Hamiltonian Simulation by Qubitization

We present the problem of approximating the time-evolution operator e − i H ^ t to error ϵ , where the Hamiltonian H ^ = ( ⟨ G | ⊗ I ^ ) U ^ ( | G ⟩ ⊗ I ^ ) is the projection of a unitary oracle U ^ onto the state | G ⟩ created by another unitary oracle. Our algorithm solves this with a query complexity O ( 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 are d -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 ^ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any H ^ in an invariant SU ( 2 ) subspace. A large class of operator functions of H ^ can then be computed with optimal query complexity, of which e − i H ^ t is a special case.