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.
Highlights
Quantum computers were originally envisioned as machines for efficiently simulating quantum Hamiltonian dynamics
The cost of simulating the time-evolution operator e−iHt depends on several factors: the number of system qubits n, evolution time t, target error, and how information on the Hamiltonian His accessed by the quantum computer
Our general procedure for Hamiltonian simulation in Theorem 1 extends the scope of possible useful formulations of Hamiltonian simulation
Summary
Quantum computers were originally envisioned as machines for efficiently simulating quantum Hamiltonian dynamics. The quantum signal processor exploits this structure to attack the often-considered problem of designing a quantum circuit Qthat queries Gand Usuch that in the standard-form, ( G|a ⊗ Is)Q(|G a ⊗ Is) = f [H ] for some target operator f [·] Though this is accomplished in prior art using the linear-combination-of-unitaries algorithm [25], that approach requires a case-by-case detailed analysis of f to obtain the L1 norm of its coefficients in a Taylor expansion, and has a success probability that decays with the inverse square of this norm. Our quantum signal processor computes f [H ] with no such restrictions and with an optimal query complexity that exactly matches polynomial lower bounds for a large class of functions We call this ‘quantum signal processing’, which generalizes our previous results [7] for d-sparse oracles to the standard-form and a larger class of functions. Though we focus on qubitization of Hermitian matrices in the body of this paper, we describe the extension to normal matrices in Appendix A
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