Quadratically Shallow Quantum Circuits for Hamiltonian Functions

• URL: http://arxiv.org/abs/2510.04059v1
• Date: Sun, 05 Oct 2025 06:43:18 GMT
• Title: Quadratically Shallow Quantum Circuits for Hamiltonian Functions
• Authors: Youngjun Park, Minhyeok Kang, Chae-Yeun Park, Joonsuk Huh,
• Abstract summary: Many quantum algorithms for ground-state preparation and energy estimation require the implementation of high-degrees of a Hamiltonian to achieve better convergence rates.<n>We show that various Hamiltonian functions for ground-state preparation and energy estimation can be implemented with quadratically shallow circuits.
• Score: 3.218714138503326
• License: http://creativecommons.org/licenses/by/4.0/
• 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 cosine and sine can also be $\delta$-approximated by quadratically reduced-degree Laurent polynomials. We demonstrate that various Hamiltonian functions for quantum ground-state preparation and energy estimation can be implemented with quadratically shallow circuits.

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