Two-Sided Bounds on Ground State Properties in Lattice Gauge Theories

AI summary

Overview

Research area: Quantum many-body physics and lattice gauge theory, specifically rigorous, certification-style bounds on ground-state energies and observables using semidefinite programming (SDP) relaxations combined with tensor-network methods.

Technical level: Advanced. The paper assumes familiarity with non-commutative polynomial optimization, the Navascués–Pironio–Acín (NPA) hierarchy, matrix product states/DMRG, and Hamiltonian formulations of lattice gauge theory.

Scope (one sentence): The paper builds a hierarchy of SDP relaxations that lower-bound the ground-state energy of lattice gauge theories, combines these with variational tensor-network upper bounds, and propagates the resulting certified energy interval to physical observables such as Wilson loops and mesonic strings in a (1+1)-dimensional Z₂ theory with dynamical fermions and a (2+1)-dimensional pure-gauge Z₂ theory.

What This Paper Is About

Lattice gauge theories are hard to simulate: classical Monte Carlo can fail because of the sign problem, and variational methods — classical or quantum, including tensor networks — only ever give upper bounds on the ground-state energy, with accuracy that depends on the chosen ansatz and no rigorous certificate of convergence. The authors' goal is to supply the missing other side of the bracket, namely provable lower bounds from convex SDP relaxations, and then to reuse those energy bounds to certify the values of physically important observables rather than just the energy itself.

Key Contributions

1. A two-sided certification pipeline. A variational matrix-product-state (MPS) calculation supplies an upper bound E_max on the ground-state energy, an SDP relaxation of the non-commutative polynomial optimization problem supplies a lower bound E_min, and the resulting interval is then imposed as a constraint in further SDPs that bound observables. The MPS choice is explicitly described as non-fundamental — any suitable variational model could be substituted.

2. Gauge-structured SDP relaxations. The NPA hierarchy is used up to level three with a restricted interaction range to exploit Hamiltonian sparsity, and the feasible set is restricted to the physical gauge sector by imposing ⟨G_i⟩ = 1 for all gauge generators. The problem is formulated in a gauge-adapted, block-diagonal basis: using the group generated by the gauge operators, the symmetry-adapted basis matrix turns out to be a permutation matrix, so a suitable ordering ("block-order") of the generating sequence makes the moment matrix block-diagonal, with blocks falling into five distinct classes (three per matter site, one per field site, and one residual class).

3. Certified intervals in two different theories. Results are given for a (1+1)-dimensional Z₂ Kogut–Susskind model with dynamical fermionic matter and for its pure-gauge version in (2+1) dimensions including a plaquette term. For the one-dimensional system with 129 sites the certified interval has a relative spread of 0.04%; for the two-dimensional system with 97 sites (a 7 × 8 lattice) the relative spread is 2.08%.

4. Bounds beyond the energy. The energy interval is propagated to short- and long-range observables, including one-body matter observables, Wilson loops spanning multiple plaquettes, and — under mild continuity assumptions — the modulus of non-Hermitian mesonic string operators, which are central to characterizing confinement. The authors note that Hermitian operators involving up to eight-body terms are bounded by adding specific monomials to the generating sequence to increase expressivity.

Main Findings

• Certified energy windows are tight in 1D: for a one-dimensional system of 129 sites, the interval bracketing the ground-state energy has a relative spread of 0.04%.
• Certified energy windows in 2D: for a 7 × 8 lattice (97 sites) in the pure-gauge (2+1)-dimensional theory, the relative spread is 2.08%.
• Hierarchy level and interaction range trade off differently: increasing the relaxation level or the interaction range systematically improves the SDP lower bound, but raising the level typically enlarges the generating sequence while raising the interaction range does not — so level increase usually gives the better bound, while range restriction mainly buys computational tractability at the cost of tightness.
• Gauge-sector constraints tighten the relaxation: imposing the extensive set of gauge-sector constraints gives a tighter relaxation than the unconstrained formulation, because the extra linear constraints shrink the SDP feasible set.
• Sign structure of the bounds: MPS energies always lie above the exact values while SDP relaxations provide lower bounds; this distinction is not visible in the paper's error plots because absolute values of errors are shown on a logarithmic scale.
• Benchmarking against exact results: for small systems, exact diagonalization is used as the reference; for larger systems, the MPS energy serves as the reference. The grey shaded region of the error figure marks errors below the guaranteed numerical precision of the SDP solver.
• Basis structure is simple in practice: after computations with dedicated software (RepLAB and RepnDecomp), the symmetry-adapted basis is observed to be a permutation matrix, so block-diagonalization reduces to correctly ordering the sequence; merging of subsequences follows a simple criterion based on whether g(s_i s_j) = s_i s_j or −s_i s_j.
• Long-range observables are certifiable: the method yields rigorous bounds on Wilson loops over multiple plaquettes and on the magnitude of the mesonic string, not just on local correlators.

Methodology in Plain English

The ground-state energy problem is rewritten as an optimization over the expectation values of Pauli strings. Asking directly for the true set of achievable expectation values is intractable, so the authors replace it with a larger, convex set described by a positive-semidefinite "moment matrix" built from a chosen list (sequence) of Pauli operators — the NPA hierarchy. Because enlarging the feasible set can only lower the minimum, the resulting semidefinite program gives a guaranteed lower bound on the energy, solvable with standard interior-point solvers (the paper uses MOSEK).

To keep the SDPs tractable, two simplifications are used. First, the sequence of operators is truncated by interaction range, so only Pauli strings spanning nearby sites enter the moment matrix. Second, symmetries are exploited: if the SDP is invariant under a group — here the group generated by the lattice gauge transformations — one may restrict to symmetric solutions, and in a symmetry-adapted basis the large positive-semidefinite constraint splits into several smaller blocks, sharply reducing cost. Restricting to the physical gauge sector (⟨G_i⟩ = 1) is what makes the bounds physically relevant rather than bounds on an unconstrained theory.

The other side of the bracket comes from DMRG on matrix product states (via tenpy, with the gauge symmetry enforced by a penalty term), which variationally minimizes the energy and therefore cannot go below the true value. With an interval [E_min, E_max] in hand, the authors solve further SDPs minimizing and maximizing an observable subject to the energy lying inside that interval, which yields certified upper and lower bounds on the observable — bounds that are looser than the true ground-state-manifold range but rigorously valid. For degenerate ground states, this range formulation is the well-defined substitute for a single expectation value.

Why This Matters

Variational simulations of gauge theories — tensor networks today, quantum simulators soon — produce numbers without error bars. This work supplies the error bars: a convex, certificate-producing counterpart that turns "we think the energy is about X" into "the energy lies in this interval, provably." That is a qualitative change in what numerical lattice gauge theory can claim.

• Confinement studies: rigorous bounds on Wilson loops over multiple plaquettes and on the mesonic string magnitude target the observables used to diagnose confinement, rather than only the energy.
• Quantum-simulator certification: because the bounds are derived from the Hamiltonian and the operator algebra alone, they can be used to certify or falsify the output of an analog or digital quantum simulation of a gauge theory.
• Sign-problem regimes: the approach is formulated in a Hamiltonian setting and does not rely on Monte Carlo sampling, so it is in principle applicable where classical Monte Carlo is obstructed.
• Benchmarking variational and tensor-network pipelines: the certified window quantifies exactly how much a given bond dimension or ansatz is leaving on the table.

Industry relevance: quantum-hardware and quantum-simulation groups needing device-independent or assumption-light validation of their outputs; scientific-software and optimization vendors, since the method leans on state-of-the-art semidefinite programming solvers (MOSEK is used here); and high-energy and nuclear-physics computational groups interested in non-perturbative, sign-problem-free routes to strongly coupled gauge dynamics.

Future Directions

• Exploiting more symmetries: the authors state that additional symmetries such as translation invariance or mirror symmetry could also be exploited and leave this to future work, which would shrink the moment matrix further and could allow higher levels or larger lattices.
• Scaling to larger and higher-dimensional systems: the reported 2.08% spread on a 7 × 8 lattice versus 0.04% on 129 sites in 1D shows the cost of going up in dimension; extending the pipeline to bigger (2+1)- and (3+1)-dimensional systems is the obvious open challenge.
• Going beyond the truncated observable results: the provided text cuts off inside the section on bounding few-body observables, so the full set of numerical bounds on Wilson loops and mesonic strings, and the precise continuity assumptions needed for the non-Hermitian string operator, are not reported in the available content.
• Coupling to other variational ansätze and to real data: since the upper bound is explicitly not tied to MPS, the framework invites substitution of other variational models, and the cited prior work on supplementing SDP relaxations with measurement data suggests certification of actual experimental or simulator data as a next step.

Target Audience

Quantum many-body and lattice gauge theory researchers, especially those already using tensor networks or quantum simulators who want rigorous error control; practitioners of semidefinite programming and non-commutative polynomial optimization looking for a demanding many-body application; and quantum-computing researchers interested in certification tools for gauge-theory simulations. Readers without a background in SDP hierarchies or Hamiltonian lattice gauge theory will find the methods sections demanding.

Authors’ abstract

Gauge theories describe fundamental interactions in the standard model of particle physics and build effective theories in condensed matter physics. Being notoriously hard to simulate, these theories are commonly regularized as lattice gauge theories which can be evaluated with high precision through Monte Carlo algorithms or variational methods. However, Monte Carlo algorithms are not applicable in all regimes due to the sign problem; while variational methods, quantum and classical alike, only give upper-bounds on the ground-state energy. Their precision crucially depends on the chosen ansatz. Here, we present a framework based on a hierarchy of semidefinite programs to yield increasingly good lower-bounds on the ground-state energy and propagate these bounds to extended observables like Wilson loops and mesonic strings. We demonstrate the numerical capabilities by applying the algorithm to a (1+1)-dimensional $\mathbb{Z}_2$ theory with dynamic fermionic matter and its pure-gauge version in (2+1)-dimensions. In both cases, we obtain certified intervals for the ground-state energy. For a one-dimensional system with 129 sites, the certified interval has a relative spread of $0.04\%$. For a two-dimensional system with 97 sites (corresponding to a $7 \times 8$ lattice), the relative spread is $2.08\%$. Beyond energy estimates, the proposed method provides rigorous bounds on both short- and long-range observables such as Wilson loops over multiple plaquettes and the magnitude of the mesonic string. The method can be both used as a computational microscope into many-body physics and a way to certify results of quantum simulators.