arXiv subjects
Publications and source records attributed to Jack Raymond.
This paper presents an overview of the topology of D-Wave's next-generation quantum processors. It provides examples of minor embeddings and discusses performance of embedding algorithms for the new topology compared to the existing Chimera topology. It also presents some initial performance results for simple, standard Ising model classes of problems.
We investigate the computational hardness of spin-glass instances on a square lattice, generated via a recently introduced tunable and scalable approach for planting solutions. The method relies on partitioning the problem graph into edge-disjoint subgraphs, and planting frustrated, elementary subproblems that share a common local ground state, which guarantees that the ground state of the entire problem is known a priori. Using population annealing Monte Carlo, we compare the typical hardness of problem classes over a large region of the multi-dimensional tuning parameter space. Our results show that the problems have a wide range of tunable hardness. Moreover, we observe multiple transitions in the hardness phase space, which we further corroborate using simulated annealing and simulated quantum annealing. By investigating thermodynamic properties of these planted systems, we demonstrate that the harder samples undergo magnetic ordering transitions which are also ultimately responsible for the observed hardness transitions on changing the sample composition.
The promise of quantum computing lies in harnessing programmable quantum devices for practical applications such as efficient simulation of quantum materials and condensed matter systems. One important task is the simulation of geometrically frustrated magnets in which topological phenomena can emerge from competition between quantum and thermal fluctuations. Here we report on experimental observations of relaxation in such simulations, measured on up to 1440 qubits with microsecond resolution. By initializing the system in a state with topological obstruction, we observe quantum annealing (QA) relaxation timescales in excess of one microsecond. Measurements indicate a dynamical advantage in the quantum simulation over the classical approach of path-integral Monte Carlo (PIMC) fixed-Hamiltonian relaxation with multiqubit cluster updates. The advantage increases with both system size and inverse temperature, exceeding a million-fold speedup over a CPU. This is an important piece of experimental evidence that in general, PIMC does not mimic QA dynamics for stoquastic Hamiltonians. The observed scaling advantage, for simulation of frustrated magnetism in quantum condensed matter, demonstrates that near-term quantum devices can be used to accelerate computational tasks of practical relevance.
The celebrated work of Berezinskii, Kosterlitz and Thouless in the 1970s revealed exotic phases of matter governed by topological properties of low-dimensional materials such as thin films of superfluids and superconductors. Key to this phenomenon is the appearance and interaction of vortices and antivortices in an angular degree of freedom---typified by the classical XY model---due to thermal fluctuations. In the 2D Ising model this angular degree of freedom is absent in the classical case, but with the addition of a transverse field it can emerge from the interplay between frustration and quantum fluctuations. Consequently a Kosterlitz-Thouless (KT) phase transition has been predicted in the quantum system by theory and simulation. Here we demonstrate a large-scale quantum simulation of this phenomenon in a network of 1,800 in situ programmable superconducting flux qubits arranged in a fully-frustrated square-octagonal lattice. Essential to the critical behavior, we observe the emergence of a complex order parameter with continuous rotational symmetry, and the onset of quasi-long-range order as the system approaches a critical temperature. We use a simple but previously undemonstrated approach to statistical estimation with an annealing-based quantum processor, performing Monte Carlo sampling in a chain of reverse quantum annealing protocols. Observations are consistent with classical simulations across a range of Hamiltonian parameters. We anticipate that our approach of using a quantum processor as a programmable magnetic lattice will find widespread use in the simulation and development of exotic materials.
Sampling from a Boltzmann distribution is NP-hard and so requires heuristic approaches. Quantum annealing is one promising candidate. The failure of annealing dynamics to equilibrate on practical time scales is a well understood limitation, but does not always prevent a heuristically useful distribution from being generated. In this paper we evaluate several methods for determining a useful operational temperature range for annealers. We show that, even where distributions deviate from the Boltzmann distribution due to ergodicity breaking, these estimates can be useful. We introduce the concepts of local and global temperatures that are captured by different estimation methods. We argue that for practical application it often makes sense to analyze annealers that are subject to post-processing in order to isolate the macroscopic distribution deviations that are a practical barrier to their application.
A recent Google study [Phys. Rev. X, 6:031015 (2016)] compared a D-Wave 2X quantum processing unit (QPU) to two classical Monte Carlo algorithms: simulated annealing (SA) and quantum Monte Carlo (QMC). The study showed the D-Wave 2X to be up to 100 million times faster than the classical algorithms. The Google inputs are designed to demonstrate the value of collective multiqubit tunneling, a resource available to D-Wave QPUs but not to simulated annealing. But the computational hardness in these inputs is highly localized in gadgets, with only a small amount of complexity coming from global interactions, meaning that the relevance to real-world problems is limited. In this study we provide a new synthetic problem class that addresses the limitations of the Google inputs while retaining their strengths. We use simple clusters instead of more complex gadgets and more emphasis is placed on creating computational hardness through frustrated global interactions like those seen in interesting real-world inputs. The logical problems used to generate these inputs can be solved in polynomial time [J. Phys. A, 15:10 (1982)]. However, for general heuristic algorithms that are unaware of the planted problem class, the frustration creates meaningful difficulty in a controlled environment ideal for study. We use these inputs to evaluate the new 2000-qubit D-Wave QPU. We include the HFS algorithm---the best performer in a broader analysis of Google inputs---and we include state-of-the-art GPU implementations of SA and QMC. The D-Wave QPU solidly outperforms the software solvers: when we consider pure annealing time (computation time), the D-Wave QPU reaches ground states up to 2600 times faster than the competition. In the task of zero-temperature Boltzmann sampling from challenging multimodal inputs, the D-Wave QPU holds a similar advantage as quantum sampling bias does not seem significant.
Inference methods are often formulated as variational approximations: these approximations allow easy evaluation of statistics by marginalization or linear response, but these estimates can be inconsistent. We show that by introducing constraints on covariance, one can ensure consistency of linear response with the variational parameters, and in so doing inference of marginal probability distributions is improved. For the Bethe approximation and its generalizations, improvements are achieved with simple choices of the constraints. The approximations are presented as variational frameworks; iterative procedures related to message passing are provided for finding the minima.
Variational inference is a powerful concept that underlies many iterative approximation algorithms; expectation propagation, mean-field methods and belief propagations were all central themes at the school that can be perceived from this unifying framework. The lectures of Manfred Opper introduce the archetypal example of Expectation Propagation, before establishing the connection with the other approximation methods. Corrections by expansion about the expectation propagation are then explained. Finally some advanced inference topics and applications are explored in the final sections.
Ground state entropy of the network source location problem is evaluated at both the replica symmetric level and one-step replica symmetry breaking level using the entropic cavity method. The regime that is a focus of this study, is closely related to the vertex cover problem with randomly quenched covered nodes. The resulting entropic message passing inspired decimation and reinforcement algorithms are used to identify the optimal location of sources in single instances of transportation networks. The conventional belief propagation without taking the entropic effect into account is also compared. We find that in the glassy phase the entropic message passing inspired decimation yields a lower ground state energy compared to the belief propagation without taking the entropic effect. Using the extremal optimization algorithm, we study the ground state energy and the fraction of frozen hubs, and extend the algorithm to collect statistics of the entropy. The theoretical results are compared with the extremal optimization results.
We introduce a new mean-field approximation based on the reconciliation of maximum entropy and linear response for correlations in the cluster variation method. Within a general formalism that includes previous mean-field methods, we derive formulas improving upon, e.g., the Bethe approximation and the Sessak-Monasson result at high temperature. Applying the method to direct and inverse Ising problems, we find improvements over standard implementations.
Approximating marginals of a graphical model is one of the fundamental problems in the theory of networks. In a recent paper a method was shown to construct a variational free energy such that the linear response estimates, and maximum entropy estimates (for beliefs) are in agreement, with implications for direct and inverse Ising problems[1]. In this paper we demonstrate an extension of that method, incorporating new information from the response matrix, and we recover the adaptive-TAP equations as the first order approximation[2]. The method is flexible with respect to applications of the cluster variational method, special cases of this method include Naive Mean Field (NMF) and Bethe. We demonstrate that the new framework improves estimation of marginals by orders of magnitude over standard implementations in the weak coupling limit. Beyond the weakly coupled regime we show there is an improvement in a model where the NMF and Bethe approximations are known to be poor for reasons of frustration and short loops.
This paper develops results for the next nearest neighbour Ising model on random graphs. Besides being an essential ingredient in classic models for frustrated systems, second neighbour interactions interactions arise naturally in several applications such as the colour diversity problem and graphical games. We demonstrate ensembles of random graphs, including regular connectivity graphs, that have a periodic variation of free energy, with either the ratio of nearest to next nearest couplings, or the mean number of nearest neighbours. When the coupling ratio is integer paramagnetic phases can be found at zero temperature. This is shown to be related to the locked or unlocked nature of the interactions. For anti-ferromagnetic couplings, spin glass phases are demonstrated at low temperature. The interaction structure is formulated as a factor graph, the solution on a tree is developed. The replica symmetric and energetic one-step replica symmetry breaking solution is developed using the cavity method. We calculate within these frameworks the phase diagram and demonstrate the existence of dynamical transitions at zero temperature for cases of anti-ferromagnetic coupling on regular and inhomogeneous random graphs.
Compressed sensing of sparse sources can be improved by incorporating prior knowledge of the source. In this paper we demonstrate a method for optimal selection of weights in weighted $L_1$ norm minimization for a noiseless reconstruction model, and show the improvements in compression that can be achieved.
Typical properties of computing circuits composed of noisy logical gates are studied using the statistical physics methodology. A growth model that gives rise to typical random Boolean functions is mapped onto a layered Ising spin system, which facilitates the study of their ability to represent arbitrary formulae with a given level of error, the tolerable level of gate-noise, and its dependence on the formulae depth and complexity, the gates used and properties of the function inputs. Bounds on their performance, derived in the information theory literature via specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. The framework is employed for deriving results on error-rates, function-depth and sensitivity, and their dependence on the gate-type and noise model used that are difficult to obtain via the traditional methods used in this field.
Computing circuits composed of noisy logical gates and their ability to represent arbitrary Boolean functions with a given level of error are investigated within a statistical mechanics setting. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. This framework paves the way for obtaining new results on error-rates, function-depth and sensitivity, and their dependence on the gate-type and noise model used.
Balancing efficiency of bandwidth use and complexity of detection involves choosing a suitable load for a multi-access channel. In the case of synchronous CDMA, with random codes, it is possible to demonstrate the existence of a threshold in the load beyond which there is an apparent jump in computational complexity. At small load unit clause propagation can determine a jointly optimal detection of sources on a noiseless channel, but fails at high load. Analysis provides insight into the difference between the standard dense random codes and sparse codes, and the limitations of optimal detection in the sparse case.
This thesis includes analysis of disordered spin ensembles corresponding to Exact Cover, a multi-access channel problem, and composite models combining sparse and dense interactions. The satisfiability problem in Exact Cover is addressed using a statistical analysis of a simple branch and bound algorithm. The algorithm can be formulated in the large system limit as a branching process, for which critical properties can be analysed. Far from the critical point a set of differential equations may be used to model the process, and these are solved by numerical integration and exact bounding methods. The multi-access channel problem is formulated as an equilibrium statistical physics problem for the case of bit transmission on a channel with power control and synchronisation. A sparse code division multiple access method is considered and the optimal detection properties are examined in typical case by use of the replica method, and compared to detection performance achieved by iterative decoding methods. These codes are found to have phenomena closely resembling the well-understood dense codes. The composite model is introduced as an abstraction of canonical sparse and dense disordered spin models. The model includes couplings due to both dense and sparse topologies simultaneously. Through an exact replica analysis at high temperature, and variational approaches at low temperature, several phenomena uncharacteristic of either sparse or dense models are demonstrated. An extension of the composite interaction structure to a code division multiple access method is presented. The new type of codes are shown to outperform sparse and dense codes in some regimes both in optimal performance, and in performance achieved by iterative detection methods in finite systems.
Code Division Multiple Access (CDMA) in which the spreading code assignment to users contains a random element has recently become a cornerstone of CDMA research. The random element in the construction is particular attractive as it provides robustness and flexibility in utilising multi-access channels, whilst not making significant sacrifices in terms of transmission power. Random codes are generated from some ensemble, here we consider the possibility of combining two standard paradigms, sparsely and densely spread codes, in a single composite code ensemble. The composite code analysis includes a replica symmetric calculation of performance in the large system limit, and investigation of finite systems through a composite belief propagation algorithm. A variety of codes are examined with a focus on the high multi-access interference regime. In both the large size limit and finite systems we demonstrate scenarios in which the composite code has typical performance exceeding sparse and dense codes at equivalent signal to noise ratio.