Pattern recognition on a quantum computer

PACS: 03.67.Lx, 03.67.-a, 42.30.Sy, 89.70.+c. Introduction Pattern recognition is one of the basic problems in artificial intelligence, see, e.g., [1]. For example, generally a short look at a picture like the one in Fig. 1 suffices for the human brain to spot the region with the pattern. However, it is a rather non-trivial task to accomplish the same performance with a computer – in particular if the orientation and the structure of the pattern are not known a priori. Besides the detection and localization of pattern (for example identifying seismic waves in the outputs of seismographs) the comparison and matching of the observed pattern to a set of templates (such as face recognition) is another interesting question. Usually these problems are solved with special classifiers, such as neuronal networks or Fourier analysis, etc., cf. [1]. The specific properties of the task of pattern recognition (one may consider many combinations simultaneously and is interested in global features only) give raise to the hope that quantum algorithms may be advantageous in comparison with classical (local) computational methods (with a unique entry). During the last decade the topic of quantum computing has attracted increasing interest, see, e.g., [2] for a review. It has been shown that quantum algorithms can be enormously faster that the best (known) classical techniques: Shor’s factoring algorithm [3], which exhibits an exponential speed-up relative to the best known classical method; Grover’s search routine [4] with a quadratic speed-up; and several black-box problems [5–8], some of which also exhibit an exponential speed-up. In the following a quantum algorithm for the detection and localization of certain patterns in an otherwise random data set is presented. It turns out that this method is also exponentially faster than its classical counterpart. The idea of using quantum computers for the aforementioned task of template matching (which is different from pattern detection/localization) has been elaborated in [9]. More generally, Ref. [10] points out the advantages of a quantum memory in this respect. Note, however, that the necessity of loading the complete data set into a quantum memory may represent a drawback. In [11] an algorithm for data clustering (in pattern recognition problems) is developed, which is based on/inspired by principles of quantum mechanics – but does not involve quantum computation. Description of the Problem Let us consider a N ×M array containing P = %NM points with a homogeneous density % < 1 (for example % = 1/2). Without loss of generality (w.l.o.g.) we may assume % ≤ 1/2 – otherwise we could just consider the complementary (negative) picture %→ 1− %. A small fraction χ of these points (say χ = 1/10) forms a pattern in a region of the size χNM , cf. Fig. 1. For simplicity we restrict our consideration to linear patterns, i.e., the angles within the pattern do not change.