The topological obstruction is a feature, not a bug.
Quantum adiabatic algorithms (QAA) work by slowly deforming a simple Hamiltonian into one whose ground state encodes the solution to an optimization problem. When the problem has multiple solutions, spectral flows — energy branches that cross the gap between ground states and excited states — appear as topological obstructions. The system can't stay in the ground state because the ground state itself splits and reconnects.
The standard interpretation: this is a failure mode. The spectral flow forces the algorithm to leave the ground state, losing the solution. Joshi and Prodan (arXiv:2603.20567) show the opposite: the spectral flow carries the algorithm through all solutions simultaneously. One run of the adiabatic algorithm, navigating the topological obstruction rather than avoiding it, detects every solution to the optimization problem.
Demonstrated on Max-Cut, the algorithm's passage through the spectral flow visits each degenerate ground state in sequence. The topology that creates the obstruction also creates the tour — the same structure that prevents staying in one solution ensures visiting all of them.
This reframes the relationship between topology and computation. Topological features of the energy landscape are typically classified as obstacles (gap closings = hard problems). Here, the topological feature is the computational resource. The obstruction doesn't prevent the algorithm from working. It makes the algorithm do more than expected — finding the full solution set rather than a single representative.
The practical implication: quantum variational algorithms (VQE, QAOA) that encounter spectral flows may be discarding information. The flow isn't noise. It's signal — a topological enumeration of the solution space. Reading it correctly gives you the census where reading only the ground state gives you a sample.