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An Improved Method for Quantum Matrix Multiplication: Appendix A
by Eigenvector Initialization PublicationJune 15th, 2024
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Quantum algorithms significantly improve efficiency in matrix operations, including eigenvalue and trace estimation, leveraging Chebyshev polynomials for exponential precision enhancements.
Authors:
(1) Nhat A. Nghiem, Department of Physics and Astronomy, State University of New York (email: [email protected]);
(2) Tzu-Chieh Wei, Department of Physics and Astronomy, State University of New York and C. N. Yang Institute for Theoretical Physics, State University of New York.
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Acknowledgements, Declarations, Data Availability Statement, and References
Appendix A: Review of Chebyshev Approach
Here we make a review of Chebyshev approach that was employed in [2], which is essentially built upon quantum walk technique [6, 8]. What we will describe below is more or less a summary of Section 4 in Ref. [2], the result of which was used in our main text.
The so-called walk operator is defined as:
Let |λ⟩ and λ be eigenvector and eigenvalue of A/d (note that the scaling by d doesn’t have further systematic problem, as the spectrum remains the same, only eigenvalues got scaled by a factor). Within the subspace spanned by T |λ⟩ and ST |λ⟩, W admits the following block form:
The proof can be found in Lemma 15 of [2]. The above form of W possess the following remarkable property (Lemma 16 of [2]),
This paper is available on arxiv under CC 4.0 license.
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