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Algebra for Optical Computing and Quantum Computing
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Thomas Beth, Markus Grassl, Joern Mueller-Quade

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Date:  July 17th (Wednesday)\\
Time: 14:30-15:00\\

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Abstract
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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 diffractive elements. 
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. 

