Hamiltonian simulation with nearly optimal dependence on spectral norm

TL;DR

This paper introduces a quantum algorithm for simulating Hamiltonian dynamics with nearly optimal dependence on spectral norm, achieving polynomial speedups and improving query complexities for unitary implementation and linear systems.

Contribution

It presents a nearly optimal quantum Hamiltonian simulation algorithm that leverages spectral norm knowledge, improving query complexity bounds for unitary implementation and linear system solving.

Findings

Achieves nearly optimal query complexity for Hamiltonian simulation.

Provides a polynomial speedup in sparsity for spectral norm known cases.

Improves query bounds for sparse linear systems with condition number .

Abstract

We present a quantum algorithm for approximating the real time evolution $e^{-iHt}$ of an arbitrary $d$-sparse Hamiltonian to error $\epsilon$, given black-box access to the positions and $b$-bit values of its non-zero matrix entries. The complexity of our algorithm is $\mathcal{O}((t\sqrt{d}\|H\|_{1 \rightarrow 2})^{1+o(1)}/\epsilon^{o(1)})$ queries and a factor $\mathcal{O}(b)$ more gates, which is shown to be optimal up to subpolynomial factors through a matching query lower bound. This provides a polynomial speedup in sparsity for the common case where the spectral norm $\|H\|\ge\|H\|_{1 \rightarrow 2}$ is known, and generalizes previous approaches which achieve optimal scaling, but with respect to more restrictive parameters. By exploiting knowledge of the spectral norm, our algorithm solves the black-box unitary implementation problem -- $\mathcal{O}(d^{1/2+o(1)})$ queries suffice to approximate any $d$-sparse unitary in the black-box setting, which matches the quantum search lower bound of $\Omega(\sqrt{d})$ queries and improves upon prior art [Berry and Childs, QIP 2010] of $\tilde{\mathcal{O}}(d^{2/3})$ queries. Combined with known techniques, we also solve systems of sparse linear equations with condition number $\kappa$ using $\mathcal{O}((\kappa \sqrt{d})^{1+o(1)}/\epsilon^{o(1)})$ queries, which is a quadratic improvement in sparsity.