Efficient solvability of Hamiltonians and limits on the power of some quantum computational models

Published

January 4, 2006

Author(s)

Emanuel Knill, Rolando Somma, Howard Barnum, Gerardo Ortiz

Abstract

We consider quantum computational models defined via a Lie-algebraic theory, where a class of initial states is acted on by Lie-algebraic quantum gates, and the expectation value of a Lie algebra element is measured at the end. We show that these models can be efficiently simulated on a classical computer in time polynomial in the dimension of the algebra, regardless of the dimension of the Hilbert space where the algebra acts. Similar results hold for the computation of the expectation value of operators implemented by a gate-sequence. We introduce a Lie-algebraic notion of generalized mean-field Hamiltonians and show that they are efficiently (exactly) solvable by means of a Jacobi-like diagonalization method. Our results generalize earlier ones on fermionic linear optics computation and provide insight into the source of the power of the conventional model of quantum computation.

Citation

Physical Review Letters

Pub Type

Journals

Keywords

Lie algebra, mean field, quantum computing, solvability

Citation

Knill, E. , Somma, R. , Barnum, H. and Ortiz, G. (2006), Efficient solvability of Hamiltonians and limits on the power of some quantum computational models, Physical Review Letters (Accessed July 30, 2025)