Topological Obstructions for Quantum Adiabatic Algorithms: Evidence from MaxCut Instances

• URL: http://arxiv.org/abs/2601.02255v1
• Date: Mon, 05 Jan 2026 16:35:31 GMT
• Title: Topological Obstructions for Quantum Adiabatic Algorithms: Evidence from MaxCut Instances
• Authors: Prathamesh S. Joshi,
• Abstract summary: We show that degeneracy alone imposes unavoidable global constraints on spectral flow, even in instances where adiabatic algorithms succeed with high probability.<n>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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Parallel Complex Diffusion for Scalable Time Series Generation [50.01609741902786]
  PaCoDi is a spectral-native architecture that decouples generative modeling in the frequency domain.<n>We show that PaCoDi outperforms existing baselines in both generation quality and inference speed.
  arXiv  Detail & Related papers  (2026-02-10T14:31:53Z)
• Quantum Algorithm for Low Energy Effective Hamiltonian and Quasi-Degenerate Eigenvalue Problem [7.107390133525336]
  Quasi-degenerate eigenvalue problems are central to quantum chemistry and condensed-matter physics.<n>We propose a quantum algorithm that diagonalizes such quasi-degenerate manifold by solving an effective-Hamiltonian eigenproblem in a low-dimensional subspace.<n>Our analysis provides provable bounds on eigenvalue accuracy and subspace fidelity, together with total query complexity.
  arXiv  Detail & Related papers  (2025-10-09T11:20:15Z)
• Self-concordant Schrödinger operators: spectral gaps and optimization without condition numbers [2.027398351960778]
  We study Schr"odinger operators associated with self-concordant barriers over convex domains.<n>We find that the spectral gap does not display any condition-number dependence when the usual Laplacian is replaced by the Laplace--Beltrami operator.
  arXiv  Detail & Related papers  (2025-10-07T16:50:42Z)
• Mechanisms for Quantum Advantage in Global Optimization of Nonconvex Functions [6.135587835061064]
  We show new theoretical mechanisms for quantum speedup in the global optimization of non-asymotic functions.<n>We formalize these ideas by proving that a real-space quantum algorithm (RsAA) achieves provably on-time runtimes.
  arXiv  Detail & Related papers  (2025-10-03T17:40:31Z)
• Avoiding spectral pollution for transfer operators using residuals [0.6116681488656472]
  We present algorithms for computing spectral properties of transfer operators without spectral pollution.<n>Case studies range from families of Blaschke maps with known spectrum to a molecular dynamics model of protein folding.<n>Our methods offer robust tools for spectral estimation across a broad range of applications.
  arXiv  Detail & Related papers  (2025-07-22T18:01:05Z)
• Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking [45.147301546565316]
  This paper considers the problem of decentralized optimization on compact submanifolds.<n>We propose an algorithm where agents update variables using quantized variables.<n>To the best of our knowledge, this is the first algorithm to achieve an $mathcalO (1/K)$ convergence rate in the presence of quantization.
  arXiv  Detail & Related papers  (2025-06-09T01:57:25Z)
• Avoided-crossings, degeneracies and Berry phases in the spectrum of quantum noise through analytic Bloch-Messiah decomposition [49.1574468325115]
  "analytic Bloch-Messiah decomposition" provides approach for characterizing dynamics of quantum optical systems.<n>We show that avoided crossings arise naturally when a single parameter is varied, leading to hypersensitivity of the singular vectors.<n>We highlight the possibility of programming the spectral response of photonic systems through the deliberate design of avoided crossings.
  arXiv  Detail & Related papers  (2025-04-29T13:14:15Z)
• QAMA: Scalable Quantum Annealing Multi-Head Attention Operator for Deep Learning [48.12231190677108]
  Quantum Annealing Multi-Head Attention (QAMA) is proposed, a novel drop-in operator that reformulates attention as an energy-based Hamiltonian optimization problem.<n>In this framework, token interactions are encoded into binary quadratic terms, and quantum annealing is employed to search for low-energy configurations.<n> Empirically, evaluation on both natural language and vision benchmarks shows that, across tasks, accuracy deviates by at most 2.7 points from standard multi-head attention.
  arXiv  Detail & Related papers  (2025-04-15T11:29:09Z)
• Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space [54.13671100638092]
  Holistic Physics Mixer (HPM) is a framework for integrating spectral and physical information in a unified space.<n>We show that HPM consistently outperforms state-of-the-art methods in both accuracy and computational efficiency.
  arXiv  Detail & Related papers  (2024-10-15T08:19:39Z)
• Spectral chaos bounds from scaling theory of maximally efficient quantum-dynamical scrambling [44.99833362998488]
  A key conjecture about the evolution of complex quantum systems towards an ergodic steady state, known as scrambling, is that this process acquires universal features when it is most efficient.<n>We develop a single- parameter scaling theory for the spectral statistics in this scenario, which embodies exact self-similarity of the spectral correlations along the complete scrambling dynamics.<n>We establish that scaling predictions are matched by a privileged process and serve as bounds for other dynamical scrambling scenarios, allowing one to quantify inefficient or incomplete scrambling on all time scales.
  arXiv  Detail & Related papers  (2023-10-17T15:41:50Z)
• Spectra of generators of Markovian evolution in the thermodynamic limit: From non-Hermitian to full evolution via tridiagonal Laurent matrices [0.0]
  generators of single-particle, translation-invariant Lindblad operators on the infinite line are unitarily equivalent to direct integrals of finite-range bi-infinite Laurent operator with finite-range perturbations.
  arXiv  Detail & Related papers  (2022-06-20T16:32:14Z)
• Spectral clustering under degree heterogeneity: a case for the random walk Laplacian [83.79286663107845]
  This paper shows that graph spectral embedding using the random walk Laplacian produces vector representations which are completely corrected for node degree. In the special case of a degree-corrected block model, the embedding concentrates about K distinct points, representing communities.
  arXiv  Detail & Related papers  (2021-05-03T16:36:27Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.