No abstract

The modular (or entanglement) Hamiltonian correspondent to the half-space-bipartition of a quantum state uniquely characterizes its entanglement properties. However, in the context of lattice models, its explicit form is analytically known only for the Ising chain and certain free theories in one-dimension. In this work, we provide a throughout investigation of entanglement Hamiltonians in lattice models obtained via the Bisognano-Wichmann theorem, which provides an explicit functional form for the entanglement Hamiltonian itself in quantum field theory. Our study encompasses a variety of one-and two-dimensional models, supporting diverse quantum phases and critical points, and, most importantly, scanning several universality classes, including Ising, Potts, and Luttinger liquids. We carry out extensive numerical simulations based on the density-matrixrenormalization-group method, exact diagonalization, and quantum Monte Carlo. In particular, we compare the exact entanglement properties and correlation functions to those obtained applying the Bisognano-Wichmann theorem on the lattice. We carry out this comparison on both the eigenvalues and eigenvectors of the entanglement Hamiltonian, and expectation values of correlation functions and order parameters. Our results evidence that, as long as the low-energy description of the lattice model is well-captured by a Lorentz-invariant quantum field theory, the Bisognano-Wichmann theorem provides a qualitatively and quantitatively accurate description of the lattice entanglement Hamiltonian. The resulting framework paves the way to direct studies of entanglement properties utilizing well-established statistical mechanics methods and experiments. arXiv:1807.01322v1 [cond-mat.stat-mech] 3 Jul 2018

In simple ferromagnetic quantum Ising models characterized by an effective double-well energy landscape the characteristic tunneling time of path-integral Monte Carlo (PIMC) simulations has been shown to scale as the incoherent quantum-tunneling time, i.e., as 1/∆ 2 , where ∆ is the tunneling gap. Since incoherent quantum tunneling is employed by quantum annealers (QAs) to solve optimization problems, this result suggests there is no quantum advantage in using QAs w.r.t. quantum Monte Carlo (QMC) simulations. A counterexample is the recently introduced shamrock model, where topological obstructions cause an exponential slowdown of the PIMC tunneling dynamics with respect to incoherent quantum tunneling, leaving the door open for potential quantum speedup, even for stoquastic models. In this work, we investigate the tunneling time of projective QMC simulations based on the diffusion Monte Carlo (DMC) algorithm without guiding functions, showing that it scales as 1/∆, i.e., even more favorably than the incoherent quantum-tunneling time, both in a simple ferromagnetic system and in the more challenging shamrock model. However a careful comparison between the DMC ground-state energies and the exact solution available for the transverse-field Ising chain points at an exponential scaling of the computational cost required to keep a fixed relative error as the system size increases.Difficult optimization problems are ubiquitous in science and in engineering. Relevant examples are protein folding, the traveling salesman problem, and portfolio optimization. Such problems can often be formulated as the search of the lowest-energy spin configuration in an Ising glass [1], a task that has been proven to be NP-hard in the case of non-planar graphs [2]. While exact classical algorithms are believed to require computational times that exponentially grow with the problem size (unless P = NP), various heuristic methods can often provide quite accurate (but possibly not exact) solutions in a feasible time. Perhaps, the most versatile of such heuristic methods is simulated classical annealing (SCA) [3], which exploits thermal fluctuations in a Markov chain Monte Carlo simulation to escape local minima and, hopefully, find the lowest energy state at the end of the annealing process when the temperature has been reduced to zero.Also adiabatic quantum computers, such as the quantum annealers (QAs) built using superconducting flux qubits [4-6] -or, potentially, with Rydberg atoms trapped in arrays of optical tweezers [7] -can be used to solve complex combinatorial optimization problems. They implement a quantum annealing process [8][9][10], in which quantum mechanical tunneling through tall barriers is used to escape local minima, and quantum fluctuations are gradually removed by reducing to zero the transverse field of a quantum Ising model. While in problems with energy landscapes characterized by tall but thin barriers quantum tunneling definitely makes QAs more efficient than classical optimization methods such as SCA [11,12], cert...