Computational complexity continuum within Ising formulation of NP problems

A promising approach to achieve computational supremacy over the classical von Neumann architecture explores classical and quantum hardware as Ising machines. The minimisation of the Ising Hamiltonian is known to be NP-hard problem yet not all problem instances are equivalently hard to optimise. Given that the operational principles of Ising machines are suited to the structure of some problems but not others, we propose to identify computationally simple instances with an ‘optimisation simplicity criterion’. Neuromorphic architectures based on optical, photonic, and electronic systems can naturally operate to optimise instances satisfying this criterion, which are therefore often chosen to illustrate the computational advantages of new Ising machines. As an example, we show that the Ising model on the Möbius ladder graph is ‘easy’ for Ising machines. By rewiring the Möbius ladder graph to random 3-regular graphs, we probe an intermediate computational complexity between P and NP-hard classes with several numerical methods. Significant fractions of polynomially simple instances are further found for a wide range of small size models from spin glasses to maximum cut problems. A compelling approach for distinguishing easy and hard instances within the same NP-hard class of problems can be a starting point in developing a standardised procedure for the performance evaluation of emerging physical simulators and physics-inspired algorithms.

Delocalization transition in low energy excitation modes of vector spin glasses

We study the energy minima of the fully-connected mm-components vector spin glass model at zero temperature in an external magnetic field for m\ge 3m≥3. The model has a zero temperature transition from a paramagnetic phase at high field to a spin glass phase at low field. We study the eigenvalues and eigenvectors of the Hessian in the minima of the Hamiltonian. The spectrum is gapless both in the paramagnetic and in the spin glass phase, with a pseudo-gap behaving as \lambda^{m-1}λm−1 in the paramagnetic phase and as \sqrt{\lambda}λ at criticality and in the spin glass phase. Despite the long-range nature of the model, the eigenstates close to the edge of the spectrum display quasi-localization properties. We show that the paramagnetic to spin glass transition corresponds to delocalization of the edge eigenvectors. We solve the model by the cavity method in the thermodynamic limit. We also perform numerical minimization of the Hamiltonian for N\le 2048N≤2048 and compute the spectral properties, that show very strong corrections to the asymptotic scaling approaching the critical point.

Parallel Computing of Edwards—Anderson Model

A scheme for parallel computation of the two-dimensional Edwards—Anderson model based on the transfer matrix approach is proposed. Free boundary conditions are considered. The method may find application in calculations related to spin glasses and in quantum simulators. Performance data are given. The scheme of parallelisation for various numbers of threads is tested. Application to a quantum computer simulator is considered in detail. In particular, a parallelisation scheme of work of quantum computer simulator.

The de Almeida–Thouless Line in Hierarchical Quantum Spin Glasses

We determine explicitly and discuss in detail the effects of the joint presence of a longitudinal and a transversal (random) magnetic field on the phases of the Random Energy Model and its hierarchical generalization, the GREM. Our results extent known results both in the classical case of vanishing transversal field and in the quantum case for vanishing longitudinal field. Following Derrida and Gardner, we argue that the longitudinal field has to be implemented hierarchically also in the Quantum GREM. We show that this ensures the shrinking of the spin glass phase in the presence of the magnetic fields as is also expected for the Quantum Sherrington–Kirkpatrick model.

Parallel computing of Edwards–Anderson model

An algorithm for parallel exact calculation of the ground state of a two-dimensional Edwards–Anderson model with free boundary conditions is given. The running time of the algorithm grows exponentially as the side of the lattice square increases. If one side of the lattice is fixed, the running time grows polynomially with increasing size of the other side. The method may find application in the theory of spin glasses, in the field of quantum computing. Performance data for the bimodal distribution is given. The distribution of spin bonds can be either bimodal or Gaussian. The method makes it possible to compute systems up to a size of 40x40.

Analytical solution for low energy state estimation by quantum annealing to arbitrary Ising spin Hamiltonian

We point to the existence of an analytical solution to a general quantum annealing (QA) problem of finding low energy states of an arbitrary Ising spin Hamiltonian HI by implementing time evolution with a Hamiltonian H(t) = HI + g(t)Ht. We will assume that the nonadiabatic annealing protocol is defined by a specific decaying coupling g(t) and a specific mixing Hamiltonian Ht that make the model analytically solvable arbitrarily far from the adiabatic regime. In specific cases of HI, the solution shows the possibility of a considerable quantum speedup of finding the Ising ground state. We then compare predictions of our solution to results of numerical simulations, and argue that the solvable QA protocol produces the optimal performance in the limit of maximal complexity of the computational problem. Our solution demonstrates for the most complex spin glasses a power-law energy relaxation with the annealing time T and uncorrelated from HI annealing schedule. This proves the possibility for spin glasses of a faster than ∼ 1/logβT energy relaxation.

Field-induced vortex-like textures as a probe of the critical line in reentrant spin glasses

We study the evolution of the low-temperature field-induced magnetic defects observed under an applied magnetic field in a series of frustrated amorphous ferromagnets (Fe$$_{1-x}$$ 1 - x Mn$$_{x}$$ x )$$_{75}$$ 75 P$$_{16}$$ 16 B$$_{3}$$ 3 Al$$_{3}$$ 3 (“a-Fe$$_{1-x}$$ 1 - x Mn$$_{x}$$ x ”). Combining small-angle neutron scattering and Monte Carlo simulations, we show that the morphology of these defects resemble that of quasi-bidimensional spin vortices. They are observed in the so-called “reentrant” spin-glass (RSG) phase, up to the critical concentration $$x_{\mathrm{C}} \approx 0.36$$ x C ≈ 0.36 which separates the RSG and “true” spin glass (SG) within the low temperature part of the magnetic phase diagram of a-Fe1−xMnx. These textures systematically decrease in size with increasing magnetic field or decreasing the average exchange interaction, and they finally disappear in the SG sample ($$x = 0.41$$ x = 0.41 ), being replaced by field-induced correlations over finite length scales. We argue that the study of these nanoscopic defects could be used to probe the critical line between the RSG and SG phases.