Quantum pattern matching fast on average

The $d$-dimensional pattern matching problem is to find an occurrence of a pattern of length $m \times \dots \times m$ within a text of length $n \times \dots \times n$, with $n \ge m$. This task models various problems in text and image processing, among other application areas. This work describes a quantum algorithm which solves the pattern matching problem for random patterns and texts in time $\widetilde{O}((n/m)^{d/2} 2^{O(d^{3/2}\sqrt{\log m})})$. For large $m$ this is super-polynomially faster than the best possible classical algorithm, which requires time $\widetilde{\Omega}( (n/m)^d + n^{d/2} )$. The algorithm is based on the use of a quantum subroutine for finding hidden shifts in $d$ dimensions, which is a variant of algorithms proposed by Kuperberg.Comment: 22 pages, 2 figures; v3: further minor changes, essentially published versio

Self-stabilized Nonlinear Lateral Modes of Broad Area Lasers

The lateral modes of broad area lasers are investigated theoretically. The nonlinear interaction between optical field and effective refractive index leads to a saturable nonlinearity in the governing field equation, so that self-modulated solutions are found to be stable with increased current injection above saturation intensity. We derive approximate analytical solutions for traveling wave fields within the broad area laser. The field amplitude consists of a small ripple superimposed on a large dc value. Matching fields at the boundary determines the modulation depth and imparts an overall phase curvature to the traveling wave mode. There are multiple lateral modes for a given set of operating conditions, and modes with successively more lobes in the ripple have greater overall phase curvature. In contrast to the linear problem, several lateral modes can achieve the same modal gain, for a given injected current density, by saturating the gain to different extent. Thus, these modes would exhibit slightly different optical powers

Pattern recognition on a quantum computer

By means of a simple example it is demonstrated that the task of finding and identifying certain patterns in an otherwise (macroscopically) unstructured picture (data set) can be accomplished efficiently by a quantum computer. Employing the powerful tool of the quantum Fourier transform the proposed quantum algorithm exhibits an exponential speed-up in comparison with its classical counterpart. The digital representation also results in a significantly higher accuracy than the method of optical filtering. PACS: 03.67.Lx, 03.67.-a, 42.30.Sy, 89.70.+c.Comment: 6 pages RevTeX, 1 figure, several correction

Pattern recognition on a quantum computer

By means of a simple example it is demonstrated that the task of finding and identifying certain patterns in an otherwise (macroscopically) unstructured picture (data set) can be accomplished efficiently by a quantum computer. Employing the powerful tool of the quantum Fourier transform the proposed quantum algorithm exhibits an exponential speed-up in comparison with its classical counterpart. The digital representation also results in a significantly higher accuracy than the method of optical filtering. PACS: 03.67.Lx, 03.67.-a, 42.30.Sy, 89.70.+c.Comment: 6 pages RevTeX, 1 figure, several correction

Spectral properties of photon pairs generated by spontaneous four wave mixing in inhomogeneous photonic crystal fibers

The photonic crystal fiber (PCF) is one of the excellent media for generating photon pairs via spontaneous four wave mixing. Here we study how the inhomogeneity of PCFs affect the spectral properties of photon pairs from both the theoretical and experimental aspects. The theoretical model shows that the photon pairs born in different place of the inhomogeneous PCF are coherently superposed, and a modulation in the broadened spectrum of phase matching function will appear, which prevents the realization of spectral factorable photon pairs. In particular, the inhomogeneity induced modulation can be examined by measuring the spectrum of individual signal or idler field when the asymmetric group velocity matching is approximately fulfilled. Our experiments are performed by tailoring the spectrum of pulsed pump to satisfy the specified phase matching condition. The observed spectra of individual signal photons, which are produced from different segments of the 1.9 m inhomogeneous PCF, agree with the theoretical predictions. The investigations are not only useful for fiber based quantum state engineering, but also provide a dependable method to test the homogeneity of PCF.Comment: to appear in Phys. Rev.