Optical Computing and Computational Complexity

Abstract

This work concerns the computational complexity of a model of computation that is inspired by optical computers. The model is called the continuous space machine and operates in discrete timesteps over a number of two-dimensional images of fixed size and arbitrary spatial resolution. The (constant time) operations on images include Fourier transformation, multiplication, addition, thresholding, copying and scaling. We survey some of the work to date on the continuous space machine. This includes a characterisation of the power of an important discrete restriction of the model. Parallel time corresponds, within a polynomial, to sequential space on Turing machines, thus satisfying the parallel computation thesis. A characterisation of the complexity class NC in terms of the model is also given. Thus the model has computational power that is (polynomially) equivalent to that of many well-known parallel models. Such characterisations give a method to translate parallel algorithms to optical algorithms and facilitate the application of the complexity theory toolbox to optical computers. In the present work we improve on these results. Specifically we tighten a lower bound and present some new resource trade-offs.1 That is, finding lower and upper bounds on computational power in terms of known complexity classes.

Optical computing

“…It turns out that a wide range of optical computers that run for at most polylogarithmic time, and use at most polynomial space-like resources, solve exactly NC [97,93,95] (this can be shown to be a corollary of the PSPACE characterisation cited earlier in Section 5). In effect this means that we have an algorithmic way (in other words, a compiler) to convert existing NC algorithms into optical algorithms that use similar amounts of resources.…”

“…It is known that it is possible to solve any NP (and even any PSPACE) problem in polynomial time on optical computers, albeit with exponential use of some other, spacelike, resources [97,93,95]. These results were shown on the CSM, a general model of a wide range of optical computers.…”

“…Here we give an example CSM algorithm (taken from [95]) that makes use of the data representations described above. The algorithm squares a n × n matrix in O(log n) time and O(n 3 ) spatialRes (number of pixels), while all other CSM resources are constant.…”

“…There are a number of existing CSM algorithms, for these we point the reader to the literature [63][64][65]93,95,97,98].…”

“…This transition matrix represents all possible computations of the Turing machine and is of size O(2 S ) × O(2 S ). The matrix squaring part was already given as an example (Lemma 1), and the remainder of the algorithm is given in [95]. The algorithm uses space that is cubic in one of the matrix dimensions.…”