We prove that center-stabilized SU(N_c) Yang-Mills theory on R³×S¹ has a positive mass gap for all compactification radii. The Bravyi-Hastings-Michalakis (BHM) gap stability theorem is extended to gauge-invariant Hilbert spaces using the exact local topological quantum order (LTQO) provided by the Z_{N_c} center symmetry. For pure SU(3) Yang-Mills, the double-trace deformation decouples exponentially at large R, yielding a conditional mass gap for R > R_BHM ≈ 12/Λ_QCD. Companion paper to the SFST framework (DOI: 10.5281/zenodo.19177245). Code: https://github.com/SFST-2026/Yang-Mills
We prove that center-stabilized SU(N_c) Yang-Mills theory on R³×S¹ has a positive mass gap for all compactification radii. The Bravyi-Hastings-Michalakis (BHM) gap stability theorem is extended to gauge-invariant Hilbert spaces using the exact local topological quantum order (LTQO) provided by the Z_{N_c} center symmetry. For pure SU(3) Yang-Mills, the double-trace deformation decouples exponentially at large R, yielding a conditional mass gap for R > R_BHM ≈ 12/Λ_QCD. Companion paper to the SFST framework (DOI: 10.5281/zenodo.19177245). Code: https://github.com/SFST-2026/Yang-Mills
Abstract We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state.
We establish a proof that pure Yang-Mills theory in four-dimensional Euclidean space possesses a mass gap Delta > 0, addressing one of the Clay Millennium Prize problems. The argument combines the Freed-Hopkins-Teleman correspondence mapping the center symmetry Z_6 of SU(3) Yang-Mills to the modular tensor category SU(3)_3 tensor SU(2)_2, the Walker-Wang model construction of a 4+1D topological phase, and Bauer theorem that such models with modular input have trivial bulk. The 3+1D boundary inherits all excitations and admits a rank-1 transfer matrix, yielding a unique ground state and mass gap Delta = 2pi/(9R) > 0. Lattice glueball spectrum agreement to 5.5% provides empirical validation. Remaining gaps toward the full Clay Prize criterion are identified.
Quantum simulations of the dynamics of QCD have been limited by the complexities of mapping the continuous gauge fields onto quantum computers. By parametrizing the gauge invariant Hilbert space in terms of plaquette degrees of freedom, we show how the Hilbert space and interactions can be expanded in inverse powers of N_{c}. At leading order in this expansion, the Hamiltonian simplifies dramatically, both in the required size of the Hilbert space as well as the type of interactions involved. Adding a truncation of the resulting Hilbert space in terms of local energy states we give explicit constructions that allow simple representations of SU(3) gauge fields on qubits and qutrits. This formulation allows a simulation of the real time dynamics of a SU(3) lattice gauge theory on a 5×5 and 8×8 lattice on ibm_torino with a CNOT depth of 113.
We develop a constructive, entanglement-first derivation of quantum field theory in which all physical structure arises from a single operator relation, ρ = e −S(H) . The fundamental object is the correlation matrix Cij = ⟨λi |ρ|λj ⟩, from which locality, spectral geometry, renormalization flow, and interaction structure are derived without prior assumptions of spacetime, fields, or gauge symmetry. We show that the conventional kernel formulation is a projection of the full entanglement structure and is therefore insufficient to reconstruct physical dynamics, establishing operator entanglement as the unique fundamental layer. Within this framework, non-Abelian gauge structure emerges from symmetry-induced degeneracies of the entanglement operator, yielding a natural construction of gauge connections and curvature. The Yang–Mills action is derived as the leading gauge-invariant functional consistent with entanglement-induced locality and spectral scaling, rather than postulated. In the strong-coupling regime, exponential decay of correlations implies clustering, a nonzero mass gap, and an area law for Wilson loops, providing a nonperturbative mechanism for confinement. We further construct the full quantum theory via an entanglement-induced measure, verifying all Osterwalder–Schrader axioms and establishing Hilbert space reconstruction, uniqueness of the continuum limit, universality, vacuum uniqueness, and operator-domain control. The resulting framework provides a unified and constructive foundation for quantum field theory in which Yang–Mills dynamics and nonperturbative phenomena arise directly from operator entanglement.
This paper presents a novel approach to solving the Yang–Mills existence and mass gap problem using quantum information theory. We develop a rigorous mathematical framework that reformulates the Yang–Mills theory in terms of quantum circuits and entanglement structures. Our method provides a concrete realization of the Yang–Mills theory that is manifestly gauge-invariant and satisfies the Wightman axioms. We demonstrate the existence of a mass gap by analyzing the entanglement spectrum of the vacuum state, establishing a direct connection between the mass gap and the minimum non-zero eigenvalue of the entanglement Hamiltonian. Our approach also offers new insights into non-perturbative phenomena such as confinement and asymptotic freedom. We introduce new mathematical tools, including entanglement renormalization for gauge theories and quantum circuit complexity measures for quantum fields. The implications of our work extend beyond the Yang–Mills theory, suggesting new approaches to quantum gravity, strongly coupled systems, and cosmological problems. This quantum information perspective on gauge theories opens up exciting new directions for research at the intersection of quantum field theory, quantum gravity, and quantum computation.
Abstract We prove that uniformly small short-range perturbations do not close the bulk gap above the ground state of frustration-free quantum spin systems that satisfy a standard local topological quantum order condition. In contrast with earlier results, we do not require a positive lower bound for finite-system spectral gaps uniform in the system size. To obtain this result, we extend the Bravyi–Hastings–Michalakis strategy so it can be applied to perturbations of the GNS Hamiltonian of the infinite-system ground state.