A scaling cost landscape analysis of the variational quantum eigensolver for hydrogen chains

The variational quantum eigensolver (VQE) is a potential avenue to quantum advantage in near-term, noise-susceptible quantum hardware. VQE uses a classical algorithm to optimize an 'ansatz' quantum circuit and compute the energy eigenstate for a given physical system. We present a scaling study of the VQE cost function landscape for the unitary-coupled-cluster (UCC) ansatz, using variable-length hydrogen chains as our scaling model and obtaining data for up to 28 qubits. We focus in particular on two potential limitations of VQE: noise susceptibility and the barren plateaus phenomenon, in which the average magnitude of gradients decreases exponentially with the number of qubits in the system to be analyzed. Using a sparse wave function simulation of the UCC ansatz on Perlmutter, we find local minima for hydrogen chains and compare gradients near chemical accuracy to average gradients sampled uniformly throughout the cost function. We compare the exponential decrease in gradients for UCC to the previously documented behavior for variable-depth hardware efficient ansatze. Additionally, we use a sparse density matrix simulator to implement noise channels between gates in the UCC ansatz, focusing on depolarization noise, amplitude damping, and phase damping. We compute the effects of these noise channels on gradients and optimizability, again scaling the test system up to 28 qubits.