Optimizing Quantum Embedding using Genetic Algorithm for QML Applications

Quantum Embeddings (QE) is an important component of Quantum Machine Learning (QML) algorithms to load classical data present in Euclidean space onto quantum Hilbert space, which are then later forwarded to the Parametric Quantum Circuit (PQC) for training and finally measured to compute the cost classically. The performance of the QML algorithm can vary according to the type of QE used, and also on mapping of features within the embedding (i.e., onto the qubits). This provides motivation to search for the optimal quantum embedding (i.e., feature to qubit mapping). Typically this problem is presented as an optimization problem, where the quantum embedding has trainable parameters and the optimal embedding is found out by training the embedding. In this work, we present identification of the optimal embedding as a search problem rather than an optimization problem. We show that for fixed number of qubits and model initialization of a Quantum Neural Network (QNN), different mapping of features onto qubits via QE changes the final performance of the QML algorithm. We propose Genetic Algorithm (GA) based search to find the optimal mapping of the features to the qubits. We perform experiments to find the optimal QE for binary classes of MNIST and Tiny Imagenet datasets and compare the results with randomly selected feature to qubit mapping (under identical number of runs). Our results show that GA-based approach performs better than random selection for MNIST (Tiny Imagenet) by 0.33-3.33 (0.5-3.36) higher average fitness score with up to 8.1%-15% (for MNIST) and 5.3%-8.8% (for Tiny Imagenet) less runtime. For both the datasets increasing the qubit counts marginally affects the GA fitness implying that the GA is scalable both in terms of dataset and in terms of QNN size. Compared to existing methodologies such as Quantum Embedding Kernel (QEK), Quantum Approximate Optimization Algorithm (QAOA)-based embedding and Quantum Random Access Code (QRAC), GA performs better by 1.003X, 1.03X and 1.06X respectively.