Semidefinite programs can be constructed to provide a nonperturbative view of the zero-temperature behavior of quantum systems. Past work has found that small semidefinite programs (relative to the dimension of Hilbert space) yield quantitatively accurate estimates for the ground-state energy and expectation values. This paper examines the properties of these semidefinite programs when applied to lattice-regulated field theories exhibiting fermion sign problems. Specifically on the finite-density Thirring model, small semidefinite programs still yield quantitatively accurate results, and there is no indication that these methods encounter exponential costs (analogous to the fermion sign problem) at finite chemical potential.
Semidefinite programs can be constructed to provide a nonperturbative view of the zero-temperature behavior of quantum systems. Past work has found that small semidefinite programs (relative to the dimension of Hilbert space) yield quantitatively accurate estimates for the ground-state energy and expectation values. This paper examines the properties of these semidefinite programs when applied to lattice-regulated field theories exhibiting fermion sign problems. Specifically on the finite-density Thirring model, small semidefinite programs still yield quantitatively accurate results, and there is no indication that these methods encounter exponential costs (analogous to the fermion sign problem) at finite chemical potential.
This paper demonstrates the application of semidefinite programming to lattice field theories, showcasing spin chains and lattice scalar field theory. Requiring expectation values of manifestly positive semi-definite operators to be non-negative results in a lower bound on the ground-state energy of any quantum mechanical system, which can be made arbitrarily tight for systems described by finite-dimensional Hilbert spaces. Such bounds can be obtained directly in the infinite-volume limit. The process of optimizing these lower bounds also yields estimates for a chosen set of expectation values in the ground state.
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 Z2 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 7×8 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.
We study semidefinite programming (SDP) methods to analyze spin chain Hamiltonians. We examine the ground state energy, the first excited charged states and ground state correlators in two simple models: the Ising model in a transverse magnetic field and the closely related 3-state Potts model. Our goal is to understand precisely what the SDP program is doing and when it works well, why it does so. We focus on the following novel ingredients: using charge constraints to obtain excited states and to see if additional constraints from integrable models are effective at improving the method. At criticality we also explore to what extent we can use approximate Virasoro correlators to extract conformal data: the central charge and some critical exponents of charged states. We also use these to identify the location of the phase transition. In the special case where the system is made of free fermions we prove that the SDP finds the exact energy of the ground state and produces the correct two point functions of the fermions. Away from free fermion theories, the SDP gets progressively worse at estimating data beyond the value of the ground state energy (like correlation functions), although it qualitatively matches these. In order to be effective, the SDP seems to run into scaling issues where the amount of input needed scales poorly with the lattice volume.
We study the semidefinite programs (SDPs) of quantum information whose variables are density operators, measurement operators, or unbounded positive semidefinite operators, together with their thermodynamic regularizations by the von Neumann, Fermi-Dirac, and Bose-Einstein entropies, which have been associated with Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, respectively. We find that this correspondence is exact: the Fermi-Dirac and Bose-Einstein regularized SDPs coincide with the constrained free-energy minimization of an ideal Fermi or Bose gas whose Hamiltonian and non-commuting charges are the second quantizations of the SDP data and whose one-body density matrix is the SDP variable, while the Boltzmann framework corresponds to the single-particle case. We show that the three frameworks can be unified under one framework, in which each entropy-regularized dual is, up to a term linear in the chemical potentials, the grand potential of the corresponding system; the duals are unconstrained and concave in the chemical-potential vector, with gradient and Hessian given by thermal expectation values that hybrid quantum-classical algorithms can estimate. We further identify the Fermi-Dirac and Bose-Einstein relative entropies with the Umegaki relative entropy between quasi-free states, which yields their monotonicity under gauge-covariant Gaussian channels. The result places a wide class of SDPs and their solvers within the thermodynamics of ideal quantum gases and points to extensions to interacting gases.
We present a generic approach to the condensed-matter ground-state problem which is complementary to variational techniques and works directly in the thermodynamic limit. Relaxing the ground-state problem, we obtain semidefinite programs (SDP). These can be solved efficiently, yielding strict lower bounds to the ground-state energy and approximations to the few-particle Green's functions. As the method is applicable for all particle statistics, it represents, in particular, a novel route for the study of strongly correlated fermionic and frustrated spin systems in D>1 spatial dimensions. It is demonstrated for the XXZ model and the Hubbard model of spinless fermions. The results are compared against exact solutions, quantum Monte Carlo calculations, and Anderson bounds, showing the competitiveness of the SDP method.
Wave function methods have offered a robust, systematically improvable means to study the ground-state properties in quantum many-body systems. Theories like coupled cluster and their derivatives provide highly accurate approximations to the energy landscape at a reasonable computational cost. Analogs of such methods to study thermal properties, though highly desirable, have been lacking because evaluating thermal properties involve a trace over the entire Hilbert space. Approximating every state in the Hilbert space is an impossible task. Besides, excited-state theories are not as well studied as ground-state ones. In this thesis, we overcome these difficulties by employing thermofield dynamics, a theory that allows the purification of the ensemble density matrix and constructs a pure wave function that encodes the equilibrium thermal behavior of the system. Ensemble averages become expectation values over this so-called thermal state. Around this thermal state, we develop a framework to extend ground-state wave function theories to non-zero temperatures. We discuss explicit formulations of mean-field, configuration interaction, and coupled cluster theories for thermal properties of fermions in the grand canonical ensemble. For fermions in the canonical ensemble, we present mean-field, configuration interaction, and perturbation theories. We also discuss thermal mean-field, configuration interaction, and coupled cluster formalisms for \SU systems. To assess the quality of these approximations, we study both model and small atomic and molecular as benchmark electronic problems while comparing against exact results. For spins, we use the Lipkin-Meshkov-Glick and the one-dimensional transverse field Ising models as our reference systems. Indeed, the thermal methods perform reasonably well while merely adding a prefactor to the computational cost. They also inherit all the properties, good or bad, from their ground-state counterparts, signifying the robustness of our formalism.