Developing Quantum Algorithms Research Articles

The quantum realization of Arnold and Fibonacci image scrambling

The quantum Fourier transform, the quantum wavelet transform, etc., have been shown to be a powerful tool in developing quantum algorithms. However, in classical computing, there is another kind of transforms, image scrambling, which are as useful as Fourier transform, wavelet transform, etc. The main aim of image scrambling, which is generally used as the preprocessing or postprocessing in the confidentiality storage and transmission, and image information hiding, was to transform a meaningful image into a meaningless or disordered image in order to enhance the image security. In classical image processing, Arnold and Fibonacci image scrambling are often used. In order to realize these two image scrambling in quantum computers, this paper proposes the scrambling quantum circuits based on the flexible representation for quantum images. The circuits take advantage of the plain adder and adder modulo $$N$$N to factor the classical transformations into basic unitary operators such as Control-NOT gates and Toffoli gates. Theoretical analysis indicates that the network complexity grows linearly with the size of the number to be operated.

Histogram-based segmentation of quantum images

In this paper we investigate the use of quantum computing systems in the field of image processing. We consider histogram-based image processing operations and develop quantum algorithms for histogram computation and threshold-based segmentation. The underlying principle used for constructing the proposed quantum algorithms is to reformulate them in order to exploit the performance of the quantum Fourier transform and of quantum amplitude amplification. We show that, compared to the classical correspondents, a significant speedup can be achieved by expressing parts of the computational process in terms of problems that can be solved using these quantum techniques.

Noninteracting multiparticle quantum random walks applied to the graph isomorphism problem for strongly regular graphs

We investigate the quantum dynamics of particles on graphs ("quantum random walks"), with the aim of developing quantum algorithms for determining if two graphs are isomorphic (related to each other by a relabeling of vertices). We focus on quantum random walks of multiple non-interacting particles on strongly regular graphs (SRGs), a class of graphs with high symmetry that is known to have pairs of graphs that are hard to distinguish. Previous work has already demonstrated analytically that two-particle non-interacting quantum walks cannot distinguish non-isomorphic SRGs of the same family. Here, we demonstrate numerically that three-particle non-interacting quantum walks have significant, but not universal, distinguishing power for pairs of SRGs, proving a fundamental difference between the distinguishing power of two-particle and three-particle non-interacting walks. We analytically show why this distinguishing power is possible, whereas it is forbidden for two-particle non-interacting walks. Based on sampling of SRGs with up to 64 vertices, we find no difference in the distinguishing power of bosonic and fermionic walks. In addition, we find that the four-fermion non-interacting walk has greater distinguishing power than the three-particle walks on SRGs, showing that increasing particle number increases distinguishing power. However, we also analytically show that no non-interacting walk with a fixed number of particles can distinguish all SRGs, thus demonstrating a potential fundamental difference between the distinguishing power of interacting and noninteracting walks.

Optimal query error of quantum approximation on some Sobolev classes

We study the approximation of the imbedding of functions from anisotropic and generalized Sobolev classes into Lq([0, 1]d) space in the quantum model of computation. Based on the quantum algorithms for approximation of finite imbedding from LpN to LqN, we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(Wpr ([0, 1]d)) to Lq([0, 1]d) space for all 1 ⩽ q,p ⩽ ∞ and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup roughly up to a squaring of the rate in the classical deterministic and randomized settings.

Quantum Algorithms for Learning and Testing Juntas

In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: (1) whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; (2) with no access to any classical or quantum membership (“black-box”) queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; (3) which require only a few quantum examples but possibly many classical random examples (which are considered quite “cheap” relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: (1) We give an algorithm for testing k-juntas to accuracy ε that uses O(k/ϵ) quantum examples. This improves on the number of examples used by the best known classical algorithm. (2) We establish the following lower bound: any FS-based k-junta testing algorithm requires $$\Omega(\sqrt{k})$$ queries. (3) We give an algorithm for learning k-juntas to accuracy ϵ that uses O(ϵ−1 k log k) quantum examples and O(2 k log(1/ϵ)) random examples. We show that this learning algorithm is close to optimal by giving a related lower bound.

Quantum integration in Sobolev classes

We study high-dimensional integration in the quantum model of computation. We develop quantum algorithms for integration of functions from Sobolev classes W p r ([0,1] d ) and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of Novak on integration of functions from Hölder classes.

Quantum Summation with an Application to Integration

We study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences that satisfy a p-summability condition and for integration of functions from Lebesgue spaces Lp([0, 1]d), and analyze their convergence rates. We also prove lower bounds showing that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of G. Brassard et al. (2000, “Quantum Amplitude Amplification and Estimation,” Technical Report, http://arXiv.org/abs/quant-ph/0005055) on computing the mean for bounded sequences and complements results of E. Novak (2001, J. Complexity17, 2–16) on integration of functions from Hölder classes. The analysis requires an appropriate model of quantum computation, capable of covering the typical features of numerical problems such as dealing with real numbers and real-valued functions and with vector and function spaces. We develop and study such a model, which can be viewed as a quantum setting for information-based complexity theory.