Decision Preservation Across Sequential Quantum-Classical Representations: A Risk-Based Mathematical Framework

Source abstract

The process of transforming quantum states into classical data in hybrid quantum-classical learning pipelines consists of three stages: measurement, feature extraction, and processing of the extracted features. While this process can simplify representation, it can also discard valuable information necessary for making predictions. In this paper, we propose a decision-theoretic approach for analysis of information preservation during sequential processing of quantum and classical information. We represent the learning pipeline as a sequence of quantum states, measurements, and classical representations, and evaluate the performance of each step in terms of change in the optimal predictive risk. We identify conditions when risk cannot decrease after the transformation reducing the amount of information, prove a composable identity for the sum of risk changes and characterise the equality conditions for logarithmic loss. We also differentiate between decision preservation and quantum state reconstruction and show their relevance for hybrid quantum machine learning. These results are obtained without imposing any assumptions about the relationship between information about the quantum state and task-related information. Our framework relates to several well-known results about statistical sufficiency, quantum information bottleneck, and generalisation in quantum machine learning. The contribution of this work is therefore in providing a systematic mathematical treatment of these principles in the context of sequential information processing.