Fig. 1: Summary of our main results.
From: Cost function dependent barren plateaus in shallow parametrized quantum circuits
McClean et al.9 proved that a barren plateau can occur when the depth D of a hardware-efficient ansatz is \(D\in {\mathcal{O}}(\mathrm{poly}\,(n))\). Here we extend these results by providing bounds for the variance of the gradient of global and local cost functions as a function of D. In particular, we find that the barren plateau phenomenon is cost-function dependent. a For global cost functions (e.g., Eq. (1)), the landscape will exhibit a barren plateau essentially for all depths D. b For local cost functions (e.g., Eq. (2)), the gradient vanishes at worst polynomially and hence is trainable when \(D\in {\mathcal{O}}(\mathrm{log}\,(n))\), while barren plateaus occur for \(D\in {\mathcal{O}}(\mathrm{poly}\,(n))\), and between these two regions the gradient transitions from polynomial to exponential decay.
