Quantum Circuits, Feature Maps, and Expanded Pseudo-Entropy: Analysis of Encoding Real-World Data into a Quantum Computer

TL;DR

This paper introduces a new method called operator pseudo-entropy to analyze quantum feature maps, enabling efficient assessment of their nonlinearity and state concentration, with implications for encoding real-world data into quantum computers.

Contribution

It develops a rigorous, complex-valued operator pseudo-entropy technique to evaluate quantum feature maps, extending von Neumann entropy and pseudo-entropy concepts.

Findings

Operator pseudo-entropy effectively characterizes quantum feature maps.

The method distinguishes maps with high state concentration.

Experimental comparisons show advantages over existing entropy measures.

Abstract

This manuscript introduces a computationally efficient method to calculate the nonlinearity of a quantum feature map, as well as a method for determining whether a quantum feature map will have a high concentration of quantum states. The technique analyzes quantum operators, through an extension of the functions of von Neumann entropy and state-transition pseudo-entropy, by deriving a method to extract the entropy of an operator. The technique is denoted as operator pseudo-entropy, is rigorously derived, and is generally complex valued; as with state-transition pseudo-entropy, complex values contain a lot of information about entanglement or nonlinearity. The characteristics of a class of quantum feature maps are rigorously shown. The operator pseudo-entropy is illuminated through experiments and compared with von Neumann entropy and state-transition pseudo-entropy. We end the manuscript with open questions and potential directions for further research.