to unitary matrices. We will finish with
some results about the encrypted execution of encrypted quantum computations.

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