Quadratically Shallow Quantum Circuits For Hamiltonian Functions

Abstract

Many quantum algorithms for ground-state preparation and energy estimation require the implementation of high-degree polynomials of a Hamiltonian to achieve better convergence rates. Their circuit implementation typically relies on quantum signal processing (QSP), whose circuit depth is proportional to the degree of the polynomial. Previous studies exploit the Chebyshev polynomial approximation, which requires a Chebyshev series of degree $O(\sqrt{\{}n\ln (1/\delta )\})$ for an $n$-degree polynomial, where $\delta$ is the approximation error. However, the approximation is limited to only a few functions, including monomials, truncated exponential, Gaussian, and error functions. In this work, we present the most generalized function approximation methods for $\delta$-approximating linear combinations or products of polynomial-approximable functions with quadratically reduced-degree polynomials. We extend the list of polynomial-approximable functions by showing that the functions of