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arXiv:1606.03873 · Entropy

Dimensional Lifting through the Generalized Gram–Schmidt Process

Hans Havlicek, Karl Svozil

Published Apr 14, 2018• 7 citations
arXivHTMLPDF

Abstract

A new way of orthogonalizing ensembles of vectors by "lifting" them to higher dimensions is introduced. This method can potentially be utilized for solving quantum decision and computing problems.
DOI
10.3390/e20040284

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Vector set orthogonal normalization and matrix QR decomposition are fundamental problems in matrix analysis with important applications in many fields. We know that the Gram-Schmidt process is a widely used method to solve these two problems. However, existing methods, including the Gram-Schmidt process, have problems of high complexity, scaling O(N3)O({N}^{3})O(N3) in the system dimension NNN, which leads to difficulties when calculating large-scale or ill-conditioned problems. With the development of quantum information processing, a series of quantum algorithms have been proposed, providing advantages and speedups over classical algorithms in many fields. In this paper we propose quantum algorithms to solve these two problems based on the idea of the Gram-Schmidt process and quantum phase estimation. The complexity of proposed quantum algorithms is theoretically and numerically analyzed. We find that our algorithms provide polynomial acceleration over the best-known quantum algorithms and a potential polynomial acceleration over the best-known classical algorithms on these two problems, scaling O(N2poly(logN))O({N}^{2}\mathrm{poly}(logN))O(N2poly(logN)) in the dimension NNN of the system when ignoring the complexity of state readout or O(N2poly(logN))O({N}^{2}\mathrm{poly}(logN))O(N2poly(logN)) when considering the complexity of state readout.
Abstract Factorization methods such as the Hamiltonian hierarchy have been useful to find eigenfunctions for Schrödinger equations, in particular, for potentials that are partially or approximately solvable. In this paper, an alternative approach is proposed to study excited states via the variational method. The trial functions are built from the exact or approximate superpotential for the ground state combined with the Gram–Schmidt process to ensure orthogonalization between the functions. The results found variationally for one dimensional potentials are compared with previous results from the literature. The energy eigenvalues obtained agree with previous ones and, for most of the results, the percentage difference between the proposed approach and others in the literature is less than 0.1%. The method introduced is an effective and intuitive approach to determine trial wave functions for the excited states. This approach can be useful in studying the Schrödinger equation and related problems which can be mapped onto a Schrödinger type-equation as, for example, the Fokker–Planck equation.