Barren Plateaus

A variational algorithm trains by following the gradient of a cost $C(\boldsymbol{\theta}) = \langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$ . A barren plateau is a landscape in which that gradient is exponentially small almost everywhere: $\partial C/\partial\theta_k \approx 0$ to within the precision you can afford to measure. When this happens the optimizer wanders aimlessly on a flat plain, and no amount of classical cleverness rescues it. McClean and collaborators identified this as a fundamental obstacle to scaling VQE and quantum neural networks.

The variance argument

The diagnosis is statistical. For a sufficiently random ansatz the gradient component $\partial_k C$ has mean zero when averaged over parameters,

\mathbb{E}\bigl[\partial_k C\bigr] = 0,

so the issue is not bias but vanishing variance. McClean et al. showed that when the ansatz forms (approximately) a 2-design — which deep, generic hardware-efficient circuits do — the variance of each gradient component shrinks exponentially in the number of qubits:

\mathrm{Var}\bigl[\partial_k C\bigr] \sim \frac{1}{2^{\,n}}.

Because the typical gradient magnitude is $\sqrt{\mathrm{Var}}$ , you would need on the order of $2^{n}$ shots just to resolve the slope from shot noise. At, say, $n = 50$ qubits that is hopeless.

Three things that cause plateaus

Research since 2018 has mapped several distinct mechanisms, all of which flatten the landscape:

Expressibility / circuit depth. The more closely an ansatz approximates a Haar-random unitary (the lesson on expressibility), the flatter its average landscape. Depth that buys expressibility costs trainability.
Global cost functions. Cerezo et al. proved that observables acting on all qubits at once (global cost) give plateaus even at shallow, logarithmic depth, whereas local cost functions — sums of few-qubit terms — keep gradients polynomially large for shallow circuits.
Hardware noise. Noise itself induces noise-induced barren plateaus, where the gradient decays exponentially in circuit depth independent of the ansatz structure.
Excess entanglement. Ansätze that generate volume-law entanglement across a bipartition concentrate the cost, again flattening gradients.

A small illustration

Imagine measuring a single Pauli $Z_0$ on a state produced by a deep random circuit. The reduced state of qubit 0 is, on average, the maximally mixed state $I/2$ , for which $\langle Z_0\rangle = 0$ exactly. Perturbing one angle nudges this expectation by an amount that, averaged over the random parameters, has variance $\sim 1/2^n$ because the perturbation is "diluted" across the exponentially large Hilbert space. The expectation sits pinned near zero, and so does its derivative.

Mitigation strategies (previewed here, used later)

You do not simply give up. The rest of this module is, in part, a toolbox against plateaus:

Structured, shallow ansätze — symmetry-preserving or problem-inspired circuits restrict the search to a smaller, trainable manifold.
Local cost functions — reformulate the objective so each term touches few qubits.
Adaptive construction (ADAPT-VQE) — grow the circuit one well-chosen operator at a time so it never becomes generically random.
Smart initialization — start near the identity, or use layerwise / warm-start schemes that keep early gradients large.
Natural gradient — rescale steps by the quantum Fisher information to move efficiently even where raw gradients are small.

Key takeaways

A barren plateau is an exponentially flat cost landscape: $\mathrm{Var}[\partial_k C]\sim 2^{-n}$ .
It is caused by high expressibility, global cost functions, hardware noise, and excess entanglement.
It cannot be cured by restarts; it must be designed around with structure, locality, and adaptive or warm-started training.