Double-Logarithmic Depth Block-Encodings of Simple Finite Difference Method's Matrices

Solving differential equations is one of the most computationally expensive problems in classical computing, occupying the vast majority of high-performance computing resources devoted towards practical applications in various fields of science and engineering. Despite recent progress made in the field of quantum computing and quantum algorithms, its end-to-end application towards practical realization still remains unattainable. In this article, we tackle one of the primary obstacles towards this ultimate objective, specifically the encoding of matrices derived via finite difference method solving Poisson partial differential equations in simple boundary-value problems. To that end, we propose a novel methodology called block-diagonalization, which provides a common decomposition form for our matrices, and similarly a common procedure for block-encoding these matrices inside a unitary operator of a quantum circuit. The depth of these circuits is double-logarithmic in the matrix size, which is an exponential improvement over existing quantum methods and a superexponential improvement over existing classical methods. These improvements come at the price of a constant multiplicative overhead on the number of qubits and the number of gates. Combined with quantum linear solver algorithms, we can utilize these quantum circuits to produce a quantum state representation of the solution to the Poisson partial differential equations and their boundary-value problems.