Hamiltonian Simulation by Qubitization

Abstract

We present the problem of approximating the time-evolution operatore−iH^tto errorϵ, where the HamiltonianH^=(⟨G|⊗I^)U^(|G⟩⊗I^)is the projection of a unitary oracleU^onto the state|G⟩created by another unitary oracle. Our algorithm solves this with a query complexityO(t+log⁡(1/ϵ))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 ared-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 whereH^is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed anyH^in an invariantSU(2)subspace. A large class of operator functions ofH^can then be computed with optimal query complexity, of whiche−iH^tis a special case.