A block encoding circuit for a D-dimensional Laplacian with periodic boundary conditions

[virajd98]

Hi,

I'll be implementing most parts of this paper.

Block encoding is a nice technique to input non-unitary matrices on a quantum computer by embedding them in larger unitaries. But, this process is not trivial for random matrices. However, for certain structured and sparse matrices this can be done In efficient ways by constructing appropriate oracles that exploit the structure and sparsity in the matrices. This paper explicitly provides such construction for the finite difference discretization of a d-dimensional Laplacian operator with periodic boundary conditions. This efficient embedding is useful in several fields like computational fluid dynamics that require embedding non unitary matrices that are structured and sparse for several quantum algorithms. Notably, the technique provided in the paper allows for embedding large matrices using fewer resources compared to the widely known LCU approach for block encoding.

Technical Approach for Implementation:

I'll construct a general quantum circuit via Qmod for block encoding a d-dimensional laplacian as per the construction given in the paper and then benchmark the algorithm. Also, beyond the paper, I'll provide estimates as to how this approach scales efficiently compared to linear combination of unitaries. I'll try to complete implementing all the aspects of the paper. The approach and final deliverable would be similar as one of my past contributions on block encodings to classiq.

Deliverables

A detailed Jupyter notebook with all implementations and benchmarks.

Thanks

[TomerGoldfriend]

Thank you @virajd98 for this issue. Sounds good! Note that in many cases, when dealing with Laplacians, one can apply quantum cosine or sine transforms, and then you need to encode a diagonal matrix (see here). It might be nice to discuss this direction/usecase as well.

Please note that we accept high-quality implementations to our repository and will be glad to accept a PRs that meets our standards. Looking forward for your notebook.