Learning the N-Input Parity Function with a Single-Qubit and Single-Measurement Sampling

The parity problem, a generalization of the XOR problem to higher-dimensional inputs, is a challenging benchmark for evaluating learning algorithms, due to its increased complexity as the number of dimensions of the feature space grows. In this work, a single-qubit classifier is developed, which can efficiently learn the parity function from input data. Despite the qubit model’s simplicity, the solution landscape created in the context of the parity problem offers an attractive test bed for exploring optimization methods for quantum classifiers. We propose a new optimization method called Ensemble Stochastic Gradient Descent (ESGD), with which density matrices describing batches of quantum states are incorporated into the loss function. We demonstrate that ESGD outperforms both Gradient Descent and Stochastic Gradient Descent given the aforementioned problem. Additionally, we show that applying ESGD with only one measurement per data input does not lead to any performance degradation. Our findings not only highlight the potential of a single-qubit model, but also offer valuable insights into the use of density matrices for optimization. Further to this, we complement the outcome with interesting results arising by the employment of a Doubly Stochastic Gradient Descent for training quantum variational circuits.