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Matrix Encoding Method in Variational Quantum Singular Value Decomposition Cover

Matrix Encoding Method in Variational Quantum Singular Value Decomposition

Quantum Information & Computation
Volume 25 (2025): Issue 4 (August 2025)
By: Alexander I. Zenchuk,  Wentao Qi and  Junde Wu  
Open Access
|Aug 2025

Figures & Tables

Figure 1.

Structure of subsystems required for performing the quantum part of variational SVD. It includes five n-qubit subsystems for encoding the matrix M(subsystems R, C), orthonormal eigenvectors 〈ψj| and |ψj〉 (subsystems χ, ψ) and weights qi (subsystem q). In addition, the one-qubit subsystem K serves as a controlling qubit in controlled operators of the algorithm and two one-qubit ancillae B and B˜{\tilde B} serve for the controlled measurement.
Structure of subsystems required for performing the quantum part of variational SVD. It includes five n-qubit subsystems for encoding the matrix M(subsystems R, C), orthonormal eigenvectors 〈ψj| and |ψj〉 (subsystems χ, ψ) and weights qi (subsystem q). In addition, the one-qubit subsystem K serves as a controlling qubit in controlled operators of the algorithm and two one-qubit ancillae B and B˜{\tilde B} serve for the controlled measurement.

Figure 2.

The circuit for the quantum part of the variational SVD algorithm. The depth of this circuit can be estimated as O(Qlog(N)/ε). (a) The circuit for creating the state |ψout〉K given in (27); the operatorsW(j), j = 0, …,6 are presented without subscripts for brevity. (b) The operators applied to the state |ψout〉KB to probabilistically obtain the normalization G and the objective function L(α, β).
The circuit for the quantum part of the variational SVD algorithm. The depth of this circuit can be estimated as O(Qlog(N)/ε). (a) The circuit for creating the state |ψout〉K given in (27); the operatorsW(j), j = 0, …,6 are presented without subscripts for brevity. (b) The operators applied to the state |ψout〉KB to probabilistically obtain the normalization G and the objective function L(α, β).

Figure 3.

The circuit for the operatorsWψK(2)(α)W_{\chi K}^{(2)}(\alpha ) (and WψK(2)(β)W_{\psi K}^{(2)}(\beta )). Here the set of parameters γ is either α (for WψK(2)(α)W_{\chi K}^{(2)}(\alpha )) or β (for WψK(2)(β)W_{\psi K}^{(2)}(\beta )), Z ≡ σ(z).
The circuit for the operatorsWψK(2)(α)W_{\chi K}^{(2)}(\alpha ) (and WψK(2)(β)W_{\psi K}^{(2)}(\beta )). Here the set of parameters γ is either α (for WψK(2)(α)W_{\chi K}^{(2)}(\alpha )) or β (for WψK(2)(β)W_{\psi K}^{(2)}(\beta )), Z ≡ σ(z).

Figure 4.

The hybrid algorithm for calculating SVD. The matrices Û, D and V^{\hat V} are defined in terms of U(α*) and U(β*) according to (4), ΔL = |L[k+1] – L[k]|. For simplicity, we indicate only the function L and first derivatives ∂γL to be transferred from the output of the quantum algorithm to the input of the classical algorithm. However, the higher order derivatives might be required as well.
The hybrid algorithm for calculating SVD. The matrices Û, D and V^{\hat V} are defined in terms of U(α*) and U(β*) according to (4), ΔL = |L[k+1] – L[k]|. For simplicity, we indicate only the function L and first derivatives ∂γL to be transferred from the output of the quantum algorithm to the input of the classical algorithm. However, the higher order derivatives might be required as well.
DOI: https://doi.org/10.2478/qic-2025-0020 | Journal eISSN: 3106-0544 | Journal ISSN: 1533-7146
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Language: English
Page range: 356 - 368
Submitted on: Apr 14, 2025
Accepted on: Jul 1, 2025
Published on: Aug 22, 2025
Published by: Cerebration Science Publishing Co., Limited
In partnership with: Paradigm Publishing Services
Publication frequency: 1 issue per year
Keywords:
quantum singular value decomposition,
matrix encoding,
controlled measurement,
probabilistic method,
gradient method
Related subjects:
Physics,
Quantum physics,
Mathematics,
Applied mathematics,
Computer sciences,
Computer sciences in the humanities,
Computer sciences,
Computer sciences in natural sciences

© 2025 Alexander I. Zenchuk, Wentao Qi, Junde Wu, published by Cerebration Science Publishing Co., Limited
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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