Absence of poor local minima in matrix product states

Abstract:Quantum circuits suffer from severe trainability issues: even shallow circuits are swamped with poor local minima. Yet matrix product states (MPS), which can be prepared by sequential circuits, are remarkably trainable in practice -- as demonstrated by decades of successful density matrix renormalization group calculations. In this work, we resolve this apparent paradox by proving that the energy landscapes of MPS are free from poor local minima, under the same setting where brickwork circuits are not. The key insight is that the gauge freedom of MPS creates an effective local overparametrization that causes local minima to concentrate near the global minimum, analogous to overparametrized classical neural networks. We rigorously prove that the local minimum distribution of the MPS energy landscape is invariant under moves of the orthogonality center. Numerical experiments further confirm that the optimization of sequential circuits converges to near-optimal solutions even for random Hamiltonians, in stark contrast to brickwork circuits. Our findings highlight the pivotal role of effective local overparametrization in determining trainability, providing a valuable guide for overcoming the trainability bottleneck of variational quantum algorithms.

Quantum Physics (quant-ph); Disordered Systems and Neural Networks (cond-mat.dis-nn); Strongly Correlated Electrons (cond-mat.str-el); Computational Physics (physics.comp-ph)

arXiv:2606.09988 [quant-ph]

(or arXiv:2606.09988v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.09988 arXiv-issued DOI via DataCite (pending registration)