Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging algorithmic task, particularly for highly regular structures. Here we introduce a new algorithmic approach based on ideas from spectral graph theory and geometry that constructs candidate correspondences between vertices using their curvatures. Any correspondence produced by the algorithm is explicitly verified, ensuring that non-isomorphic graphs are never incorrectly identified as isomorphic. Although the method does not yet guarantee success on all inputs, we find that it correctly resolves every instance tested in deterministic polynomial time, including a broad collection of graphs known to be difficult for classical techniques. These results demonstrate that enriched spectral methods can be far more powerful than previously understood, and suggest a promising direction for the practical resolution of the complexity of the graph isomorphism problem.
Amirhossein Rezaei, Mahmood Hasani, Alireza Rezaei
et al.
In this study, we present a novel analytical approach to solving large-scale Ising problems by reformulating the discrete Ising Hamiltonian into a continuous framework. This transformation enables us to derive exact solutions for a non-trivial class of fully connected Ising models. To validate our method, we conducted numerical experiments comparing our analytical solutions with those obtained from a quantum-inspired Ising algorithm and a quantum Ising machine. The results demonstrate that the quantum-inspired algorithm and brute-force method successfully align with our solutions, while the quantum Ising machine exhibits notable deviations. Our method offers promising avenues for analytically solving diverse Ising problem instances, while the class of Ising problems addressed here provides a robust framework for assessing the fidelity of Ising machines.
The use of quantum computers to calculate the ground state (lowest) energies of a spin lattice of electrons described by the Fermi-Hubbard model of great importance in condensed matter physics has been studied. The ability of quantum bits (qubits) to be in a superposition state allows quantum computers to perform certain calculations that are not possible with even the most powerful classical (digital) computers. This work has established a method for calculating the ground state energies of small lattices which should be scalable to larger lattices that cannot be calculated by classical computers. Half-filled lattices of sizes 1x4, 2x2, 2x4, and 3x4 were studied. The calculated energies for the 1x4 and 2x2 lattices without Coulomb repulsion between the electrons and for the 1x4 lattice with Coulomb repulsion agrees with the true energies to within 0.60%, while for the 2x2 lattice with Coulomb repulsion the agreement is within 1.50% For the 2x4 lattice, the true energy without Coulomb repulsion was found to agree within 0.18%.
The Klein-Gordon equation for a scalar field sourced by a static spherically symmetric background is an interesting second-order differential equation with applications in particle physics, astrophysics, and elsewhere. Here we present static solutions for generic source density profiles in the case where the scalar field has no interactions or a mass term. For a $λφ^4$ self-interaction term, we develop the techniques that are necessary numerical computation. We also provide code to perform the numerical calculations that can be adapted for arbitrary density profiles.
Thies Romig, Vladislav Kochetov, Sergey I. Bokarev
Recent experimental advances in ultrafast science put different processes occurring on the electronic timescale below a few femtoseconds in focus. In the present theoretical work, we demonstrate how the transformation and propagation of the density matrix in the basis of irreducible spherical tensors can be conveniently used to study sub-few fs spin-flip dynamics in the core-excited transition metal compounds. With the help of the Wigner-Eckart theorem, such a transformation separates the essential dynamical information from the geometric factors governed by the angular momentum algebra. We show that an additional reduction can be performed by the physically motivated truncation of the spherical tensor basis. In particular, depending on the degree of coherence, the ultrafast dynamics can be considered semi-quantitative in the notably reduced spherical basis when only total populations of the basis states of the given spin are of interest. Such truncation should be especially beneficial when the number of the high-spin basis states is vast, as it substantially reduces computational costs.
We investigate how neural networks (NNs) understand physics using 1D quantum mechanics. After training an NN to accurately predict energy eigenvalues from potentials, we used it to confirm the NN's understanding of physics from four different aspects. The trained NN could predict energy eigenvalues of different kinds of potentials than the ones learned, predict the probability distribution of the existence of particles not used during training, reproduce untrained physical phenomena, and predict the energy eigenvalues of potentials with an unknown matter effect. These results show that NNs can learn physical laws from experimental data, predict the results of experiments under conditions different from those used for training, and predict physical quantities of types not provided during training. Because NNs understand physics in a different way than humans, they will be a powerful tool for advancing physics by complementing the human way of understanding.
Quantum matter, the research field studying phases of matter whose properties are intrinsically quantum mechanical, draws from areas as diverse as hard condensed matter physics, materials science, statistical mechanics, quantum information, quantum gravity, and large-scale numerical simulations. Recently, researchers interested quantum matter and strongly correlated quantum systems have turned their attention to the algorithms underlying modern machine learning with an eye on making progress in their fields. Here we provide a short review on the recent development and adaptation of machine learning ideas for the purpose advancing research in quantum matter, including ideas ranging from algorithms that recognize conventional and topological states of matter in synthetic an experimental data, to representations of quantum states in terms of neural networks and their applications to the simulation and control of quantum systems. We discuss the outlook for future developments in areas at the intersection between machine learning and quantum many-body physics.
OPENMMF is a numerical library designed to evaluate the Time-Evolution Operator of quantum systems with a discrete spectrum, and driven by an arbitrary combination of harmonic couplings. The time-evolution operator is calculated as a multifrequency Fourier expansion, which results from expressing the time-dependent Schrödinger equation in the frequency domain (Ho, Chu and Tietz, Ch. Phys. Lett 96, 464 (1983)). The library provides a generic tool to study systems with arbitrary spectral composition, limited only by the available computational resources. OPENMMF includes functionalities to build dense and sparse matrix representations of the system Hamiltonian and various functions to calculate physical quantities such as the micromotion operator and time/phase average of state populations. The library uses a generalised notion of dressed state for systems with polychromatic driving. In this paper, we describe the design and functionality of OPENMMF, provide examples of its use and discuss its range of applicability. The library is written in object-oriented style Fortran90 and includes a set of wrappers for C++ and Python.
A lesser-known but powerful application of parabolic equation methods is to the target scattering problem. In this paper, we use noncanonically shaped objects to establish the limits of applicability of the traditional approach, and introduce wide-angle and multiple-scattering approaches to allow accurate treatment of concave scatterers. The PE calculations are benchmarked against finite-element results, with good agreement obtained for convex scatterers in the traditional approach, and for concave scatterers with our modified approach. We demonstrate that the PE-based method is significantly more computationally efficient than the finite-element method at higher frequencies where objects are several or more wavelengths long.
We propose an effective approach to rapid estimation of the energy spectrum of quantum systems with the use of machine learning (ML) algorithm. In the ML approach (back propagation), the wavefunction data known from experiments is interpreted as the attributes class (input data), while the spectrum of quantum numbers establishes the label class (output data). To evaluate this approach, we employ two exactly solvable models with the random modulated wavefunction amplitude. The random factor allows modeling the incompleteness of information about the state of quantum system. The trial wave functions fed into the neural network, with the goal of making prediction about the spectrum of quantum numbers. We found that in such configuration, the training process occurs with rapid convergence if the number of analyzed quantum states is not too large. The two qubits entanglement is studied as well. The accuracy of the test prediction (after training) reached 98 percent. Considered ML approach opens up important perspectives to plane the quantum measurements and optimal monitoring of complex quantum objects.
The rectangular collocation approach makes it possible to solve the Schrödinger equation with basis functions that do not have amplitude in all regions in which wavefunctions have significant amplitude. Collocation points can be restricted to a small region of space. As no integrals are computed, there are no problems due to discontinuities in the potential, and there is no need to use integrable basis functions. In this paper, we show, for the Kohn-Sham equation, that machine learning can be used to drastically reduce the size of the collocation point set. This is demonstrated by solving the Kohn-Sham equations for CO and H2O. We solve the Kohn-Sham equation on a given effective potential which is a critical part of all DFT calculations, and monitor orbital energies and orbital shapes. We use a combination of Gaussian process regression and a genetic algorithm to reduce the collocation point set size by more than an order of magnitude (from about 51,000 points to 2,000 points) while retaining mHartree accuracy.
Here is presented an original program based on molecular Schrodinger equations. It is dedicated to target specific states of infrared vibrational spectrum in a very precise way with a minimal usage of memory. An eigensolver combined with a new probing technique accumulates information along the iterations so that desired eigenpairs rapidly tend towards the variational limit. Basis set is augmented from the maximal components of residual vectors that usually require the construction of a big matrix block that here is bypassed with a new factorisation of the Hamiltonian. The latest borrows the mathematical concept of duality and the second quantization formalism of quantum theory.
A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. The method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within $0.015~{E}_\text{h}$ of the exact basis-set-limit value with a statistical uncertainty of $10^{-5}~{E}_\text{h}$ without an importance sampling or Jastrow factor.