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

• Main Author: Juan José García-Ripoll
• Format: Article
• Language: English
• Published: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften 2021-04-01
• Series: Quantum
• Online Access: https://quantum-journal.org/papers/q-2021-04-15-431/pdf/

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.