Block-encoding structured matrices for data input in quantum computing

Christoph Sünderhauf1, Earl Campbell1,2, and Joan Camps1

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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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.

Quantum computers promise to solve certain problems much, much faster than even the largest classical supercomputers. Data describing the problem to be solved must be loaded onto the quantum computer; this data loading can be a bottleneck, even thwarting the entire speed-up promised by quantum computation. A particularly widespread method of representing numerical data matrices on quantum computers is in the form of “block encodings”.

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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