Optimal Hamiltonian Simulation by Quantum Signal Processing

The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a $d$-sparse Hamiltonian $\hat{H}$ for time-interval $t$ with error $\epsilon$ is $\mathcal{O}(td\|\hat{H}\|_{\text{max}}+\frac{\log{(1/\epsilon)}}{\log{\log{(1/\epsilon)}}})$, which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (1) transducing eigenvalues of $\hat{H}$ into a single ancilla qubit, (2) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (3) projecting this ancilla with near unity success probability.