Abrupt Transitions in Variational Quantum Circuit Training

TL;DR

This paper disproves the layer-wise trainability conjecture in variational quantum circuits by demonstrating abrupt transitions in training success, revealing limitations in piece-wise training approaches for quantum algorithms.

Contribution

It provides a rigorous proof that the layer-wise trainability conjecture is false and identifies critical thresholds causing abrupt changes in trainability in quantum circuits.

Findings

Counterexamples close to the identity matrix

Abrupt transition at a critical layer depth

Training near the target becomes possible beyond the threshold

Abstract

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.