Spectrally Corrected Polynomial Approximation for Quantum Singular Value Transformation

2026-03-04

AI Summary

This work addresses the limitations of existing quantum singular value transformation (QSVT) approaches, whose polynomial approximations ignore spectral information of the matrix, leading to excessive circuit depth and constrained accuracy. The authors propose a spectral correction method that, given partial knowledge of eigenvalues, modifies any base polynomial to interpolate exactly at those points while preserving the original error bounds, parity, and generality. This is the first technique to incorporate partial spectral information directly into QSVT construction without increasing polynomial degree, thereby enhancing approximation accuracy. Compatible with classical strategies such as Remez approximation, the method achieves up to a 5× reduction in circuit depth with unit fidelity for the 1D Poisson equation, and in 2D cases attains fidelity exceeding 0.999 by correcting only a few eigenvalues, demonstrating marked gains in efficiency and robustness.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Quantum Singular Value Transformation (QSVT) provides a unified framework for applying polynomial functions to the singular values of a block-encoded matrix. QSVT prepares a state proportional to $\bA^{-1}\bb$ with circuit depth $O(d\cdot\mathrm{polylog}(N))$, where $d$ is the polynomial degree of the $1/x$ approximation and $N$ is the size of $\bA$. Current polynomial approximation methods are over the continuous interval $[a,1]$, giving $d = O(\sqrt{\kap}\log(1/\varepsilon))$, and make no use of any properties of $\bA$. We observe here that QSVT solution accuracy depends only on the polynomial accuracy at the eigenvalues of $\bA$. When all $N$ eigenvalues are known exactly, a pure spectral polynomial $p_{S}$ can interpolate $1/x$ at these eigenvalues and achieve unit fidelity at reduced degree. But its practical applicability is limited. To address this, we propose a spectral correction that exploits prior knowledge of $K$ eigenvalues of $\bA$. Given any base polynomial $p_0$, such as Remez, of degree $d_0$, a $K\times K$ linear system enforces exact interpolation of $1/x$ only at these $K$ eigenvalues without increasing $d_0$. The spectrally corrected polynomial $p_{SC}$ preserves the continuous error profile between eigenvalues and inherits the parity of $p_0$. QSVT experiments on the 1D Poisson equation demonstrate up to a $5\times$ reduction in circuit depth relative to the base polynomial, at unit fidelity and improved compliance error. The correction is agnostic to the choice of base polynomial and robust to eigenvalue perturbations up to $10\%$ relative error. Extension to the 2D Poisson equation suggests that correcting a small fraction of the spectrum may suffice to achieve fidelity above $0.999$.

Problem

• Quantum Singular Value Transformation
• polynomial approximation
• spectral information
• eigenvalue interpolation
• circuit depth

Innovation

• Quantum Singular Value Transformation
• Spectral Correction
• Polynomial Approximation
• Eigenvalue Interpolation
• Circuit Depth Reduction