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...
Previous studies revealed a crucial effect of symmetries on the properties of a single particle moving in a disorder potential. More recently, a phenomenon of many-body localization (MBL) has been attracting much theoretical and experimental interest. MBL systems are characterized by the emergence of quasi-local integrals of motion, and by the area-law entanglement entropy scaling of its eigenstates. In this paper, we investigate the effect of a non-Abelian SU (2) symmetry on the dynamical properties of a disordered Heisenberg chain. While SU (2) symmetry is inconsistent with the conventional MBL, a new non-ergodic regime is possible. In this regime, the eigenstates exhibit faster than area-law, but still a strongly sub-thermal scaling of entanglement entropy. Using extensive exact diagonalization simulations, we establish that this non-ergodic regime is indeed realized in the strongly disordered Heisenberg chains. We use real-space renormalization group (RSRG) to construct tree-tensor-network approximation to excited eigenstates, and demonstrate the accuracy of this procedure for systems of size up to L = 26. As the effective disorder strength is decreased, a crossover to the thermalizing phase occurs. To establish the ultimate fate of the nonergodic regime in the thermodynamic limit, we develop a novel approach for describing many-body processes that are usually neglected by RSRG. This approach is capable of describing systems of size L 2000. We characterize the resonances that arise due to such processes, finding that they involve an ever growing number of spins as the system size is increased. Crucially, the probability of finding resonances grows with the system's size. Even at strong disorder, we can identify a large lengthscale beyond which resonances proliferate. Presumably, this eventually would drive the system to a thermalizing phase. However, the extremely long thermalization time scales indicate that a broad non-ergodic regime will be observable experimentally. Our study demonstrates that, similar to the case of single-particle localization, symmetries control dynamical properties of disordered, many-body systems. The approach introduced here provides a versatile tool for describing a broad range of disordered many-body systems, well beyond sizes accessible in previous studies.arXiv :1902.09236v1 [cond-mat.str-el]
We investigate the ground-state properties of a disorderd Ising model with uniform transverse field on the Bethe lattice, focusing on the quantum phase transition from a paramagnetic to a glassy phase that is induced by reducing the intensity of the transverse field. We use a combination of quantum Monte Carlo algorithms and exact diagonalization to compute Rényi entropies, quantum Fisher information, correlation functions and order parameter. We locate the transition by means of the peak of the Rényi entropy and we find agreement with the transition point estimated from the emergence of finite values of the Edwards-Anderson order parameter and from the peak of the correlation length. We interpret the results by means of a mean-field theory in which quantum fluctuations are treated as massive particles hopping on the interaction graph. We see that the particles are delocalized at the transition, a fact that points towards the existence of possibly another transition deep in the glassy phase where these particles localize, therefore leading to a many-body localized phase.arXiv:1606.06462v1 [quant-ph]
We study numerically the population transfer protocol on the Quantum Random Energy Model and its relation to quantum computing, for system sizes of n ≤ 20 quantum spins. We focus on the energy matching problem, i.e. finding multiple approximate solutions to a combinatorial optimization problem when a known approximate solution is provided as part of the input. We study the delocalization process induced by the population transfer protocol by observing the saturation of the Shannon entropy of the time-evolved wavefunction as a measure of its spread over the system. The scaling of the value of this entropy at saturation with the volume of the system identifies the three known dynamical phases of the model. In the non-ergodic extended phase, we observe that the time necessary for the population transfer to complete follows a long-tailed distribution. We devise two statistics to quantify how effectively and uniformly the protocol populates the target energy shell. We find that population transfer is most effective if the transverse-field parameter Γ is chosen close to the critical point of the Anderson transition of the model. In order to assess the use of population transfer as a quantum algorithm we perform a comparison with random search. We detect a "black box" advantage in favour of PT, but when the running times of population transfer and random search are taken into consideration we do not see strong indications of a speedup at the system sizes that are accessible to our numerical methods. We discuss these results and the impact of population transfer on NISQ devices.
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