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Quadratically Shallow Quantum Circuits for Hamiltonian Functions

Youngjun Park, Minhyeok Kang, Chae-Yeun Park, Joonsuk Huh

TL;DR

This work develops a generalized framework for approximating Hamiltonian functions with quadratically reduced polynomial degrees using Chebyshev and newly formulated Laurent polynomial approximations. By extending to cosine/sine functions via Laurent polynomials and proving two key theorems for linear combinations and products, it enables δ-accurate approximations with degrees that scale as the square root of the original degrees. The approximations are then embedded into quantum circuits through QSP and GQSP, achieving significant depth reductions from linear to near-quadratic scaling in the polynomial degree for a broad class of Hamiltonian filters. The result is a versatile approach to depth-efficient Hamiltonian-function synthesis, with direct implications for quantum ground-state preparation and energy estimation.

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 $O(\sqrt{n\ln(1/δ)})$ for an $n$-degree polynomial, where $δ$ 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 $δ$-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 $δ$-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.

Quadratically Shallow Quantum Circuits for Hamiltonian Functions

TL;DR

This work develops a generalized framework for approximating Hamiltonian functions with quadratically reduced polynomial degrees using Chebyshev and newly formulated Laurent polynomial approximations. By extending to cosine/sine functions via Laurent polynomials and proving two key theorems for linear combinations and products, it enables δ-accurate approximations with degrees that scale as the square root of the original degrees. The approximations are then embedded into quantum circuits through QSP and GQSP, achieving significant depth reductions from linear to near-quadratic scaling in the polynomial degree for a broad class of Hamiltonian filters. The result is a versatile approach to depth-efficient Hamiltonian-function synthesis, with direct implications for quantum ground-state preparation and energy estimation.

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 for an -degree polynomial, where 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 -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 -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.

Paper Structure

This paper contains 11 sections, 11 theorems, 82 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Chebyshev and Laurent Polynomial Approximation
  3. Efficient Approximations to a Linear Combination and a Product of Polynomial-approximable Functions
  4. Synthesizing Quadratically Shallow Circuits for the Polynomial-Approximable Functions
  5. Conclusion
  6. The Chebyshev Polynomial Approximation Theorems and Proofs
  7. The Laurent Polynomial Approximations to Sine Functions
  8. Proofs to the Main Theorems
  9. Review of QSP
  10. Review of GQSP
  11. Depth Analysis of GQSP Circuits Implementing Laurent Polynomials

Key Result

Theorem 1

Let $F: [-1,1] \times \mathbb{R} \rightarrow \mathbb{R}$ be a linear combination of elements $f_i \in \mathcal{F}_{\mathrm{all}}$ and $g_j \in \mathcal{G}_{\mathrm{all}}$, i.e., for real coefficients $a_i, b_j$, where $\mathcal{N} \geq 0$ and $\mathcal{M} \geq 0$ represent the constant number of functions. Let $h_j(z)$ denote the Laurent polynomial representation of each $g_j(y)$. The degree of $

Figures (2)

  • Figure 1: The QSP circuit for implementing a real polynomial $f(\mathcal{H})$ of degree $d$ using a block-encoding $U_{\mathcal{H}}$dong2021efficient. Each signal-processing operator $\mathrm{e}^{\mathrm{i} \varphi_j U_{\Pi}}$ is realized using a gate sequence consisting of an $(M+1)$-qubit Toffoli gate, a single-qubit $Z$-rotation, and another $(M+1)$-qubit Toffoli gate.
  • Figure 2: GQSP circuit for implementing a Laurent polynomial $L(U)$ with real coefficients. The rotation gate $R(\theta)$ is defined in Eq. \ref{['eq:lqsp_spo']}.

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • ...and 8 more