Algebra for Optical Computing and Quantum Computing
Date: July 17th (Wednesday)
Optical and quantum mechanical systems enjoy a great intrinsic
parallelism that can be used to speed-up computations. As both types
of systems are linear, designing these systems leads to the problem of
factoring a given matrix into matrices corresponding to the
computational primitives of the physical system.
We present an algebraic approach to this "programming" problem for
optical computing and quantum computing.
For optical systems, the computational primitives are diffractive
elements corresponding to diagonal matrices and wave propagation
modelled by the Fresnel transformation. We show that, in principle,
any linear transformation can be realized optically up to a scalar
factor, but possibly with many diffractive elements. In practice,
optical setups with a small number of diffractive elements are of
interest. We present a divide and conquer algorithm for the design of
8f-systems, i. e., systems with up to four lenses and three arbitrary
Computations in quantum mechanical systems are specified by unitary
matrices acting on a Hilbert space which is a tensor product of
smaller subspaces. The computational primitives are unitary
transformations acting on these subspaces. In principle, any unitary
transformation can be factored into such basic transformations. For
complexity reasons we are interested in a factorization with a small
number of factors. We present an algebraic approach to find a
"factorization" similar to that for optical systems.
Solving the equations arising in the design problem is equivalent to
computing the inverse of a rational mapping. For this mapping, a
decomposition can be derived from a tower of unirational function
fields. We discuss how to compute the intermediate fields using
Groebner bases and primary decomposition. Algorithms using such a
functional decomposition solve the nonlinear equations faster than
more universal methods.