Rodeo Algorithm for Quantum Computing

We present a stochastic quantum computing algorithm that can prepare any eigenvector of Hamiltonian within selected energy interval $[E\ensuremath{-}\ensuremath{\epsilon},E+\ensuremath{\epsilon}]$. In order to reduce the spectral weight all other eigenvectors by suppression factor $\ensuremath{\delta}$, required computational effort scales as $O[|\mathrm{log}\ensuremath{\delta}|/(p\ensuremath{\epsilon})]$, where $p$ is squared overlap initial state with target eigenvector. The method, which we call rodeo algorithm, uses auxiliary qubits control time evolution minus some tunable parameter $E$. With each qubit measurement, amplitudes are multiplied depends on proximity their this manner, converge exponential accuracy in number measurements. addition preparing eigenvectors, method also compute full spectrum Hamiltonian. illustrate performance several examples. For eigenvalue determination error $\ensuremath{\epsilon}$, scaling $O[(\mathrm{log}\ensuremath{\epsilon}{)}^{2}/(p\ensuremath{\epsilon})]$. eigenstate preparation, $O(\mathrm{log}\mathrm{\ensuremath{\Delta}}/p)$, $\mathrm{\ensuremath{\Delta}}$ magnitude orthogonal component residual vector. speed for preparation exponentially faster than phase estimation or adiabatic evolution.