Quantum Cross Entropy and Maximum Likelihood Principle
This work addresses a foundational challenge in quantum machine learning for researchers, but it is incremental as it extends classical concepts to the quantum domain without major breakthroughs.
The paper tackles the problem of generalizing cross entropy to quantum machine learning by defining quantum cross entropy and proving its lower bounds, showing that minimizing it is equivalent to maximizing likelihood only when quantum data is undisturbed by measurements, with information loss occurring otherwise.
Quantum machine learning is an emerging field at the intersection of machine learning and quantum computing. Classical cross entropy plays a central role in machine learning. We define its quantum generalization, the quantum cross entropy, prove its lower bounds, and investigate its relation to quantum fidelity. In the classical case, minimizing cross entropy is equivalent to maximizing likelihood. In the quantum case, when the quantum cross entropy is constructed from quantum data undisturbed by quantum measurements, this relation holds. Classical cross entropy is equal to negative log-likelihood. When we obtain quantum cross entropy through empirical density matrix based on measurement outcomes, the quantum cross entropy is lower-bounded by negative log-likelihood. These two different scenarios illustrate the information loss when making quantum measurements. We conclude that to achieve the goal of full quantum machine learning, it is crucial to utilize the deferred measurement principle.