Fermions--elementary particles such as electrons--are classified as Dirac, Majorana or Weyl. Majorana and Weyl fermions had not been observed experimentally until the recent discovery of condensed matter systems such as topological superconductors and semimetals, in which they arise as low-energy excitations. Here we propose the existence of a previously overlooked type of Weyl fermion that emerges at the boundary between electron and hole pockets in a new phase of matter. This particle was missed by Weyl because it breaks the stringent Lorentz symmetry in high-energy physics. Lorentz invariance, however, is not present in condensed matter physics, and by generalizing the Dirac equation, we find the new type of Weyl fermion. In particular, whereas Weyl semimetals--materials hosting Weyl fermions--were previously thought to have standard Weyl points with a point-like Fermi surface (which we refer to as type-I), we discover a type-II Weyl point, which is still a protected crossing, but appears at the contact of electron and hole pockets in type-II Weyl semimetals. We predict that WTe2 is an example of a topological semimetal hosting the new particle as a low-energy excitation around such a type-II Weyl point. The existence of type-II Weyl points in WTe2 means that many of its physical properties are very different to those of standard Weyl semimetals with point-like Fermi surfaces.

The challenge posed by the many-body problem in quantum physics originates from the difficulty of describing the non-trivial correlations encoded in the exponential complexity of the many-body wave function. Here we demonstrate that systematic machine learning of the wave function can reduce this complexity to a tractable computational form, for some notable cases of physical interest. We introduce a variational representation of quantum states based on artificial neural networks with variable number of hidden neurons. A reinforcement-learning scheme is then demonstrated, capable of either finding the ground-state or describing the unitary time evolution of complex interacting quantum systems. We show that this approach achieves very high accuracy in the description of equilibrium and dynamical properties of prototypical interacting spins models in both one and two dimensions, thus offering a new powerful tool to solve the quantum many-body problem.The wave function Ψ is the fundamental object in quantum physics and possibly the hardest to grasp in a classical world. Ψ is a monolithic mathematical quantity that contains all the information on a quantum state, be it a single particle or a complex molecule. In principle, an exponential amount of information is needed to fully encode a generic many-body quantum state. However, Nature often proves herself benevolent, and a wave function representing a physical many-body system can be typically characterized by an amount of information much smaller than the maximum capacity of the corresponding Hilbert space. A limited amount of quantum entanglement, as well as the typicality of a small number of physical states, are then the blocks on which modern approaches build upon to solve the many-body Schrödinger's equation with a limited amount of classical resources.Numerical approaches directly relying on the wave function can either sample a finite number of physically relevant configurations or perform an efficient compression of the quantum state. Stochastic approaches, like quantum Monte Carlo (QMC) methods, belong to the first category and rely on probabilistic frameworks typically demanding a positive-semidefinite wave function. [1][2][3]. Compression approaches instead rely on efficient representations of the wave function, and most notably in terms of matrix product states (MPS) [4][5][6] or more general tensor networks [7,8]. Examples of systems where existing approaches fail are however numerous, mostly due to the sign problem in QMC [9], and to the inefficiency of current compression approaches in high-dimensional systems. As a result, despite the striking success of these methods, a large number of unexplored regimes exist, including many interesting open problems. These encompass fundamental questions ranging from the dynamical properties of high-dimensional systems [10,11] to the exact ground-state properties of strongly interacting fermions [12,13]. At the heart of this lack of understanding lyes the difficulty in finding a general strategy to reduce the exp...

Quantum impurity models describe an atom or molecule embedded in a host material with which it can exchange electrons. They are basic to nanoscience as representations of quantum dots and molecular conductors and play an increasingly important role in the theory of "correlated electron" materials as auxiliary problems whose solution gives the "dynamical mean field" approximation to the self energy and local correlation functions. These applications require a method of solution which provides access to both high and low energy scales and is effective for wide classes of physically realistic models. The continuous-time quantum Monte Carlo algorithms reviewed in this article meet this challenge. We present derivations and descriptions of the algorithms in enough detail to allow other workers to write their own implementations, discuss the strengths and weaknesses of the methods, summarize the problems to which the new methods have been successfully applied and outline prospects for future applications. 16 1. Measurement of the Green's function 16 2. Role of the parameter K -potential energy 16 V. Hybridization expansion solvers CT-HYB 16 A. The hybridization expansion representation 16 B. Density -density interactions 17 C. Formulation for general interactions

Quantum Monte Carlo simulations, while being efficient for bosons, suffer from the "negative sign problem" when applied to fermions -causing an exponential increase of the computing time with the number of particles. A polynomial time solution to the sign problem is highly desired since it would provide an unbiased and numerically exact method to simulate correlated quantum systems. Here we show, that such a solution is almost certainly unattainable by proving that the sign problem is NP-hard, implying that a generic solution of the sign problem would also solve all problems in the complexity class NP (nondeterministic polynomial) in polynomial time.Half a century after the seminal paper of Metropolis et al.[1] the Monte Carlo method has widely been established as one of the most important numerical methods and as a key to the simulation of many-body problems. Its main advantage is that it allows phase space integrals for many-particle problems, such as thermal averages, to be evaluated in a time that scales only polynomially with the particle number N although the configuration space grows exponentially with N . This enables the accurate simulation of large systems with millions of particles.Monte Carlo simulations of quantum systems, such as fermions, bosons, or quantum spins, can be performed after mapping the quantum system to an equivalent classical system. For fermionic or frustrated models this mapping may yield configurations with negative Boltzmann weights, resulting in an exponential growth of the statistical error and hence the simulation time with the number of particles, defeating the advantage of the Monte Carlo method. A polynomial time solution of this "sign problem" of negative weights would revolutionize electronic structure calculations by providing an unbiased and approximation-free method to study correlated fermionic systems. This would be of invaluable help, for example, in finding the mechanism for high-temperature superconductivity or in determining the properties of dense nuclear matter and quark matter.The difficulties in finding polynomial time solutions to the sign problem are reminiscent of the apparent impossibility to find polynomial time algorithms for nondeterministic polynomial (NP)-complete decision problems, which could be solved in polynomial time on a hypothetical non-deterministic machine, but for which no polynomial time algorithm is known for deterministic classical computers. A hypothetical non-deterministic machine can always follow both branches of an if-statement simultaneously, but can never merge the branches again. It can, equivalently, be viewed as having exponentially many processors, but without any communication between them. In addition, it must be possible to check a positive answer to a problem in NP on a classical computer in polynomial time.Many important computational problems in the complexity class NP, including the traveling salesman problem and the problem of finding ground states of spin glasses have the additional property of being NP-hard, forming...

We present a new continuous-time solver for quantum impurity models such as those relevant to dynamical mean field theory. It is based on a stochastic sampling of a perturbation expansion in the impurity-bath hybridization parameter. Comparisons with Monte Carlo and exact diagonalization calculations confirm the accuracy of the new approach, which allows very efficient simulations even at low temperatures and for strong interactions. As examples of the power of the method we present results for the temperature dependence of the kinetic energy and the free energy, enabling an accurate location of the temperature-driven metal-insulator transition.

We present an open-source software package WannierTools, a tool for investigation of novel topological materials. This code works in the tight-binding framework, which can be generated by another software package Wannier90 [1]. It can help to classify the topological phase of a given materials by calculating the Wilson loop, and can get the surface state spectrum which is detected by angle resolved photoemission (ARPES) and in scanning tunneling microscopy (STM) experiments . It also identifies positions of Weyl/Dirac points and nodal line structures, calculates the Berry phase around a closed momentum loop and Berry curvature in a part of the Brillouin zone(BZ).

Quantum technology is maturing to the point where quantum devices, such as quantum communication systems, quantum random number generators and quantum simulators, may be built with capabilities exceeding classical computers. A quantum annealer, in particular, solves hard optimisation problems by evolving a known initial configuration at non-zero temperature towards the ground state of a Hamiltonian encoding a given problem. Here, we present results from experiments on a 108 qubit D-Wave One device based on superconducting flux qubits. The strong correlations between the device and a simulated quantum annealer, in contrast with weak correlations between the device and classical annealing or classical spin dynamics, demonstrate that the device performs quantum annealing. We find additional evidence for quantum annealing in the form of small-gap avoided level crossings characterizing the hard problems. To assess the computational power of the device we compare it to optimised classical algorithms.Annealing a material by slow cooling is an ancient technique to improve the properties of glasses, metals and steel that has been used for more than seven millennia [1]. Mimicking this process in computer simulations is the idea behind simulated annealing as an optimisation method [2], which views the cost function of an optimisation problem as the energy of a physical system. Its configurations are sampled in a Monte Carlo simulation using the Metropolis algorithm [3], escaping from local minima by thermal fluctuations to find lower energy configurations. The goal is to find the global energy minimum (or at least a close approximation) by slowly lowering the temperature and thus obtain the solution to the optimisation problem.The phenomenon of quantum tunneling suggests that it can be more efficient to explore the state space quantum mechanically in a quantum annealer [4][5][6]. In simulated quantum annealing [7,8], one makes use of this effect by adding quantum fluctuations, which are slowly reduced while keeping the temperature constant and positive -ultimately ending up in a low energy configuration of the optimisation problem. Simulated quantum annealing, using a quantum Monte Carlo algorithm, has been observed to be more efficient than thermal annealing for certain spin glass models [8], although the opposite has been observed for k-satisfiability problems [9]. Further speedup may be expected in physical quantum annealing, either as an experimental technique for annealing a quantum spin glass [10], or -and this is what we will focus on here -as a computational technique in a programmable quantum device.In this work we report on computer simulations and experimental tests on a D-Wave One device [11] in order to address central open questions about quantum annealers: is the device actually a quantum annealer, i.e., do the quantum effects observed on 8 [11,12] and 16 qubits [13] persist when scaling problems up to more than 100 qubits, or do short coherence times turn the device into a classical, thermal annealer? Which ...

The preparation of quantum states using short quantum circuits is one of the most promising near-term applications of small quantum computers, especially if the circuit is short enough and the fidelity of gates high enough that it can be executed without quantum error correction. Such quantum state preparation can be used in variational approaches, optimizing parameters in the circuit to minimize the energy of the constructed quantum state for a given problem Hamiltonian. For this purpose we propose a simple-to-implement class of quantum states motivated by adiabatic state preparation. We test its accuracy and determine the required circuit depth for a Hubbard model on ladders with up to 12 sites (24 spin-orbitals), and for small molecules. We find that this ansatz converges faster than previously proposed schemes based on unitary coupled clusters. While the required number of measurements is astronomically large for quantum chemistry applications to molecules, applying the variational approach to the Hubbard model (and related models) is found to be far less demanding and potentially practical on small quantum computers. We also discuss another application of quantum state preparation using short quantum circuits, to prepare trial ground states of models faster than using adiabatic state preparation.