Hamiltonian Simulation by Qubitization

TL;DR

This paper introduces a qubitization-based algorithm for Hamiltonian simulation that achieves optimal query complexity and low overhead, improving efficiency for various Hamiltonian types including density matrices.

Contribution

It presents a novel qubitization technique enabling optimal and resource-efficient Hamiltonian simulation for a broad class of operators.

Findings

Achieves query complexity $ig(t+ ext{log}(1/ ext{epsilon})ig)$, optimal in all parameters.

Reduces space and gate complexity, providing quadratic speed-up for precision simulations.

Enables simulation of Hamiltonians represented as density matrices and other operator functions.

Abstract

We present the problem of approximating the time-evolution operator $e^{-i\hat{H}t}$ to error $\epsilon$, where the Hamiltonian $\hat{H}=(\langle G|\otimes\hat{\mathcal{I}})\hat{U}(|G\rangle\otimes\hat{\mathcal{I}})$ is the projection of a unitary oracle $\hat{U}$ onto the state $|G\rangle$ created by another unitary oracle. Our algorithm solves this with a query complexity $\mathcal{O}\big(t+\log({1/\epsilon})\big)$ 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 $d$-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 $\hat{H}$ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any $\hat{H}$ in an invariant $\text{SU}(2)$ subspace. A large class of operator functions of $\hat{H}$ can then be computed with optimal query complexity, of which $e^{-i\hat{H}t}$ is a special case.