Quantum Krylov Algorithm for Szegö Quadrature

• URL: http://arxiv.org/abs/2509.19195v1
• Date: Tue, 23 Sep 2025 16:13:08 GMT
• Title: Quantum Krylov Algorithm for Szegö Quadrature
• Authors: William Kirby, Yizhi Shen, Daan Camps, Anirban Chowdhury, Katherine Klymko, Roel Van Beeumen,
• Abstract summary: We present a quantum algorithm to evaluate matrix elements of functions of unitary operators.<n>The method is based on calculating quadrature nodes and weights using data collected from a quantum processor.
• Score: 0.8158532237212478
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $|\psi_0\rangle$, $|\psi_1\rangle$, the resulting quadrature rules form a functional that can then be used to classically approximate $\langle\psi_1|f(U)|\psi_0\rangle$ for any function $f$. In particular, the algorithm calculates Szeg\"o 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.

Related papers

• Quantum oracles for the finite element method [45.200826131319815]
  This study examines the quantum routines required for the implementation of oracles used in the block-encoding of the $N times N stiffness and mass matrices.<n>We show how to construct the necessary oracles, which require the calculation of element geometry, square root and the implementation of conditional operations.
  arXiv  Detail & Related papers  (2025-04-28T14:28:31Z)
• Arbitrary state creation via controlled measurement [49.494595696663524]
  This algorithm creates an arbitrary $n$-qubit pure quantum superposition state with precision of $m$-decimals.<n>The algorithm uses one-qubit rotations, Hadamard transformations and C-NOT operations with multi-qubit controls.
  arXiv  Detail & Related papers  (2025-04-13T07:23:50Z)
• Quantum algorithm for the gradient of a logarithm-determinant [0.0]
  The inverse of a sparse-rank input operator may be determined efficiently.<n>The algorithm is envisioned for fully error-corrected quantum computers.<n>We discuss how this algorithm can be used for kernel-based quantum machine-learning.
  arXiv  Detail & Related papers  (2025-01-16T09:39:31Z)
• Quantum Algorithms for Stochastic Differential Equations: A Schrödingerisation Approach [29.662683446339194]
  We propose quantum algorithms for linear differential equations.<n>The gate complexity of our algorithms exhibits an $mathcalO(dlog(Nd))$ dependence on the dimensions.<n>The algorithms are numerically verified for the Ornstein-Uhlenbeck processes, Brownian motions, and one-dimensional L'evy flights.
  arXiv  Detail & Related papers  (2024-12-19T14:04:11Z)
• Resource-efficient algorithm for estimating the trace of quantum state powers [1.5133368155322298]
  Estimating the trace of quantum state powers, $textTr(rhok)$, for $k$ identical quantum states is a fundamental task.<n>We introduce an algorithm that requires only $mathcalO(widetilder)$ qubits and $mathcalO(widetilder)$ multi-qubit gates.<n>We extend our algorithm to the estimation of $textTr(Mrhok)$ for arbitrary observables and $textTr(rho
  arXiv  Detail & Related papers  (2024-08-01T06:23:52Z)
• A Novel Quantum-Classical Hybrid Algorithm for Determining Eigenstate Energies in Quantum Systems [1.9714447272714082]
  This paper presents a novel quantum algorithm, XZ24, for efficiently computing the eigen-energy spectra of arbitrary quantum systems. XZ24 has three key advantages: It removes the need for eigenstate preparation, requiring only a reference state with non-negligible overlap. It enables simultaneous computation of multiple eigen-energies, depending on the reference state.
  arXiv  Detail & Related papers  (2024-06-01T04:31:43Z)
• Calculating response functions of coupled oscillators using quantum phase estimation [40.31060267062305]
  We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer.<n>Our proposed quantum algorithm operates in the standard $s-sparse, oracle-based query access model.<n>We show that a simple adaptation of our algorithm solves the random glued-trees problem in time.
  arXiv  Detail & Related papers  (2024-05-14T15:28:37Z)
• Non-Linear Transformations of Quantum Amplitudes: Exponential Improvement, Generalization, and Applications [0.0]
  Quantum algorithms manipulate the amplitudes of quantum states to find solutions to computational problems. We present a framework for applying a general class of non-linear functions to the amplitudes of quantum states. Our work provides an important and efficient building block with potentially numerous applications in areas such as optimization, state preparation, quantum chemistry, and machine learning.
  arXiv  Detail & Related papers  (2023-09-18T14:57:21Z)
• Quantum Subroutine Composition [0.59829224684009]
  In quantum algorithms, a subroutine may be called on a superposition of different inputs, which complicates things.<n>We prove this by using the technique of multidimensional quantum walks, recently introduced in arXiv:2208.13492.<n>The same technique that allows us to compose quantum subroutines in quantum walks can also be used to compose in any quantum algorithm.
  arXiv  Detail & Related papers  (2022-09-28T14:52:13Z)
• Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ans\"atze [13.651587339535961]
  We develop a framework for convolutional quantum circuits with SU$(d)$symmetry. We prove Harrow's statement on equivalence between $nameSU(d)$ and $S_n$ irrep bases.
  arXiv  Detail & Related papers  (2021-12-14T18:03:43Z)
• Quantum Algorithm for Fidelity Estimation [8.270684567157987]
  For two unknown mixed quantum states $rho$ and $sigma$ in an $N$-dimensional space, computing their fidelity $F(rho,sigma)$ is a basic problem. We propose a quantum algorithm that solves this problem in $namepoly(log (N), r, 1/varepsilon)$ time.
  arXiv  Detail & Related papers  (2021-03-16T13:57:01Z)
• Quantum algorithms for spectral sums [50.045011844765185]
  We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory.
  arXiv  Detail & Related papers  (2020-11-12T16:29:45Z)
• Emulating Quantum Interference with Generalized Ising Machines [0.0]
  This paper presents an exact and general procedure for mapping any sequence of quantum gates onto a network of probabilistic p-bits. We can view this structure as a Boltzmann machine whose states each represent a Feynman path leading from an initial configuration of qubits to a final configuration. Our results for mapping an arbitrary quantum circuit to a Boltzmann machine with a complex energy function should help push the boundaries of the simulability of quantum circuits with probabilistic resources.
  arXiv  Detail & Related papers  (2020-07-14T22:10:29Z)
• Quantum Legendre-Fenchel Transform [6.643082745560234]
  We present a quantum algorithm to compute the discrete Legendre-Fenchel transform. We show that our quantum algorithm is optimal up to polylogarithmic factors.
  arXiv  Detail & Related papers  (2020-06-08T18:00:05Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.