Quantum computation as geometry
ABSTRACT Quantum computers hold great promise for solving interesting computational problems, but it remains a challenge to find efficient quantum circuits that can perform these complicated tasks. Here we show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms or to prove limitations on the power of quantum computers.
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ABSTRACT: The relationship between efficient quantum gate synthesis and control theory has been a topic of interest in the quantum control literature. Motivated by this work, we describe in the present article how the dynamic programming technique from optimal control may be used for the optimal synthesis of quantum circuits. We demonstrate simulation results on an example system on SU(2), to obtain plots related to the gate complexity and sample paths for different logic gates.07/2008;
Article: Numerical Solution of the Dynamic Programming Equation for the Optimal Control of Quantum Spin Systems[show abstract] [hide abstract]
ABSTRACT: The purpose of this paper is to describe the numerical solution of the Hamilton-Jacobi-Bellman (HJB) for an optimal control problem for quantum spin systems. This HJB equation is a first order nonlinear partial differential equation defined on a Lie group. We employ recent extensions of the theory of viscosity solutions from Euclidean space to Riemannian manifolds to interpret possibly non-differentiable solutions to this equation. Results from differential topology on the triangulation of manifolds are then used to develop a finite difference approximation method, which is shown to converge using viscosity solution techniques. An example is provided to illustrate the method.02/2010;
Conference Proceeding: Minimum time control of spin systems via dynamic programming[show abstract] [hide abstract]
ABSTRACT: In this article we show how dynamic programming can be applied to the time optimal control of spin systems. This is done by recasting the system in two ways: (i) as an adjoint system along the lines of, (ii) as an impulsive control problem. We illustrate the dynamic programming methodology using numerical examples.Decision and Control, 2008. CDC 2008. 47th IEEE Conference on; 01/2009