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

TL;DR

This work introduces operator pseudo-entropy $S(U)=-\frac{1}{\sqrt{\dim(U)}}\operatorname{Tr}(\Lambda\log\Lambda)$ for special unitary operators $U$, enabling efficient assessment of nonlinearity and entanglement induced by quantum feature maps without full state reconstruction. It defines a normalized distance between operators and contrasts operator pseudo-entropy with von Neumann and state-transition pseudo-entropy, highlighting when $S(U)$ is real versus complex. Through experiments across linear, spiral, and real-world datasets using Angle, Amplitude, and IQP encodings, the paper links real-valued $S(U)$ to less state concentration and better separability, while large imaginary parts indicate more pronounced nonlinear transformations; SVD and eigenvalue patterns are connected to exponential concentration phenomena. The authors argue that operator pseudo-entropy can subsume or approximate existing measures like expressivity, expressibility, and symmetric encoding, and discuss a potential category-theoretic framework to relate entropy and pseudo-entropy. Overall, the approach offers a computationally efficient, information-rich lens for evaluating how real-world data are encoded into quantum circuits and how this affects learning performance and entanglement structure.

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.

Figures (8)

• Figure 1: Visualization of the features for each data set.
• Figure 2: Operator pseudo-entropy, von Neumann entropy, and the average state-transition pseudo-entropy values for three different data sets with the Angle encoding, Amplitude encoding, and IQP encoding schemes. The IQP averaged state-transition pseudo-entropy values are quite large, and to adjust the imaginary values have been log-scaled and the scatter plots are separated for a more granular visual analysis. The entropy values for the Angle encoding scheme are excluded since all values are zero.
• Figure 3: To view how the operator pseudo-entropy value for a data point differs between each encoding scheme, for the linearly separable data for dimension $10$, the distribution of distances is displayed. The distances were calculated from Definition \ref{['def:diff-metric']} and the modulus for the final value.
• Figure 4: Average pseudo-entropy values of linearly separable data over varying dimensions of the features.
• Figure 5: Classification with the QSVM and LightGBM algorithm on different dimensions of the linearly separable.

Theorems & Definitions (4)

• Proposition 4
• proof
• Remark 7
• Definition 8