A Study on Expressibility and Entangling Capacity in Parametrized Quantum Circuits

Abstract

In Hybrid Quantum Classical (HQC) Algorithms, the Parameterized Quantum Circuits (PQC) play a very important role. However one of the biggest challenges in implementing these HQC Algorithms is choosing an effective circuit which can represent the solution space properly and at the same time is feasible in implementation (having low circuit complexity, depth and number of parameters). Expressiblity and Entangling Capacity are two such measures which quantifies the extent of solution space that can be covered by a particular PQC. Expressibility quantifies the ability of the PQC to generate pure states that are well representative the Hilbert Space. On the other hand the Entangling Capacity quantifies the ability of the PQC to generate entangled states. The advantage of highly entangled states in PQCs of low depth can be the ability to efficiently represent the solution space for tasks such as ground state preparation, or data classification or capturing non trivial correlation in quantum data. Both of these metrics can be estimated statistically. In this dissertation, we have taken 19 different Parameterized Quantum Circuits (all of 4 qubits) of different structures having varying circuit complexity, computed Expressibility and entangling capacity. We studied the effect of topology of the machine and the error in the computed Expressibility. We also found the limitation of Entangling Capacity that we can’t compute it for Circuits on a Quantum Channel. Finally we considered a Data Science Problem on Fraud Transaction Detection to see how the Quantum Neural Network with all the considered Parameterized Quantum Circuits to see the importance of Expressibility and Entangling Capacity on the training capacity of the circuit.