V. M. Bastidas, I. Omelchenko, A. Zakharova, E. Schöll, T. Brandes
备注：
This book chapter appear in Control of Self-Organizing Nonlinear Systems (conference proceedings, Springer-Verlag). This chapter is an extended version of our article arXiv:1505.02639 (2015)
Classical chimera states are paradigmatic examples of partial synchronization patterns emerging in nonlinear dynamics. These states are characterized by the spatial coexistence of two dramatically different dynamical behaviors, i.e., synchronized and desynchronized dynamics. Our aim in this contribution is to discuss signatures of chimera states in quantum mechanics.
We study a network with a ring topology consisting of N coupled quantum Van der Pol oscillators. We describe the emergence of chimera-like quantum correlations in the covariance matrix. Further, we establish the connection of chimera states to quantum information theory by describing the quantum mutual information for a bipartite state of the network.
Chimera states are complex spatiotemporal patterns in networks of identical oscillators, characterized by the coexistence of synchronized and desynchronized dynamics. Here we propose to extend the phenomenon of chimera states to the quantum regime, and uncover intriguing quantum signatures of these states. We calculate the quantum fluctuations about semiclassical trajectories and demonstrate that chimera states in the quantum regime can be characterized by bosonic squeezing, weighted quantum correlations, and measures of mutual information. Our findings reveal the relation of chimera states to…
We study a theoretical model of closed quasi-hermitian chain of spins which exhibits quantum analogues of chimera states, i.e. long life classical states for which a part of an oscillator chain presents an ordered dynamics whereas another part presents a disordered chaotic dynamics. For the quantum analogue, the chimera behavior deals with the entanglement between the spins of the chain. We discuss the entanglement properties, quantum chaos, quantum disorder and semi-classical similarity of our quantum chimera system. The quantum chimera concept is novel and induces new perspectives concerning…
Erratum to Phys. Rev. X 4, 021008 (2014): The critical exponent associated with the ferromagnetic susceptibility was computed incorrectly. Furthermore, Ising ferromagnets on the Chimera topology have the same universality class as two-dimensional Ising ferromagnets.
We analyze a continuous-time quantum walk on a chimera graph, which is a graph of choice for designing quantum annealers, and we discover beautiful quantum-walk features such as localization that starkly distinguishes classical from quantum behavior. Motivated by technological thrusts, we study continuous-time quantum walks on enhanced variants of the chimera graph and on a diminished chimera graph with a random removal of sites. We explain the quantum walk by constructing a generating set for a suitable subgroup of graph isomorphisms and corresponding symmetry operators that commute with the …
Recently, a programmable quantum annealing machine has been built that minimizes the cost function of hard optimization problems by adiabatically quenching quantum fluctuations. Tests performed by different research teams have shown that, indeed, the machine seems to exploit quantum effects. However experiments on a class of random-bond instances have not yet demonstrated an advantage over classical optimization algorithms on traditional computer hardware. Here we present evidence as to why this might be the case. These engineered quantum annealing machines effectively operate coupled to a dec…
We uncover a remarkable quantum scattering phenomenon in two-dimensional Dirac material systems where the manifestations of both classically integrable and chaotic dynamics emerge simultaneously and are electrically controllable. The distinct relativistic quantum fingerprints associated with different electron spin states are due to a physical mechanism analogous to chiroptical effect in the presence of degeneracy breaking. The phenomenon mimics a chimera state in classical complex dynamical systems but here in a relativistic quantum setting - henceforth the term "Dirac quantum chimera," assoc…
We estimate the critical thresholds of bond and site percolation on nonplanar, effectively two-dimensional graphs with chimera like topology. The building blocks of these graphs are complete and symmetric bipartite subgraphs of size $2n$, referred to as $K_{n,n}$ graphs. For the numerical simulations we use an efficient union-find based algorithm and employ a finite-size scaling analysis to obtain the critical properties for both bond and site percolation. We report the respective percolation thresholds for different sizes of the bipartite subgraph and verify that the associated universality c…
Symmetries are well known to have had a profound role in our understanding of nature and are a critical design concept for the realization of advanced technologies. In fact, many symmetry-broken states associated with different phases of matter appear in a variety of quantum technology applications. Such symmetries are normally broken in spatial dimension, however they can also be broken temporally leading to the concept of discrete time symmetries and their associated crystals. Discrete time crystals (DTCs) are a novel state of matter emerging in periodically-driven quantum systems. Typically…
In the paper, we investigated the influence of local and global interaction on the efficiency of continuous-time quantum spatial search. To do so, we analyzed numerically chimera graph, which is defined as 2D grid with each node replaced by complete bipartite graph. Our investigation provides a numerical evidence that with a large number of local interactions the quantum spatial search is optimal, contrary to the case with limited number of such interactions. The result suggests that relatively large number of local interactions with the marked vertex is necessary for optimal search, which in …
The Griffiths-McCoy singularity is a phenomenon characteristic of low-dimensional disordered quantum spin systems, in which the magnetic susceptibility shows singular behavior as a function of the external field even within the paramagnetic phase. We study whether this phenomenon is observed in the transverse-field Ising model with disordered ferromagnetic interactions on the quasi-two-dimensional diluted Chimera graph both by quantum Monte Carlo simulations and by extensive experiments on the D-Wave quantum annealer used as a quantum simulator. From quantum Monte Carlo simulations, evidence i…
The current generation of D-Wave quantum annealing processor is designed to minimize the energy of an Ising spin configuration whose pairwise interactions lie on the edges of a {\em Chimera} graph $\mathcal C_{M,N,L}$. In order to solve an Ising spin problem with arbitrary pairwise interaction structure, the corresponding graph must be minor-embedded into a Chimera graph. We define a combinatorial class of {\em native clique minors} in Chimera graphs with vertex images of uniform, near minimal size, and provide a polynomial-time algorithm that finds a maximum native clique minor in a given ind…
Chimera graphs define the topology of one of the first commercially available quantum computers. A variety of optimization problems have been mapped to this topology to evaluate the behavior of quantum enhanced optimization heuristics in relation to other optimizers, being able to efficiently solve problems classically to use them as benchmarks for quantum machines. In this paper we investigate for the first time the use of Evolutionary Algorithms (EAs) on Ising spin glass instances defined on the Chimera topology. Three genetic algorithms (GAs) and three estimation of distribution algorithms …
Performance of a Quantum Annealer for Ising Ground State Computations on Chimera Graphs
University of Cologne（科隆大学）；DLR German Aerospace Center（德国航空航天中心）；TU Dortmund University（多特蒙德工业大学）；Heidelberg University（海德堡大学）；University of Klagenfurt（克拉根福大学）；Istituto di Analisi dei Sistemi ed Informatica（系统与信息技术分析研究所）
Quantum annealing is a generic solver for optimization problems that uses fictitious quantum fluctuation. The most groundbreaking progress in the research field of quantum annealing is its hardware implementation, i.e., the so-called quantum annealer, using artificial spins. However, the connectivity between the artificial spins is sparse and limited on a special network known as the chimera graph. Several embedding techniques have been proposed, but the number of logical spins, which represents the optimization problems to be solved, is drastically reduced. In particular, an optimization prob…
The Sherrington-Kirkpatrick model with random $\pm1$ couplings is programmed on the D-Wave Two annealer featuring 509 qubits interacting on a Chimera-type graph. The performance of the optimizer compares and correlates to simulated annealing. When considering the effect of the static noise, which degrades the performance of the annealer, one can estimate an improvement on the comparative scaling of the two methods in favor of the D-Wave machine. The optimal choice of parameters of the embedding on the Chimera graph is shown to be associated to the emergence of the spin-glass critical temperatu…
NASA Ames Research Center（美国国家航空航天局艾姆斯研究中心）；Bay Area Environmental Research Institute（湾区环境研究所）
We present an algorithm for learning a latent variable generative model via generative adversarial learning where the canonical uniform noise input is replaced by samples from a graphical model. This graphical model is learned by a Boltzmann machine which learns low-dimensional feature representation of data extracted by the discriminator. A quantum annealer, the D-Wave 2000Q, is used to sample from this model. This algorithm joins a growing family of algorithms that use a quantum annealing subroutine in deep learning, and provides a framework to test the advantages of quantum-assisted learnin…
The article is addressing a possibility of implementation of spin network states on adiabatic quantum computer. The discussion is focused on application of currently available technologies and analyzes a concrete example of D-Wave machine. A class of simple spin network states which can be implemented on the Chimera graph architecture of the D-Wave quantum processor is introduced. However, extension beyond the currently available quantum processor topologies is required to simulate more sophisticated spin network states, which may inspire development of new generations of adiabatic quantum com…
Mappings of classical computation onto statistical mechanics models have led to remarkable successes in addressing some complex computational problems. However, such mappings display thermodynamic phase transitions that may prevent reaching solution even for easy problems known to be solvable in polynomial time. Here we map universal reversible classical computations onto a planar vertex model that exhibits no bulk classical thermodynamic phase transition, independent of the computational circuit. Within our approach the solution of the computation is encoded in the ground state of the vertex …
It is shown that the Bohmian mechanics and the Madelung quantum hydrodynamics are different theories and the latter is a better ontological interpretation of quantum mechanics. A new stochastic interpretation of quantum mechanics is proposed, which is the background of the Madelung quantum hydrodynamics. Its relation to the complex mechanics is also explored. A new complex hydrodynamics is proposed, which eliminates completely the Bohm quantum potential. It describes the quantum evolution of the probability density by a convective diffusion with imaginary transport coefficients.
Given are a first principles derivation and formulation of the probabilistic concepts that underly equilibrium quantum statistical mechanics. The transition to non-equilibrium probability is traversed briefly.
Inspired by the recent work of Carleo and Troyer[1], we apply machine learning methods to quantum mechanics in this article. The radial basis function network in a discrete basis is used as the variational wavefunction for the ground state of a quantum system. Variational Monte Carlo(VMC) calculations are carried out for some simple Hamiltonians. The results are in good agreements with theoretical values. The smallest eigenvalue of a Hermitian matrix can also be acquired using VMC calculations. Our results demonstrate that machine learning techniques are capable of solving quantum mechanical p…
Standard Quantum Mechanics (QM) predicts an anti-intuitive fenomenon here referred to as "quantum autoscattering", which is excluded by Bhomian Mechanics. The scheme of a gedanken experiment testing the QM prediction is briefly discussed.
In his recent paper arXiv:1312.7738, the author discussed $J$-Hermitian quantum mechanics and showed that $PT$-symmetric quantum mechanics is essentially $J$-Hermitian quantum mechanics. In this paper, the author discusses finite $J$-Hermitian quantum mechanics which is derived naturally from its continuum one and its relationship with finite $PT$-symmetric quantum mechanics.
In this paper we develop a discussion about PT Symmetric Quantum Mechanics, working with basics elements of this theory. In a simple case of two body system, we developed the Quantum Brachistochrone problem. Comparing the results obtained through the PT Symmetric Quantum Mechanics with that ones obtained using the standard formalism, we conclude that this new approach is not able to reveal any new effect.
This is an introductory chapter of the book in progress on quantum foundations and incompleteness of quantum mechanics. Quantum mechanics is represented as statistical mechanics of classical fields.
One of the postulates of quantum mechanics is that the Hamiltonian is Hermitian, as this guarantees that the eigenvalues are real. Recently there has been an interest in asking if $H^\dagger = H$ is a necessary condition, and has lead to the development of PT-symmetric quantum mechanics. This note shows that any finite physically acceptable non-Hermitian Hamiltonian is equivalent to doing ordinary quantum mechanics in a non-orthogonal basis. In particular, this means that there is no experimental distinction between PT-symmetric quantum mechanics and ordinary quantum mechanics for finite syste…
In spite of its popularity, it has not been possible to vindicate the conventional wisdom that classical mechanics is a limiting case of quantum mechanics. The purpose of the present paper is to offer an alternative point of view in which quantum mechanics emerges as a limiting case of classical mechanics in which the classical system is decoupled from its environment.
Quantum computation teaches us that quantum mechanics exhibits exponential complexity. We argue that the standard scientific paradigm of "predict and verify" cannot be applied to testing quantum mechanics in this limit of high complexity. We describe how QM can be tested in this regime by extending the usual scientific paradigm to include {\it interactive experiments}.
Bohm Mechanics and Nelson Stochastic Mechanics are confronted with Quantum Mechanics in presence of non-interacting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with Quantum Mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of non interacting systems at different times.
We discuss foundation of quantum mechanics (interpretations, superposition, principle of complementarity, locality, hidden variables) and quantum information theory.
The correspondence of a new form of quantum mechanics based on a quantum version of the action principle, which was proposed earlier [arXiv:0807.3508], with the ordinary quantum mechanics is established. New potentialities of the quantum action principle in the interpretation of quantum mechanics are considered.
It is shown that the apparent incompatibility of Bohmian mechanics with standard quantum mechanics, found by Akhavan and Golshani quant-ph/0305020, is an artefact of the fact that the initial wavefunction they use, being proportional to a $δ$-function, is not a regular wavefunction.
A motivation is given for expressing classical mechanics in terms of diagonal projection matrices and diagonal density matrices. Then quantum mechanics is seen to be a simple generalization in which one replaces the diagonal real matrices with suitable Hermitian matrices.
In this comment we answer to the recent critique of our article [arXiv:2208.11793] about Relational Quantum Mechanics (RQM) by Aurélien Drezet [arXiv:2209.01237]. Here we point out that our critical analysis of RQM was precisely based on the most recent formulation of RQM, and that the theses found in the critique are based on neither RQM assumptions nor on our arguments.
The author discusses a different kind of Hermitian quantum mechanics, called $J$-Hermitian quantum mechanics. He shows that $PT$-symmetric quantum mechanics is indeed $J$-Hermitian quantum mechanics, and that time evolution (in the Krein space of states) is unitary if and only if Hamiltonian is $J$-Hermitian (or equivalently $PT$-symmetric). An issue with unitarity comes up when time evolution is considered in the Hilbert space of states rather than in the Krein space of states. The author offers possible scenarios with this issue.
There is a certainty that the modern (Copenhagen's) interpretation of quantum mechanics is correct. However, the some physicist had the opinion that the modern quantum mechanics is a phenomenological theory. The suggested theory is the new quantum mechanics interpretation that is entirely according to the modern interpretation and gives a number of results, which naturally explain the postulates of the modern quantum mechanics.
Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket, and a quasidensity operator. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Classical mechanics can now be viewed as a deformation of quantum mechanics. The forms of semiquantum approximations to classical mechanics are indicated.
Classical statistical average values are generally generalized to average values of quantum mechanics, it is discovered that quantum mechanics is direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, the general…
It is shown that adopting the \emph{Quantum Field} ---extended entity in space-time build by dynamic appearance propagation and annihilation of virtual particles--- as the primary ontology the astonishing features of quantum mechanics can be rendered intuitive. This interpretation of quantum mechanics follows from the formalism of the most successful theory in physics: quantum field theory.
Supmech, an algebraic scheme of mechanics integrating noncommutative symplectic geometry and noncommutative probability, subsumes quantum and classical mechanics and permits consistent treatment of interaction of quantum and classical systems. Quantum measurements are treated in this framework; the von Neumann reduction rule (generally postulated) is derived and interpreted in physical terms.
It is argued on the basis of certain mathematical characteristics that classical mechanics is not constitutionally suited to accomodate consciousness, whereas quantum mechanics is. These mathematical characteristics pertain to the nature of the information represented in the state of the brain, and the way this information enters into the dynamics.
At the time of publication of H. Everett's Relative-State Formulation (1957) and DeWitt's Many-Worlds Interpretation (1970), quantum mechanics was available in a more modern and adequate version than the one used by these authors. We show that with the more modern quantum theory, quantum measurement could have been analyzed along more conventional lines in a one-world cosmology. Bell criticized the Everett-DeWitt theory quite sharply in 1987 but this seems not to have affected the acceptance of the old quantum mechanics as the framework for analysis of measurement.
Two questions are suggested as having priority when trying to bring together Quantum Mechanics and General Relativity. Both questions have a scope which goes well beyond Physics, and in particular Quantum Mechanics and General Relativity.
The concept of number is fundamental to the formulation of any physical theory. We give a heuristic motivation for the reformulation of Quantum Mechanics in terms of non-standard real numbers called Quantum Real Numbers. The standard axioms of quantum mechanics are re-interpreted. Our aim is to show that, when formulated in the language of quantum real numbers, the laws of quantum mechanics appear more natural, less counterintuitive than when they are presented in terms of standard numbers.
Many-Worlds quantum mechanics differs from standard quantum mechanics in that in Many-Worlds, the wave function is a relative density of universes in the multiverse amplitude rather than a probability amplitude. This means that in Many-Worlds, the Born frequencies are approached rather than given a priori. Thus in Many-Worlds the rate of approach to the final frequencies can be calculated and compared with observation. I use Many-Worlds to derive the rate of approach in the double slit experiment, and show that it agrees with observation. Standard quantum theory has never been used to derive a…
The quantum mechanical probability densities are compared with the probability densities treated by the theory of random variables. The relevance of their difference for the interpretation of quantum mechanics is commented.
In this paper we discuss and analyse the idea of trying to see (non-relativistic) quantum mechanics as a ``space-time statistical mechanics'', by using the classical statistical mechanical method on objective microscopic space-time configurations. It is argued that this could perhaps be accomplished by giving up the assumption that the objective ``state'' of a system is independent of a future measurement performed on the system. This idea is then applied in an example of quantum state estimation on a qubit system.
It is shown that probabilistic treatment of quantum mechanics can be coordinated with causality of all physical processes. The physical interpretation of quantum-mechanical phenomena such as process of measurement and collapse of quantum state is given.
The basics of the Wigner formulation of Quantum-Mechanics and few related interpretational issues are presented in a simple language. This formulation has extensive applications in Quantum Optics and in Mixed Quantum-Classical formulations.
