Limitations of some simple adiabatic quantum algorithms

Details Limitations of some simple adiabatic quantum algorithms Limitations of some simple adiabatic quantum algorithms

2007-02-26

arxiv.org/abs/quant-ph/0702241v2

International Journal of Quantum Information, Vol. 6, No. 3 (June 2008)

Physics

Quantum Physics

This work originally appeared in L. Ioannou's Master's thesis, submitted to the University of Waterloo, in 2002 (available at

Scientific paper

Let $H(t)=(1-t/T)H_0 + (t/T)H_1$, $t\in [0,T]$, be the Hamiltonian governing an adiabatic quantum algorithm, where $H_0$ is diagonal in the Hadamard basis and $H_1$ is diagonal in the computational basis. We prove that $H_0$ and $H_1$ must each have at least two large mutually-orthogonal eigenspaces if the algorithm's running time is to be subexponential in the number of qubits. We also reproduce the optimality proof of Farhi and Gutmann's search algorithm in the context of this adiabatic scheme; because we only consider initial Hamiltonians that are diagonal in the Hadamard basis, our result is slightly stronger than the original.

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