Investigating barren plateaus in variational quantum algorithms via the Clifford group

Quantum computing is a model of computing that is based on the laws of quantum mechanics. A promising path toward demonstrating quantum advantage in the near term is the use of Variational Quantum Algorithms (VQAs). VQAs are quantum algorithms that rely on a set of tunable parameters, the optimization of which is handled by a classical computer, creating a hybrid quantum-classical loop. The goal of a VQA is to find the set of parameters that minimizes a cost function that depends on them. The potential of these circuits is often hampered by the Barren Plateau (BP) phenomenon, which indicates that the cost function results being flat in vast regions of the parameter space far from the minimum, rendering the training of the VQA practically infeasible. The goal of this thesis is to relate the flatness of the cost function to the architectural features of a quantum circuit. This is the same problem addressed in [Napp, arXiv:2203.06174], however, we approach the problem in a different manner. To this aim we make use of the Pauli group, i.e. the group generated by the Pauli matrices along with the identity, and its normalizer, the Clifford group. We study the problem in two different configurations. In the first configuration, we assume that the architecture of the circuit is not fixed, as each gate is applied to randomly sampled sites at each step. We study this model with the properties of Markov chains and we derive an analytical upper bound on the expected magnitude of the gradient of the cost function, which decays exponentially with the number of gates in the circuit. We also numerically demonstrate that the bound does not exhibit exponential decay for shallow circuits. In the second configuration the architecture is fixed and we study the model as a random walk. For the same quantity we obtain an analytical lower bound that exponentially decays with the minimum number of gates that a trajectory of the random walk must pass through.