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L7-04 · Barren plateaus and trainability

Distinguish tiny true gradients from noisy gradient estimates.

Level 7 icon Level 7 · Doctoral & postdoctoral
Objective. Distinguish tiny true gradients from noisy gradient estimates.
Before this lesson. L7-02, L3-07
Hands-on lab. Variational circuits and gradients

Learn

A barren plateau is a regime where gradients concentrate near zero as problem size grows under stated circuit and cost assumptions. This is a property of a landscape distribution. One failed optimization run is not enough to establish that scaling behavior.

To test trainability, vary system size while controlling the circuit family, initialization, observable, and noise model. Estimate gradient variance across independent initializations. Compare that variance with shot noise. A flat-looking estimated landscape can result from limited measurement precision rather than a vanishing mathematical gradient.

Proposed remedies include structured circuits, local costs, informed initialization, and staged training. Their value depends on the task and assumptions. A remedy should preserve expressive power and generalization while improving measured scaling. Do not call one successful small circuit a general solution.

Practice

Solution

The variance bound is 1/4000. Its square root is about 0.0158. This bounds estimator noise only. A barren-plateau claim requires controlled gradient scaling across circuit sizes and initializations.

Check your understanding

Which evidence supports a scaling claim about barren plateaus?

Show the answer

A. Controlled gradient statistics across increasing sizes

A scaling claim needs repeated gradient statistics under controlled circuit assumptions at several sizes. A single optimization failure has many alternative explanations.

Sources

This lesson draws on McClean et al., 2018Mitarai et al., 2018.

Next: L7-05 · Identifiability before quantum causal learning

References
  1. McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R., & Neven, H. (2018). Barren plateaus in quantum neural network training landscapes. Nature Communications, 9(1). 10.1038/s41467-018-07090-4
  2. Mitarai, K., Negoro, M., Kitagawa, M., & Fujii, K. (2018). Quantum Circuit Learning. Physical Review A, 98(3). 10.1103/PhysRevA.98.032309