Quantum Krylov Algorithm for Szegö Quadrature Article Swipe

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William Kirby , Daan Camps , Anirban Chowdhury ·

YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2509.19195

We present a quantum algorithm to evaluate matrix elements of functions of unitary operators. The method is based on calculating quadrature nodes and weights using data collected from a quantum processor. Given a unitary $U$ and quantum states $|ψ_0\rangle$, $|ψ_1\rangle$, the resulting quadrature rules form a functional that can then be used to classically approximate $\langleψ_1|f(U)|ψ_0\rangle$ for any function $f$. In particular, the algorithm calculates Szegö quadrature rules, which, when $f$ is a Laurent polynomial, have the optimal relation between degree of $f$ and number of distinct quantum circuits required. The unitary operator $U$ could approximate a time evolution, opening the door to applications like estimating properties of Hamiltonian spectra and Gibbs states, but more generally could be any operator implementable via a quantum circuit. We expect this algorithm to be useful as a subroutine in other quantum algorithms, much like quantum signal processing or the quantum eigenvalue transformation of unitaries. Key advantages of our algorithm are that it does not require approximating $f$ directly, via a series expansion or in any other way, and once the output functional has been constructed using the quantum algorithm, it can be applied to any $f$ classically after the fact.

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2509.19195
PDF: https://arxiv.org/pdf/2509.19195
OA Status: green
OpenAlex ID: https://openalex.org/W4416255024

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DOI: https://doi.org/10.48550/arxiv.2509.19195

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Title: Quantum Krylov Algorithm for Szegö Quadrature

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Type: preprint

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Language: en

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Publication year: 2025

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Publication date: 2025-09-23

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Authors: William Kirby, Daan Camps, Anirban Chowdhury

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Landing page: https://arxiv.org/abs/2509.19195

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PDF URL: https://arxiv.org/pdf/2509.19195

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