Optimal Hamiltonian Simulation by Quantum Signal Processing

TLDR

Quantum mechanics underlies quantum computation, yet designing Hamiltonian simulation algorithms has remained abstract and unintuitive despite expectations that physical intuition would guide them. The authors aim to show that simple single‑qubit rotations, guided by physical intuition, yield an optimal algorithm for Hamiltonian simulation. They employ a three‑step quantum signal processing approach that transduces Hamiltonian eigenvalues into an ancilla qubit, applies an optimal sequence of single‑qubit rotations, and projects the ancilla with near‑unity success probability. They prove that the query complexity for simulating a d‑sparse Hamiltonian over time t with error ε is O(td‖H‖max + log(1/ε)/log log(1/ε)), matching all known lower bounds.

Abstract

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 $\stackrel{^}{H}$ for time-interval $t$ with error $\ensuremath{\epsilon}$ is $\mathcal{O}[td\ensuremath{\parallel}\stackrel{^}{H}{\ensuremath{\parallel}}_{\mathrm{max}}+\mathrm{log}(1/\ensuremath{\epsilon})/\mathrm{log}\mathrm{log}(1/\ensuremath{\epsilon})]$, which matches lower bounds in all parameters. This connection is made through general three-step ``quantum signal processing'' methodology, comprised of (i) transducing eigenvalues of $\stackrel{^}{H}$ into a single ancilla qubit, (ii) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii) projecting this ancilla with near unity success probability.