Hamiltonian Simulation by Qubitization [article]

Guang Hao Low, Isaac L. Chuang
2017 arXiv   pre-print
Given a Hermitian operator Ĥ=〈 G|Û|G〉 that is the projection of an oracle Û by state |G〉 created with oracle Ĝ, the problem of Hamiltonian simulation is approximating the time evolution operator e^-iĤt at time t with error ϵ. We show that this can be done with query complexity O(t+(1/ϵ)/(1/ϵ)) to Ĝ,Û that is optimal, not just in asymptotic limits, but for all values t,ϵ. Furthermore, only 2 additional ancilla qubits are required in total, together with O(1) additional single and two-qubit gates
more » ... per query. Our 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 complexity, as well as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where 〈 G|Û|G〉 is a density matrix. A key technical result is 'qubitization' which uses controlled-Û and controlled-Ĝ to embed Ĥ in an invariant SU(2) subspace. A large class of operator functions of Ĥ can then be computed with optimal query complexity, of which e^-iĤt is a special case.
arXiv:1610.06546v2 fatcat:iilep2jp5veijli4ekyexdp5qi