Quantum Krylov Algorithm for Szegö Quadrature
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914052440064000 |
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| author | Kirby, William Shen, Yizhi Camps, Daan Chowdhury, Anirban Klymko, Katherine Van Beeumen, Roel |
| author_facet | Kirby, William Shen, Yizhi Camps, Daan Chowdhury, Anirban Klymko, Katherine Van Beeumen, Roel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19195 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum Krylov Algorithm for Szegö Quadrature Kirby, William Shen, Yizhi Camps, Daan Chowdhury, Anirban Klymko, Katherine Van Beeumen, Roel Quantum Physics 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. |
| title | Quantum Krylov Algorithm for Szegö Quadrature |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2509.19195 |