  Variational quantum algorithms dominate gate-based applications of modern
quantum processors. The so called, {\it layer-wise trainability conjecture}
appears in various works throughout the variational quantum computing
literature. The conjecture asserts that a quantum circuit can be trained
piece-wise, e.g.~that a few layers can be trained in sequence to minimize an
objective function. Here we prove this conjecture false. Counterexamples are
found by considering objective functions that are exponentially close (in the
number of qubits) to the identity matrix. In the finite setting, we found
abrupt transitions in the ability of quantum circuits to be trained to minimize
these objective functions. Specifically, we found that below a critical (target
gate dependent) threshold, circuit training terminates close to the identity
and remains near to the identity for subsequently added blocks trained
piece-wise. A critical layer depth will abruptly train arbitrarily close to the
target, thereby minimizing the objective function. These findings shed new
light on the divide-and-conquer trainability of variational quantum circuits
and apply to a wide collection of contemporary literature.
