1Riverlane, St. Andrews House, 59 St. Andrews Street, Cambridge CB2 3BZ, United Kingdom
2Dept. of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom
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Abstract
The cost of data input can dominate the run-time of quantum algorithms. Here, we consider data input of arithmetically structured matrices via $textit{block encoding}$ circuits, the input model for the quantum singular value transform and related algorithms. We demonstrate how to construct block encoding circuits based on an arithmetic description of the sparsity and pattern of repeated values of a matrix. We present schemes yielding different subnormalisations of the block encoding; a comparison shows that the best choice depends on the specific matrix. The resulting circuits reduce flag qubit number according to sparsity, and data loading cost according to repeated values, leading to an exponential improvement for certain matrices. We give examples of applying our block encoding schemes to a few families of matrices, including Toeplitz and tridiagonal matrices.
Featured image: Block encoding circuit for the base scheme introduced in this article. The row and column oracles translate between a description of the matrix in terms of repeated elements (d: data index, m: multiplicity index) and a description in terms of row index i and row sparsity index s_r, or column index j and sparsity index s_c. Thereby, both sparsity and repetition of data elements can be harnessed to reduce circuit cost.
Popular summary
In this research article, we present a new set of schemes how data can be loaded into block encodings. Particularly, if the data matrices are structured, i.e. have a certain pattern and/or repeated data elements, our scheme shows how to make use of this structure in order to reduce the cost of data loading. We explain how to construct quantum circuits taking into account and optimising for such structured data. In the future, our work can help to load various data matrices into quantum computers for use in various quantum algorithms, making the most of the data’s structure to reduce the data loading bottleneck.
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