research

Hamiltonian Simulation by Qubitization

Abstract

We present the problem of approximating the time-evolution operator eiH^te^{-i\hat{H}t} to error ϵ\epsilon, where the Hamiltonian H^=(GI^)U^(GI^)\hat{H}=(\langle G|\otimes\hat{\mathcal{I}})\hat{U}(|G\rangle\otimes\hat{\mathcal{I}}) is the projection of a unitary oracle U^\hat{U} onto the state G|G\rangle created by another unitary oracle. Our algorithm solves this with a query complexity O(t+log(1/ϵ))\mathcal{O}\big(t+\log({1/\epsilon})\big) to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are dd-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where H^\hat{H} is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any H^\hat{H} in an invariant SU(2)\text{SU}(2) subspace. A large class of operator functions of H^\hat{H} can then be computed with optimal query complexity, of which eiH^te^{-i\hat{H}t} is a special case.Comment: 23 pages, 1 figure; v2: updated notation; v3: accepted versio

    Similar works

    Full text

    thumbnail-image

    Available Versions