An Improved Method for Quantum Matrix Multiplication: Appendix A

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15 Jun 2024

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.

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]),

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