Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations

Juan José García-Ripoll

Institute of Fundamental Physics, Calle Serrano 113b, 28006 Madrid, Spain

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In this work we study the encoding of smooth, differentiable multivariate functions in quantum registers, using quantum computers or tensor-network representations. We show that a large family of distributions can be encoded as low-entanglement states of the quantum register. These states can be efficiently created in a quantum computer, but they are also efficiently stored, manipulated and probed using Matrix-Product States techniques. Inspired by this idea, we present eight quantum-inspired numerical analysis algorithms, that include Fourier sampling, interpolation, differentiation and integration of partial derivative equations. These algorithms combine classical ideas – finite-differences, spectral methods – with the efficient encoding of quantum registers, and well known algorithms, such as the Quantum Fourier Transform. $\textit{When these heuristic methods work}$, they provide an exponential speed-up over other classical algorithms, such as Monte Carlo integration, finite-difference and fast Fourier transforms (FFT). But even when they don't, some of these algorithms can be translated back to a quantum computer to implement a similar task.

Computers have been traditionally an invaluable tool in the solution of complex mathematical problems, in particular in the field of numerical analysis and differential equations. In this work I continue an ongoing exploration of how quantum computers could encode multivariate functions and achieve a similar goal. This exploration reveals that many smooth differentiable functions do not require a lot of entanglement when encoded in a quantum computer. This is good, because it means that such functions can be efficiently recreated in quantum computers with short, shallow circuits. However, it also means that those quantum algorithms have an efficient classical description and can be transformed into new classical and quantum inspired methods to implement Fourier transforms, interpolation, integration or solving differential equations.

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