Zeph Landau

Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation

Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models allows stating the main open problems in quantum computation using well-studied mathematical objects such as eigenvectors and spectral gaps of sparse matrices.

Adiabatic quantum computation is equivalent to standard quantum computation

The model of adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its exact computational power has been unknown. We settle this question and describe an efficient adiabatic simulation of any given quantum algorithm. This implies that the adiabatic computation model and the standard quantum circuit model are polynomially equivalent. We also describe an extension of this result with implications to physical implementations of adiabatic computation. We believe that our result highlights the potential importance of the adiabatic computation model in the design of quantum algorithms and in their experimental realization.

Density, Overcompleteness, and Localization of Frames. I. Theory

AbstractFrames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two frames

Density, overcompleteness, and localization of frames

{\frak F} = \{f_i\}_{i\in I}

Deficits and Excesses of Frames

and

Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation

{\frak E} = \{e_j\}_{j\in G}

Exchange Relation Planar Algebras

(

Quantum Computation and the Evaluation of Tensor Networks

G

Excesses of Gabor frames

a discrete abelian group), relating the decay of the expansion of the elements of

The planar algebra associated to a Kac algebra

{\frak F}