Extracting a function encoded in amplitudes of a quantum state by tensor network and orthogonal function expansion

TL;DR

This paper introduces a quantum-classical hybrid method combining tensor networks and orthogonal function expansion to efficiently extract and evaluate functions encoded in quantum states, overcoming classical readout bottlenecks.

Contribution

It presents a novel approach for function readout from quantum states using tensor networks and orthogonal expansions, enabling efficient classical evaluation.

Findings

Successfully approximated a finance-related function

Demonstrated polynomial scalability in degrees of freedom

Validated the method through numerical experiments

Abstract

There are quantum algorithms for finding a function $f$ satisfying a set of conditions, such as solving partial differential equations, and these achieve exponential quantum speedup compared to existing classical methods, especially when the number $d$ of the variables of $f$ is large. In general, however, these algorithms output the quantum state which encodes $f$ in the amplitudes, and reading out the values of $f$ as classical data from such a state can be so time-consuming that the quantum speedup is ruined. In this study, we propose a general method for this function readout task. Based on the function approximation by a combination of tensor network and orthogonal function expansion, we present a quantum circuit and its optimization procedure to obtain an approximating function of $f$ that has a polynomial number of degrees of freedom with respect to $d$ and is efficiently evaluable on a classical computer. We also conducted a numerical experiment to approximate a finance-motivated function to demonstrate that our method works.