Barren Plateau Explorer
A variational algorithm trains a parameterised circuit by following the gradient of a cost. The trouble: for a randomly initialised circuit the gradient is zero on average, and its variance — how far from zero a typical gradient lands — can shrink exponentially with the number of qubits. Then almost every starting point is flat, and no number of shots you can afford will show you which way is down. That is a barren plateau.
This page measures it. Pick a cost and a depth, and it draws random circuits, computes one exact gradient for each, and plots the variance against qubit count. Run a second setting and the lines share an axis.
Reproduce a published result
A local observable still plateaus once the circuit is deep enough to scramble.
Cost: ⟨Z₀Z₁⟩. Gradient taken with respect to the first rotation angle, by the parameter-shift rule on the exact statevector.
What is being run
The circuit family is the one in McClean, Boixo, Smelyanskiy, Babbush & Neven, “Barren plateaus in quantum neural network training landscapes” (Nat. Commun. 9, 4812, 2018): RY(π/4) on every qubit, then layers of one rotation per qubit about a randomly chosen axis by a random angle, each followed by a ladder of CZ gates along a line.
The global-versus-local comparison is Cerezo, Sone, Volkoff, Cincio & Coles, “Cost function dependent barren plateaus in shallow parametrized quantum circuits” (Nat. Commun. 12, 1791, 2021). Their theorem is stated for a different layered ansatz; what this page shows is the same effect on the McClean ansatz at small n, which is an illustration of the result and not a reproduction of their figures.
Limits worth knowing: twelve qubits is far too few to distinguish exponential decay from a steep polynomial on the data alone, and a plateau measured without noise says nothing about noise-induced plateaus, which are a separate mechanism. For what this means for a real variational run, see the VQE lesson and the QML Reality Check.