Applications of Number Theory to the Quantum Gates Model

American Institute Of Mathematics, Pasadena CA

Investigators

Abstract

Both the research and broader activities in this award include current developments in number theory and their applications in quantum computing and quantum chaos. In terms of practical applications of the project, the PI expects that his refined version of Ross and Selinger algorithm will be used if a physical quantum computer is built. The hope is that quantum computers will eventually be able to efficiently simulate quantum physics and study many (important) computationally difficult problems inaccessible to modern-day computers. There are models such as the Quantum Gates Model that give theoretical constructions of efficient circuits to be used in quantum computers. This model is connected to the study of integral solutions to Diophantine equations, an ancient subject of interest to mathematicians. A question of interest to both quantum computer scientists as well as mathematicians is the optimal approximation of real solutions of special Diophantine equations by integral solutions. The PI has proved new (optimal) results in this direction. Furthermore, he has proved that this task is computationally hard (NP-complete) for generic inputs. On a more technical level, one of the central problems in the Quantum Gates Model is the approximation of an arbitrary qubit using a fixed set of generators called universal quantum gates. In the single-qubit case, this amounts to navigating the unitary group SU(2) by a specific set of topological generators (e.g. V-gates or the Lubotzky-Phillips-Sarnak generators) that are carefully chosen such that the associated transition matrix has the optimal spectral gap (e.g. the eigenvalues of the Hecke operators satisfy the Ramanujan bound). The PI proposes a refinement of the Ross and Selinger algorithm for approximating an arbitrary single-qubit that removes all heuristic assumptions from their algorithm. Among the new tools in this approach are the delta method, Sieve theory, and the spectral theory of modular forms and bounds on their Fourier coefficients. An objective of this project is to generalize the results of the PI to higher rank arithmetic groups which brings in the theory of the oscillator representation and the theory of automorphic representations. Motivated by Berry's conjecture in Quantum Chaos, the PI studies the statistical properties and the multiplicity of the eigenvalues of the transition matrix of the quantum gates (the Hecke operators). So far, the PI has proved power saving upper bounds as well as absolute upper bound on the multiplicity of the eigenvalues of the Hecke operators. Furthermore, the PI has proved lower bounds on the discrepancy of the spectral measure with respect to the Plancherel measure. The project brings together the deformation theory of Galois representations, Iwasawa theory, the Taylor-Wiles method, trace formulae, and other tools from the algebraic and analytic number theorists' toolbox in order to answer questions of interest to computer scientists as well as mathematicians. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.