We present several efficient implementations of the simulated annealing algorithm for Ising spin glasses on sparse graphs. In particular, we provide a generic code for any choice of couplings, an optimised code for bipartite graphs, and highly optimised implementations using multi-spin coding for graphs with small maximum degree and discrete couplings with a finite range. The latter codes achieve up to 50 spin flips per nanosecond on modern Intel CPUs. We also compare the performance of the codes to that of the special purpose D-Wave devices built for solving such Ising spin glass problems.
Using continuous-space quantum Monte Carlo methods we investigate the zero-temperature ferromagnetic behavior of a two-component repulsive Fermi gas under the influence of periodic potentials that describe the effect of a simple-cubic optical lattice. Simulations are performed with balanced and with imbalanced components, including the case of a single impurity immersed in a polarized Fermi sea (repulsive polaron). For an intermediate density below half filling, we locate the transitions between the paramagnetic, and the partially and fully ferromagnetic phases. As the intensity of the optical lattice increases, the ferromagnetic instability takes place at weaker interactions, indicating a possible route to observe ferromagnetism in experiments performed with ultracold atoms. We compare our findings with previous predictions based on the standard computational method used in material science, namely density functional theory, and with results based on tight-binding models.PACS numbers: 05.30. Fk, 03.75.Hh, 75.20.Ck Itinerant ferromagnetism, which occurs in transition metals like nickel, cobalt and iron, is an intriguing quantum mechanical phenomenon due to strong correlations between delocalized electrons. The theoretical tools allowing us to perform ab-initio simulations of the complex electronic structure of solid state systems, the most important being density functional theory (DFT) [1,2], give systematically reliable results only for simple metals and semiconductors. The extension to strongly correlated materials still represents an outstanding open challenge [3]. Our understanding of quantum magnetism is mostly based on simplified model Hamiltonians designed to capture the essential phenomenology of real materials. The first model introduced to explain itinerant ferromagnetism is the Stoner Hamiltonian [4], which describes a Fermi gas in a continuum with short-range repulsive interactions originally treated at the mean-field level. The Hubbard model, describing electrons hopping between sites of a discrete lattice with on-site repulsion, was also originally introduced to explain itinerant ferromagnetism in transition metals [5]. Despite the simplicity of these models, their zero-temperature ferromagnetic behavior is still uncertain.In recent years, ultracold atoms have emerged as the ideal experimental system to investigate intriguing quantum phenomena caused by strong correlations. Experimentalists are able to manipulate interparticle interactions and external periodic potentials independently, allowing the realization of model Hamiltonians relevant for condensed matter physics [6], or to test exchangecorrelation functionals used in DFT simulations of materials [7]. Indirect evidence consistent with itinerant (Stoner) ferromagnetism was observed in a gas of 6 Li atoms [8] when the strength of the repulsive interatomic interaction was increased following the upper branch of a Feshbach resonance. However, subsequent theoretical [9] and experimental studies [10,11] have demonstrated that three-body recomb...
We consider whether it is possible to find ground states of frustrated spin systems by solving them locally. Using spin glass physics and Imry-Ma arguments in addition to numerical benchmarks we quantify the power of such local solution methods and show that for the average low-dimensional spin glass problem outside the spinglass phase the exact ground state can be found in polynomial time. In the second part we present a heuristic, general-purpose hierarchical approach which for spin glasses on chimera graphs and lattices in two and three dimensions outperforms, to our knowledge, any other solver currently around, with significantly better scaling performance than simulated annealing.
The simplest model for itinerant ferromagnetism, the Stoner model, has so far eluded experimental observation in repulsive ultracold fermions due to rapid three-body recombination at large scattering lengths. Here we show that a ferromagnetic phase can be stabilised by imposing a moderate optical lattice. The reduced kinetic energy drop upon formation of a polarized phase in an optical lattice extends the ferromagnetic phase to smaller scattering lengths where three-body recombination is small enough to permit experimental detection of the phase. We also show, using time dependent density functional theory, that in such a setup ferromagnetic domains emerge rapidly from a paramagnetic initial state.The ground-state of a repulsive gas of fermions with contact interaction was first predicted by Stoner [1] in 1933 to be ferromagnetic and a precise value for the critical interaction strength in a homogeneous system was recently obtained with diffusion Monte Carlo simulations [2][3][4]. While ultracold fermionic gases should provide a controlled environment to study this phenomenon, the instability of a strongly repulsive gas towards molecule formation has so far prevented experimental realization of this phase. Repulsive fermions on the positive side of the Feshbach resonance live on the meta-stable repulsive branch [5]. When three atoms, one with the opposite spin of the other two, come close to each other two atoms with opposite spin will form bosonic molecules and the other one carries the binding energy away. The rate of such process increases rapidly with scattering length. The lifetime of the gas is therefore largely governed by the interaction strength and the spatial overlap between the two spin species.A recent experiment [6] presented evidence for a possible ferromagnetic state formed after rapid increase of the scattering length. The nature of this phase has, however, been questioned [7-9] as the peaks in kinetic energy, cloud size and loss rate observed in [6] are only indirect evidence for ferromagnetic domains [7], and it has been shown that molecule formation is dominant at large interaction strengths [8,9]. Several papers have proposed to reduce the recombination rate by using a polar molecular gas with dipole interactions and positive scattering range [10,11], narrower Feshbach resonances [8,9,12,13] and fermions with unequal mass [14,15]. Although these approaches might prove promising in future experiments, they all change the microscopic physics of the Stoner model.Here we suggest new strategies which achieve the same goal of stabilising the ferromagnetic phase, yet preserve the microscopic physics and thus pave the way towards experimental realization of Stoner ferromagnetism. Figure 1: Phase diagram of the homogeneous repulsive Fermi gas as a function of temperature T /T F and interaction strength k F as. The white dashed line indicates the paramagnet-ferromagnetic phase transition and the colour scale the polarization P = (n ↑ − n ↓ )/(n ↑ + n ↓ ).Firstly, the lifetime of the system can be incre...
We provide a new method for estimating spectral gaps in low-dimensional systems. Unlike traditional phase estimation, our approach does not require ancillary qubits nor does it require well characterised gates. Instead, it only requires the ability to perform approximate Haar-random unitary operations, applying the unitary whose eigenspectrum is sought out and performing measurements in the computational basis. We discuss application of these ideas to in-place amplitude estimation and quantum device calibration.
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