The bulk spectral gap is semi-decidable: a convergent family of certified upper bounds

• URL: http://arxiv.org/abs/2606.03836v1
• Date: Tue, 02 Jun 2026 16:18:19 GMT
• Title: The bulk spectral gap is semi-decidable: a convergent family of certified upper bounds
• Abstract summary: We introduce a complete family of certified upper bounds on the bulk spectral gap of quantum many-body systems.<n>As a proof of principle, we apply our algorithm to the spin-$frac12$ kagome lattice Heisenberg antiferromagnet.
• Score: 8.615139073567795
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. Existing rigorous methods are largely limited to special settings, while variational numerical approaches typically provide estimates rather than certified bounds. Here we introduce a complete family of certified upper bounds on the bulk spectral gap of quantum many-body systems. These upper bounds are obtained by solving a series of semidefinite programs and they become arbitrarily tight at the cost of more computational resources. This shows that the bulk spectral gap is semi-decidable, in contrast to undecidability results for alternative notions of spectral gap based on sequences of finite systems with prescribed boundary conditions. As a proof of principle, we apply our algorithm to the spin-$\frac{1}{2}$ kagome lattice Heisenberg antiferromagnet and obtain, to our knowledge, the first nontrivial certified upper bounds on its bulk spectral gap.

Related papers

• Generating quantum error correcting codes from topological pre-thermal scars [36.94429692322632]
  Quantum many-body scars are rare nonthermal states embedded within otherwise thermalizing spectra.<n>We develop an end-to-end framework connecting nonthermal-state discovery to certifiable quantum error correction using a many-body spectral localizer.<n>Our results establish a systematic route from identifying nonthermal many-body states to exploiting them for protecting quantum information.
  arXiv  Detail & Related papers  (2026-09-04T20:11:54Z)
• A Berry-Esseen Bound for Quantum Lattice Systems [33.82119899094064]
  We prove a version of the Berry-Esseen theorem for quantum lattice systems.<n>We provide a rigorous convergence estimate towards the normal distribution for large but finite system size.
  arXiv  Detail & Related papers  (2026-05-05T14:58:16Z)
• Hysteretic squashed entanglement in many-body quantum systems [42.085941481155295]
  Entanglement in many-body quantum systems is distributed across spatial regions.<n>We propose hysteretic entanglement $T_sq$, a conditional entanglement that measures the genuine quantum correlations between two subregions.<n>We show that $T_sq$ can detect genuine quantum correlations across both adjacent and long-range subsystems.
  arXiv  Detail & Related papers  (2026-03-10T17:00:49Z)
• Grassmann Variational Monte Carlo with neural wave functions [45.935798913942904]
  We formalize the framework introduced by Pfau et al.citepfau2024accurate in terms of Grassmann geometry of the Hilbert space.<n>We validate our approach on the Heisenberg quantum spin model on the square lattice, achieving highly accurate energies and physical observables for a large number of excited states.
  arXiv  Detail & Related papers  (2025-07-14T13:53:13Z)
• Integrability and Chaos via fractal analysis of Spectral Form Factors: Gaussian approximations and exact results [0.0]
  We identify the onset of quantum chaos and scrambling in quantum many-body systems.<n>We numerically show that Bethe Ansatz walkers fall into a category similar to the non-integrable walkers.
  arXiv  Detail & Related papers  (2025-05-08T12:56:06Z)
• Boundary Conditions for the Entanglement Cut in 2D Conformal Field Theories [0.0]
  The entanglement spectra for a subsystem in a spin chain fine-tuned to a quantum-critical point contain signatures of the underlying quantum field theory.<n>For an open chain with given boundary conditions described by a 2D conformal field theory, the entanglement spectrum of the left/right half of the system coincides with a boundary CFT spectrum.<n>The obtained entanglement spectra are in good agreement with certain boundary CFT spectra.
  arXiv  Detail & Related papers  (2025-03-16T22:21:59Z)
• A quantum algorithm for Khovanov homology [42.6618666851542]
  Khovanov homology is a knot that categorifies the Jones topological invariant, recognizes the unknot, and is conjectured to appear as an observable in $4D$ supersymmetric Yang-Mills theory.<n>Despite its rich mathematical and physical significance, the computational complexity of Khovanov homology remains largely unknown.
  arXiv  Detail & Related papers  (2025-01-21T18:54:59Z)
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems [0.0]
  We present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians in the thermodynamic limit. We demonstrate the power of the method on one-dimensional spin-chain models where we observe an improvement by several orders of magnitude over existing finite size criteria.
  arXiv  Detail & Related papers  (2024-11-06T06:04:03Z)
• Exact spectral gaps of random one-dimensional quantum circuits [0.3774866290142281]
  spectral gap of local random quantum circuits is a fundamental property that determines how close the moments of the circuit's unitaries match those of a Haar random distribution. We show that one can exactly compute the associated spectral gaps. We verify our results by numerically computing the spectral gap for systems of up to 70 qubits, as well as comparing them to gaps of random and symplectic circuits.
  arXiv  Detail & Related papers  (2024-08-20T21:23:42Z)
• Assessing the query complexity limits of quantum phase estimation using symmetry aware spectral bounds [0.0]
  computational cost of quantum algorithms for physics and chemistry is closely linked to the spectrum of the Hamiltonian. We introduce a hierarchy of symmetry-aware spectral bounds that provide a unified understanding of the performance of quantum phase estimation algorithms.
  arXiv  Detail & Related papers  (2024-03-07T18:38:49Z)
• Emergence of Fermi's Golden Rule [55.73970798291771]
  Fermi's Golden Rule (FGR) applies in the limit where an initial quantum state is weakly coupled to a continuum of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final states is discrete, with a nonzero mean level spacing.
  arXiv  Detail & Related papers  (2022-06-01T18:35:21Z)
• Finding spectral gaps in quasicrystals [0.0]
  We apply this algorithm to prove that the Hofstadter model on the Ammann-Beenker tiling of the plane has spectral gaps at certain energies. Our algorithm is applicable to more general systems with finite local complexity and eventually finds all gaps.
  arXiv  Detail & Related papers  (2022-05-21T15:21:02Z)
• Operator-algebraic renormalization and wavelets [62.997667081978825]
  We construct the continuum free field as the scaling limit of Hamiltonian lattice systems using wavelet theory. A renormalization group step is determined by the scaling equation identifying lattice observables with the continuum field smeared by compactly supported wavelets.
  arXiv  Detail & Related papers  (2020-02-04T18:04:51Z)

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.