# Quantum eigenvalue processing

arXiv: https://arxiv.org/abs/2401.06240 · HTML: https://mlsanity.com/paper/2401.06240/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jan 2024.

## Built on by (most cited first, 1 of 1)

- [Block-encodings as programming abstractions: The Eclipse Qrisp BlockEncoding Interface](https://mlsanity.com/paper/2604.18276/) (2026)

## Abstract (arXiv)

Many problems in linear algebra -- such as those arising from non-Hermitian physics and differential equations -- can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited for this task, as eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators, and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum superposition of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generate n Fourier coefficients with O(polylog(n)) gates compared to prior approaches with linear cost.