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We present a quantum algorithm for approximating the real time evolution$e^{-iHt}$ of an arbitrary $d$-sparse Hamiltonian to error $\epsilon$, givenblack-box access to the positions and $b$-bit values of its non-zero matrixentries. 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 subpolynomialfactors through a matching query lower bound. This provides a polynomialspeedup in sparsity for the common case where the spectral norm$\|H\|\ge\|H\|_{1 \rightarrow 2}$ is known, and generalizes previous approacheswhich achieve optimal scaling, but with respect to more restrictive parameters.By exploiting knowledge of the spectral norm, our algorithm solves theblack-box unitary implementation problem -- $\mathcal{O}(d^{1/2+o(1)})$ queriessuffice to approximate any $d$-sparse unitary in the black-box setting, whichmatches the quantum search lower bound of $\Omega(\sqrt{d})$ queries andimproves upon prior art [Berry and Childs, QIP 2010] of$\tilde{\mathcal{O}}(d^{2/3})$ queries. Combined with known techniques, we alsosolve systems of sparse linear equations with condition number $\kappa$ using$\mathcal{O}((\kappa \sqrt{d})^{1+o(1)}/\epsilon^{o(1)})$ queries, which is aquadratic improvement in sparsity.

Hamiltonian simulation with nearly optimal dependence on spectral norm