A Relation Between N-Square and M-Square Matrix Vector Bases of the Same Dimension - Relating Spin Matrices and Components

Claude Michael Cassano

Published Oct 1, 2016• 0 citations

Abstract

Elementary Linear Algebra theory handles transformations between sets of n-square matrix vector bases of M-dimensions well. Considering n-square & m-square matrices of M-dimensions, further theory and techniques, shown here, may be applied to yield results; including relationships between spin matrices and components.

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A Relation Between N-Square and M-Square Matrix Vector Bases of the Same Dimension - Relating Spin Matrices and Components

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Published Oct 1, 2016• 0 citations

OpenAlex

Abstract

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u\in\mathbb{C}^{n}

, we can sample from the distribution

p_u\colon\{1,\ldots,n\}\to [0,1]

defined as

p_u(i)=|u(i)|^2

for each

i\in\{1,\ldots,n\}

. Since this distribution corresponds to the distribution obtained by measuring the quantum state |

u

> in the computational basis, length-squared sampling access gives a reasonable classical analogue to the kind of quantum access considered in many quantum algorithms for linear algebra problems. In this work we investigate how robust these dequantization results are. We introduce the notion of approximate length-squared sampling, where classical algorithms are only able to sample from a distribution close to the ideal distribution in total variation distance. While quantum algorithms are natively robust against small perturbations, current techniques in dequantization are not. Our main technical contribution is showing how many techniques from randomized linear algebra can be adapted to work under this weaker assumption as well. We then use these techniques to show that the recent low-rank dequantization framework by Chia, Gilyén, Li, Lin, Tang and Wang (JACM 2022) and the dequantization framework for sparse matrices by Gharibian and Le Gall (STOC 2022), which are both based on the Quantum Singular Value Transformation, can be generalized to the case of approximate length-squared sampling access to the input. We also apply these results to obtain a robust dequantization of many quantum machine learning algorithms, including quantum algorithms for recommendation systems, supervised clustering and low-rank matrix inversion.