An Improved Method for Quantum Matrix Multiplication: Appendix A

Authors: (1) Nhat A. Nghiem, Department of Physics and Astronomy, State University of New York (email: nhatanh.nghiemvu@stonybrook.edu); (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. Table of Links Main Procedure Applications Discussion and Conclusion Acknowledgements, Declarations, Data Availability Statement, and References Appendix 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. Authors: (1) Nhat A. Nghiem, Department of Physics and Astronomy, State University of New York (email: nhatanh.nghiemvu@stonybrook.edu); (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. Authors: Authors: (1) Nhat A. Nghiem, Department of Physics and Astronomy, State University of New York (email: nhatanh.nghiemvu@stonybrook.edu); (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. Table of Links Main Procedure Main Procedure Applications Applications Discussion and Conclusion Discussion and Conclusion Acknowledgements, Declarations, Data Availability Statement, and References Acknowledgements, Declarations, Data Availability Statement, and References Appendix Appendix 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: walk operator 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. This paper is available on arxiv under CC 4.0 license. available on arxiv