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

TL;DR

This paper introduces quantum-inspired algorithms for multivariate analysis that leverage quantum encoding techniques to achieve exponential speed-ups in tasks like interpolation and solving partial differential equations, blending classical and quantum ideas.

Contribution

It presents novel quantum-inspired numerical algorithms for multivariate analysis, combining classical methods with quantum encoding to improve efficiency and potential speed-ups.

Findings

Efficient encoding of multivariate functions as low-entanglement quantum states.

Quantum-inspired algorithms can exponentially outperform classical methods in certain tasks.

Some algorithms can be adapted for quantum computers even if heuristic methods fail.

Abstract

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. {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.