Classical simulation of variational quantum classifiers using tensor rings

Abstract

In recent times, Variational Quantum Circuits (VQC) have been widely adopted to different tasks in machine learning such as Combinatorial Optimization and Supervised Learning. With the growing interest, it is pertinent to study the boundaries of the classical simulation of VQCs to effectively benchmark the algorithms. Classically simulating VQCs can also provide the quantum algorithms with a better initialization reducing the amount of quantum resources needed to train the algorithm. Even though Matrix Product State representations have been extensively used for quantum state approximation, their capacity is limited in simulating quantum circuits due to the exponential complexity in circuit depth. This manuscript proposes an algorithm that compresses the quantum state within a circuit using a noisy tensor ring representation which allows for the implementation of VQC based algorithms on a classical simulator at a fraction of the usual storage and computational complexity. Using the tensor ring approximation of the input quantum state, we propose a method that applies the parametrized unitary operations while retaining the low-rank structure of the tensor ring corresponding to the transformed quantum state, providing an exponential improvement of storage and computational time in the number of qubits and layers. This approximation is used to implement the tensor ring VQC (TRVQC) for the task of supervised learning on Iris and MNIST datasets to demonstrate the performance of the proposed method compared with the implementations from classical simulator using Matrix Product States (MPS). TRVQC has a test accuracy of 82.63% compared to the benchmark of 83.68% on Iris dataset whereas the former outperforms the latter on a reduced MNIST dataset with TRVQC having an accuracy of 83.73% compared to the benchmark 81.02%, showcasing the comparable performance of the proposed algorithm with the MPS framework.