Effects of quantum resources on the statistical complexity of quantum circuits
This work is significant for researchers in quantum machine learning and quantum complexity theory, providing theoretical bounds on the learning capabilities of quantum circuits based on their resource content.
The paper investigates how quantum resources affect the statistical complexity of quantum circuits, using Rademacher and Gaussian complexities. It derives bounds for circuits with limited resources, applying them to stabilizer circuits with T gates and IQP Clifford circuits with CCZ gates, and shows that the increase in complexity from an added quantum channel is bounded by its free robustness.
We investigate how the addition of quantum resources changes the statistical complexity of quantum circuits by utilizing the framework of quantum resource theories. Measures of statistical complexity that we consider include the Rademacher complexity and the Gaussian complexity, which are well-known measures in computational learning theory that quantify the richness of classes of real-valued functions. We derive bounds for the statistical complexities of quantum circuits that have limited access to certain resources and apply our results to two special cases: (1) stabilizer circuits that are supplemented with a limited number of T gates and (2) instantaneous quantum polynomial-time Clifford circuits that are supplemented with a limited number of CCZ gates. We show that the increase in the statistical complexity of a quantum circuit when an additional quantum channel is added to it is upper bounded by the free robustness of the added channel. Finally, we derive bounds for the generalization error associated with learning from training data arising from quantum circuits.