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arXiv:2601.02255

Topological Obstructions for Quantum Adiabatic Algorithms: Evidence from MaxCut Instances

Prathamesh S. Joshi

quant-phSubmitted Jan 5, 2026• Announced Jan 6, 2026• 0 citations

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.

Comments: 11 pages, 4 figures, 1 table

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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.

We point out that, when an optimization problem has more than one solution, the quantum adiabatic algorithms (QAA) encounter topological obstructions leading to adiabatic spectral flows where spectral branches unavoidably traverse the spectral gap above the ground states of the quantum Hamiltonians. This raises serious doubts about the validity of the algorithms in such situations. However, using the Max-Cut problem as an example, we explain and demonstrate here that QAAs correctly detect all existing solutions in one single run. This newly discovered capacity of QAAs to simultaneously detect multiple solutions to an optimization problem can have an important impact on future developments of quantum variational algorithms

We present a perturbative method to estimate the spectral gap for adiabatic quantum optimization, based on the structure of the energy levels in the problem Hamiltonian. We show that, for problems that have an exponentially large number of local minima close to the global minimum, the gap becomes exponentially small making the computation time exponentially long. The quantum advantage of adiabatic quantum computation may then be accessed only via the local adiabatic evolution, which requires phase coherence throughout the evolution and knowledge of the spectrum. Such problems, therefore, are not suitable for adiabatic quantum computation.

We establish a necessary condition for quantum heuristics to achieve nontrivial computational advantage relative to a classical problem generator A. We prove that any unitary procedure whose dynamics nearly commute with A is spectrally confined: its ability to transfer amplitude across separated spectral sectors is bounded strictly by the commutator defect divided by the spectral gap. This yields a quantitative operator-theoretic obstruction to global state-space exploration. Consequently, repeated application of such procedures (e.g., in deep variational circuits) produces apparent algorithmic activity without generating meaningful information gain. We classify this regime as a "degenerate quantum heuristic". We apply this framework to Variational Quantum Eigensolvers (VQE) and the Quantum Approximate Optimization Algorithm (QAOA), demonstrating that circuit depth cannot overcome transversality failure. The theoretical results are fully closed within trace-ideal (S₂) operator theory. We extend the spectral confinement bound to the entire resolvent geometry, proving that degenerate heuristics preserve the Birman-Schwinger kernel and regularized Fredholm determinants (det₂), thereby mathematically obstructing any computational phase transition or new eigenvalue emergence. Finally, we demonstrate that genuine quantum speedup requires structural noncommutativity, which can be deterministically enforced via an Ω-Σ dissipative control layer. The manuscript includes complete finite-dimensional numerical validation, exhibiting the exact quantitative rigidity between commutator defect, spectral transport, and resolvent deformation.

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3

-XORSAT, the accessible gauge sector has a size-independent gap, in contrast to the exponentially closing gap of a conventional transverse-field adiabatic path. For satisfiable

3

-SAT instances, finite-size benchmarks demonstrate high satisfied-clause fractions at short classical comparison budgets, without claiming an equal-cost quantum advantage. We also outline a superconducting-qutrit implementation and discuss the conceptual relationship between QPF and nondeterministic Turing machine.

It is believed that the presence of anticrossings with exponentially small gaps between the lowest two energy levels of the system Hamiltonian can render adiabatic quantum optimization inefficient. Here, we present a simple adiabatic quantum algorithm designed to eliminate exponentially small gaps caused by anticrossings between eigenstates that correspond with the local and global minima of the problem Hamiltonian. In each iteration of the algorithm, information is gathered about the local minima that are reached after passing the anticrossing nonadiabatically. This information is then used to penalize pathways to the corresponding local minima by adjusting the initial Hamiltonian. This is repeated for multiple clusters of local minima as needed. We generate 64-qubit random instances of the maximum independent set problem, skewed to be extremely hard, with between

{10}^{5}

and

{10}^{6}

highly degenerate local minima. Using quantum Monte Carlo simulations, it is found that the algorithm can trivially solve all of the instances in $\ensuremath{\sim}$10 iterations.

Understanding NP-complete problems is a central topic in computer science. This is why adiabatic quantum optimization has attracted so much attention, as it provided a new approach to tackle NP-complete problems using a quantum computer. The efficiency of this approach is limited by small spectral gaps between the ground and excited states of the quantum computer's Hamiltonian. We will discuss the statistics of the gaps using the borrowed from the theory of quantum disordered systems. It turns out that due to a phenomenon similar to Anderson localization exponentially small gaps appear close to the end of the adiabatic algorithm for large random instances of NP-complete problems. We will present the quantitative analysis of the small spectral gaps and discuss possible consequence of this phenomenon on the adiabatic optimization paradigm.