Preconditioned Multivariate Quantum Solution Extraction

TL;DR

This paper introduces a quantum technique for efficiently extracting smooth functions from quantum states, improving scalability and removing certain dependencies, with potential applications in solving PDEs.

Contribution

It presents a novel preconditioned method for extracting multivariate functions from quantum states, enhancing efficiency and applicability over previous approaches.

Findings

Achieves Heisenberg limit scaling in function extraction.

Reduces quantum complexity with respect to qubits.

Removes dependency on the minimum of the function.

Abstract

Numerically solving partial differential equations is a ubiquitous computational task with broad applications in many fields of science. Quantum computers can potentially provide high-degree polynomial speed-ups for solving PDEs, however many algorithms simply end with preparing the quantum state encoding the solution in its amplitudes. Trying to access explicit properties of the solution naively with quantum amplitude estimation can subsequently diminish the potential speed-up. In this work, we present a technique for extracting a smooth positive function encoded in the amplitudes of a quantum state, which achieves the Heisenberg limit scaling. We improve upon previous methods by allowing higher dimensional functions, by significantly reducing the quantum complexity with respect to the number of qubits encoding the function, and by removing the dependency on the minimum of the function using preconditioning. Our technique works by sampling the cumulative distribution of the given function, fitting it with Chebyshev polynomials, and subsequently extracting a representation of the whole encoded function. Finally, we trial our method by carrying out small scale numerical simulations.