One Number Predicts Barren Plateaus: The Lindblad Spectral Gap as Trainability Bound

We show that the trainability of variational quantum circuits is determined by a single number: the spectral gap \(|\lambda_1|\) of the Lindblad noise generator. The gradient variance of any parametrized circuit with \(n\) qubits at depth \(d\) satisfies \(\text{Var}[\partial_\theta C] \sim 2^{-n} \cdot \exp(-2|\lambda_1| \cdot d \cdot t_{\text{eff}})\), where \(t_{\text{eff}}\) is the effective gate time per layer. This gives an explicit critical depth:

\[d^* = \frac{\log(10^4 \cdot 2^n)}{2\,|\lambda_1| \cdot t_{\text{eff}} \cdot n}\]

Beyond \(d^\): the gradient vanishes below any useful threshold. The circuit is untrainable. Below \(d^\): gradients are usable and optimization can proceed.

For IBM Eagle (2024, \(T_1 = 300\,\mu\)s): \(d^(4\text{ qubits}) = 2{,}643\), \(d^(50\text{ qubits}) = 774\). The noise-induced barren plateau is NOT the dominant effect for shallow circuits on current hardware --- the expressibility plateau (from Haar-random initialization) dominates at depth \(\sim\)50. But for 50+ qubit systems at depth \(>\)500 --- the regime of practical quantum advantage --- the noise-induced plateau becomes the binding constraint, and \(|\lambda_1|\) is the number that predicts it.

The spectral gap \(|\lambda_1|\) is computable from a \(64 \times 64\) cluster Lindblad matrix in 0.001 seconds, without running any quantum circuit. This enables: (1) BEFORE optimization: predict whether the circuit is trainable; (2) DURING design: choose the maximum depth for a given hardware; (3) ACROSS hardware: rank quantum computers by their trainability depth \(d^*\).