Measurement-Induced Landscape Transitions and Coding Barren Plateaus in Hybrid Variational Quantum Circuits

The entanglement-induced barren plateau is an exponential vanishing of the parameter gradients with system size that limits the practical application of variational quantum algorithms (VQA). A landscape transition from barren plateau to no-barren plateau was recently observed in monitored quantum circuits, hypothesized to coincide with the measurement-induced phase transition (MIPT) that separates the area-law states from the volume-law states. We argue from an information theory perspective that these are different transitions. This hypothesis is supported by a numerical study that includes cost-gradient variances, visualizations of the optimization runs and cost-landscape, and a quantum-classical channel mutual information measure. The results are evidence for a universal measurement-induced landscape transition (MILT) at $p_c^{\text{MILT}} \approx 0.2 < p_c^{\text{MIPT}}$ and that throughout $0<p<p_c^{\text{MILT}}$, there is a finite quantum-classical channel mutual information in the limit of a large number of qubits. Unlike the barren plateau without measurements, a non-zero rate of measurements induces a coding barren plateau where, typically, information about the parameters is available to a local cost function despite a vanishing gradient.