Topological Obstructions for Quantum Adiabatic Algorithms: Evidence from MaxCut Instances

TL;DR

This paper reveals that degeneracy in quantum adiabatic algorithms imposes global spectral constraints, causing complex eigenphase interactions and topological obstructions that limit optimization success beyond local gap considerations.

Contribution

It introduces a spectral-flow perspective showing how degeneracy leads to unavoidable global spectral reordering and topological obstructions in quantum adiabatic optimization.

Findings

Eigenphase trajectories exhibit braiding and permutation of spectral bands.

Spectral congestion persists despite increasing evolution time.

Topological obstructions are rooted in global eigenstate connectivity.

Abstract

Quantum adiabatic algorithms are commonly analyzed through local spectral properties of an interpolating Hamiltonian, most notably the minimum energy gap. While this perspective captures an important constraint on adiabatic runtimes, it does not fully describe the global structure of spectral evolution in optimization problems with degenerate solution manifolds. In this work, we show that degeneracy alone imposes unavoidable global constraints on spectral flow, even in instances where adiabatic algorithms succeed with high probability. Focusing on digitized quantum adiabatic evolutions, we analyze the eigenphases of the cumulative unitary operator generated along the interpolation path. By explicitly tracking eigenphase trajectories, we demonstrate that multiple spectral bands are forced to interact, braid, and permute before coalescing into a degenerate manifold at the end of the evolution. This global reordering manifests as persistent spectral congestion and nontrivial band permutations that cannot be removed by increasing evolution time or refining the digitization. Using MaxCut instances with controlled degeneracy as a concrete setting, we extract quantitative diagnostics of spectral congestion and explicitly compute the induced band permutations. Our results show that successful adiabatic optimization can coexist with complex and constrained spectral flow, revealing a form of topological obstruction rooted in the global connectivity of eigenstates rather than in local gap closures. These findings highlight intrinsic limitations of gap-based analyses and motivate spectral-flow-based diagnostics for understanding adiabatic algorithms in degenerate optimization landscapes.