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Description
Barren plateaus are a common phenomenon in quantum machine learning which render parameterized circuits untrainable by flattening the loss landscape.
The Lie-algebraic theory of this effect expresses the loss variance exactly through the dimension of the circuit's dynamical Lie algebra, the readout purity and the purity of the encoded input.
For angle-encoding circuits, the dependence of this input factor on the input distribution has received little attention so far.
We close this gap by treating the input angle distribution as a classical preconditioning choice and by determining when it matters.
The decision is made by a simple floor criterion, namely the number of algebra basis elements that are diagonal in the encoding's clustering basis.
If such elements exist, as for the matchgate algebra, the input purity is pinned to a deterministic band and the input distribution is irrelevant.
If none exist, as for the minimal off-diagonal algebra $\mathfrak{so}(n)\oplus\mathfrak{so}(n)$, the input distribution alone decides trainability.
In this case, a uniform angle prior certifies a purity of $\Theta(n)$, while clustered inputs leave the model frozen.
This mechanism survives the transition to classically hard circuits, reached by a single non-Gaussian gate or by placing the $XY$ interaction on a branched bipartite graph.
There, clustered inputs annihilate the loss identically, but the attainable variance is capped by the exponential algebra dimension.
We conclude that floored algebras are insensitive to the input distribution, polynomial floor-free algebras are decided by it, while exponential floor-free algebras remains untrainable.
Numerical experiments validate the closed forms and bounds.
| Topic | Quantum variational circuits and their applications |
|---|---|
| Additional Topic | Theoretical foundations of quantum learning |