Optimal Universal Quantum Encoding for Statistical Inference
Farhad Farokhi
TL;DR
The paper addresses universal quantum encoding for statistical inference by introducing maximal quantum leakage $\mathcal{Q}(X\rightarrow A)_{\rho}$ as a task-agnostic performance metric. It proves the optimal universal encoder is realized by pure states, and that basis encoding is universally optimal when enough qubits are available, achieving $\mathcal{Q}=\log_2(|\mathcal{X}|)$; for other regimes, an iterative projected-subgradient ascent computes the optimal encoding. The framework shows that inference accuracy is bounded by leakage times $\max_{z} \mathbb{P}\{Z=z\}$, independent of the inference objective or input distribution, linking encoding quality directly to a universal information-theoretic quantity. Additionally, the work provides a lower bound on the required qubit budget, $\log_2(|\mathcal{X}|)/2$, and offers a scalable numerical method to compute optimal encodings across regimes, informing design of quantum inference systems.
Abstract
Optimal encoding of classical data for statistical inference using quantum computing is investigated. A universal encoder is sought that is optimal for a wide array of statistical inference tasks. Accuracy of any statistical inference is shown to be upper bounded by a term that is proportional to maximal quantum leakage from the classical data, i.e., the input to the inference model, through its quantum encoding. This demonstrates that the maximal quantum leakage is a universal measure of the quality of the encoding strategy for statistical inference as it only depends on the quantum encoding of the data and not the inference task itself. The optimal universal encoding strategy, i.e., the encoding strategy that maximizes the maximal quantum leakage, is proved to be attained by pure states. When there are enough qubits, basis encoding is proved to be universally optimal. An iterative method for numerically computing the optimal universal encoding strategy is presented.