An Analog Analogue of a Digital Quantum Computation

Abstract

We solve a problem, which while not fitting into the usual paradigm, can be viewed as a quantum computation. Suppose we are given a quantum system described by an N dimensional Hilbert space with a Hamiltonian of the form $E|w><w|$ where $|w>$ is an unknown (normalized) state. We show how to discover $|w>$ by adding a Hamiltonian (independent of $|w>$) and evolving for a time proportional to $N^{1/2}/E$. We show that this time is optimally short. This process is an analog analogue to Grover's algorithm, a computation on a conventional (!) quantum computer which locates a marked item from an unsorted list of N items in a number of steps proportional to $N^{1/2}$.

Latex, 6 pages

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