Hamiltonian Simulation by Uniform Spectral Amplification

Abstract: The exponential speedups promised by Hamiltonian simulation on a quantum computer depends crucially on structure in both the Hamiltonian $\hat{H}$, and the quantum circuit $\hat{U}$ that encodes its description. In the quest to better approximate time-evolution $e^{-i\hat{H}t}$ with error $\epsilon$, we motivate a systematic approach to understanding and exploiting structure, in a setting where Hamiltonians are encoded as measurement operators of unitary circuits $\hat{U}$ for generalized measurement. This allows us to define a \emph{uniform spectral amplification} problem on this framework for expanding the spectrum of encoded Hamiltonian with exponentially small distortion. We present general solutions to uniform spectral amplification in a hierarchy where factoring $\hat{U}$ into $n=1,2,3$ unitary oracles represents increasing structural knowledge of the encoding. Combined with structural knowledge of the Hamiltonian, specializing these results allow us simulate time-evolution by $d$-sparse Hamiltonians using $\mathcal{O}\left(t(d |\hat H|{\text{max}}|\hat H|{1})^{1/2}\log{(t|\hat{H}|/\epsilon)}\right)$ queries, where $|\hat H|\le |\hat H|1\le d|\hat H|{\text{max}}$. Up to logarithmic factors, this is a polynomial improvement upon prior art using $\mathcal{O}\left(td|\hat H|{\text{max}}+\frac{\log{(1/\epsilon)}}{\log\log{(1/\epsilon)}}\right)$ or $\mathcal{O}(t^{3/2}(d |\hat H|{\text{max}}|\hat H|{1}|\hat H|/\epsilon)^{1/2})$ queries. In the process, we also prove a matching lower bound of $\Omega(t(d|\hat H|{\text{max}}|\hat H|_{1})^{1/2})$ queries, present a distortion-free generalization of spectral gap amplification, and an amplitude amplification algorithm that performs multiplication on unknown state amplitudes.