  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.
