Topological Obstructions for Quantum Adiabatic Algorithms: Evidence from MaxCut Instances

TL;DR

This work shows that quantum adiabatic optimization in degenerate landscapes can exhibit strong global spectral constraints that are invisible to local gap diagnostics. By tracking the eigenphases of the cumulative unitary $U(0->s)$ along a digitized interpolation, the authors reveal persistent spectral congestion and nontrivial end-to-end band permutations even when optimization succeeds with high probability. The results, demonstrated on MaxCut instances with controlled degeneracy, argue for a topological obstruction in spectral flow and motivate spectral-flow–based diagnostics as essential complements to gap-based analyses. These findings suggest intrinsic limits on adiabatic analyses and invite applying unitary-spectral-flow diagnostics to a broader class of degenerate optimization problems.

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.

Figures (4)

• Figure 1: Schematic (illustration) of unitary spectral flow in a degenerate optimization problem. As the interpolation parameter $s$ varies, eigenphase bands interact and exchange ordering before coalescing into a degenerate solution manifold at $s=1$. The resulting global reordering motivates the spectral diagnostics introduced in this work.
• Figure 2: Representative final-time measurement histograms (top outcomes), showing concentration on the degenerate optimal MaxCut solutions across different graph sizes.
• Figure 3: Unitary spectral flow $U(0\to s)$ (eigenphases $\theta_j(s)$) shown across problem sizes and digitization depth.
• Figure 4: spectral congestion diagnostic $\Delta\theta_{\min}(s)$ (minimum eigenphase spacing) for the same instances shown in Fig. \ref{['fig:spectral_flow']}.