Chimera Graph Research Articles

Performance of quantum annealing inspired algorithms for combinatorial optimization problems

Two classes of quantum-annealing-inspired-algorithms (QAIA), namely different variants of simulated coherent Ising machine and simulated bifurcation, have been proposed for efficiently solving combinatorial optimization problems recently. In order to certify the superiority of these algorithms, standardized comparisons among them and against other physics-based algorithms are necessary. In this work, for Max-Cut problems up to 20,000 nodes, we benchmark QAIA against quantum annealing and other physics-based algorithms. We found that ballistic simulated bifurcation excelled for chimera and small-scale graphs, achieving nearly a 50-fold reduction in time-to-solution compared to quantum annealing. For large-scale graphs, discrete simulated bifurcation achieves the lowest time-to-target and outperforms D-Wave Advantage system when tasked with finding the maximum cut value in pegasus graphs. Our results suggest that QAIA represents a promising means for solving combinatorial optimization problems in practice, and can act as a natural baseline for competing quantum algorithms.

Minor embedding in broken chimera and derived graphs is NP-complete

The embedding is an essential step when calculating on the D-Wave machine. In this work, we show the hardness of the embedding problem for all types of currently existing hardware, represented by the Chimera and the derived Pegasus and Zephyr graphs, containing unavailable qubits. We construct certain broken Chimera graphs, where it is hard to find a Hamiltonian cycle. As the Hamiltonian cycle problem is a special case of the embedding problem, this proves the general complexity result for the Chimera graphs. By exploiting the subgraph relation between the Chimera and the derived graphs, the proof is then further extended to the Pegasus graphs and to the Zephyr graphs.

Localization effects in disordered quantum batteries.

We investigate the effect of localization on the local charging of quantum batteries (QBs) modeled by disordered spin systems. Two distinct schemes based on the transverse-field random Ising model are considered, with Ising couplings defined on a Chimera graph and on a linear chain with up to next-to-nearest-neighbor interactions. By adopting a low-energy demanding charging process driven by local fields only, we obtain that the maximum extractable energy by unitary processes (ergotropy) is highly enhanced in the ergodic phase in comparison with the many-body localization (MBL) scenario. As we turn off the next-to-nearest-neighbor interactions in the Ising chain, we have the onset of the Anderson localization phase. We then show that the Anderson phase exhibits a hybrid behavior, interpolating between large and small ergotropy as the disorder strength is increased. We also consider the splitting of total ergotropy into its coherent and incoherent contributions. This incoherent part implies in a residual ergotropy that is fully robust against dephasing, which is a typical process leading to the self-discharging of the battery in a real setup. Our results are experimentally feasible in scalable systems, such as in superconducting integrated circuits.

Comparison between a quantum annealer and a classical approximation algorithm for computing the ground state of an Ising spin glass.

Finding the ground state of an Ising spin glass on general graphs belongs to the class of NP-hard problems, widely believed to have no efficient polynomial-time algorithms to solve them. An approach developed in computer science for dealing with such problems is to devise approximation algorithms; these are algorithms, whose run time scales polynomially with the input size, that provide solutions with provable guarantees on their quality in terms of the optimal unknown solution. Recently, several algorithms for the Ising spin-glass problem on a bounded degree graph that provide different approximation guarantees were introduced. D-Wave, a Canadian-based company, has constructed a physical realization of a quantum annealer and has enabled researchers and practitioners to access it via their cloud service. D-Wave is particularly suited for computing an approximation for the ground state of an Ising spin glass on its Chimera and Pegasus graphs-both with a bounded degree. To assess the quality of D-Wave's solution, it is natural to compare it to classical approximation algorithms specifically designed to solve the same problem. In this work, we compare the performance of a recently developed approximation algorithm to solve the Ising spin-glass problem on graphs of bounded degree against the performance of the D-Wave computer. We also compared the performance of D-Wave's computer in the Chimera architecture against the performance of a heuristic tailored specifically to handle the Chimera graph. We found that the D-Wave computer was able to find better approximations for all the random instances of the problem we studied-Gaussian weights, uniform weights, and discrete binary weights. Furthermore, the convergence times of D-Wave's computer were also significantly better. These results indicate the merit of D-Wave's computer under certain specific instances. More broadly, our method is relevant to a wider class of performance comparison studies, and we suggest that it is important to compare the performance of quantum computers not only against exact classical algorithms with exponential run-time scaling, but also against approximation algorithms with polynomial run-time scaling and a provable guarantee of performance.

Embedding Overhead Scaling of Optimization Problems in Quantum Annealing

In order to treat all-to-all connected quadratic binary optimization problems (QUBO) with hardware quantum annealers, an embedding of the original problem is required due to the sparsity of the hardware's topology. Embedding fully-connected graphs - typically found in industrial applications - incurs a quadratic space overhead and thus a significant overhead in the time to solution. Here we investigate this embedding penalty of established planar embedding schemes such as minor embedding on a square lattice, minor embedding on a Chimera graph, and the Lechner-Hauke-Zoller scheme using simulated quantum annealing on classical hardware. Large-scale quantum Monte Carlo simulation suggest a polynomial time-to-solution overhead. Our results demonstrate that standard analog quantum annealing hardware is at a disadvantage in comparison to classical digital annealers, as well as gate-model quantum annealers and could also serve as benchmark for improvements of the standard quantum annealing protocol.

Template-Based Minor Embedding for Adiabatic Quantum Optimization

Quantum annealing (QA) can be used to quickly obtain near-optimal solutions for quadratic unconstrained binary optimization (QUBO) problems. In QA hardware, each decision variable of a QUBO should be mapped to one or more adjacent qubits in such a way that pairs of variables defining a quadratic term in the objective function are mapped to some pair of adjacent qubits. However, qubits have limited connectivity in existing QA hardware. This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms into the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems. In this paper, we use integer linear programming to search for an embedding of the problem graph into certain classes of minors of the Chimera graph, which we call template embeddings. One of these classes corresponds to complete bipartite graphs, for which we show the limitation of the existing approach based on minimum odd cycle transversals (OCTs). One of the formulations presented is exact and thus can be used to certify the absence of a minor embedding using that template. On an extensive test set consisting of random graphs from five different classes of varying size and sparsity, we can embed more graphs than a state-of-the-art OCT-based approach, our approach scales better with the hardware size, and the runtime is generally orders of magnitude smaller. Summary of Contribution: Our work combines classical and quantum computing for operations research by showing that integer linear programming can be successfully used as a preprocessing step for adiabatic quantum optimization. We use it to determine how a quadratic unconstrained binary optimization problem can be solved by a quantum annealer in which the qubits are coupled as in a Chimera graph, such as in the quantum annealers currently produced by D-Wave Systems. The paper also provides a timely introduction to adiabatic quantum computing and related work on minor embeddings.

Approximate optimization, sampling, and spin-glass droplet discovery with tensor networks.

We devise a deterministic algorithm to efficiently sample high-quality solutions of certain spin-glass systems that encode hard optimization problems. We employ tensor networks to represent the Gibbs distribution of all possible configurations. Using approximate tensor-network contractions, we are able to efficiently map the low-energy spectrum of some quasi-two-dimensional Hamiltonians. We exploit the local nature of the problems to compute spin-glass droplets geometries, which provides a new form of compression of the low-energy spectrum. It naturally extends to sampling, which otherwise, for exact contraction, is #P-complete. In particular, for one of the hardest known problem-classes devised on chimera graphs known as deceptive cluster loops and for up to 2048 spins, we find on the order of 10^{10} degenerate ground states in a single run of our algorithm, computing better solutions than have been reported on some hard instances. Our gradient-free approach could provide new insight into the structure of disordered spin-glass complexes, with ramifications both for machine learning and noisy intermediate-scale quantum devices.

Quantum Annealing versus Digital Computing

Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine, D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. In addition, we also evaluate the performance of a heuristic by Selby. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue. In particular, we examine how reliably the D-Wave computer can deliver true optimum solutions and present some surprising results.

Embedding of complete graphs in broken Chimera graphs

In order to solve real-world combinatorial optimization problems with a D-Wave quantum annealer, it is necessary to embed the problem at hand into the D-Wave hardware graph, namely Chimera or Pegasus. Most hard real-world problems exhibit a strong connectivity. For the worst-case scenario of a complete graph, there exists an efficient solution for the embedding into the ideal Chimera graph. However, since real machines almost always have broken qubits, it is necessary to find an embedding into the broken hardware graph. We present a new approach to the problem of embedding complete graphs into broken Chimera graphs. This problem can be formulated as an optimization problem, more precisely as a matching problem with additional linear constraints. Although being NP-hard in general, it is fixed-parameter tractable in the number of inaccessible vertices in the Chimera graph. We tested our exact approach on various instances of broken hardware graphs, both related to real hardware and randomly generated. For fixed runtime, we were able to embed larger complete graphs compared to previous, heuristic approaches. As an extension, we developed a fast heuristic algorithm which enables us to solve even larger instances. We compared the performance of our heuristic and exact approaches.

Prelude to Simulations of Loop Quantum Gravity on Adiabatic Quantum Computers

The article addresses the possibility of implementing spin network states, used in the loop quantum gravity approach to Planck scale physics on an adiabatic quantum computer. The discussion focuses on applying currently available technologies and analyzes a concrete example of a D-Wave machine. It is introduced a class of simple spin network states which can be implemented on the Chimera graph architecture of the D-Wave quantum processor. However, extension beyond the currently available quantum processor topologies is required to simulate more sophisticated spin network states. This may inspire new generations of adiabatic quantum computers. A possibility of simulating loop quantum gravity is discussed, and a method of solving a graph non-changing scalar (Hamiltonian) constraint with the use of adiabatic quantum computations is proposed. The presented results establish a basis for the future simulations of Planck scale physics, specifically quantum cosmological configurations, on quantum annealers.

Quantum-assisted associative adversarial network: applying quantum annealing in deep learning

Generative models have the capacity to model and generate new examples from a dataset and have an increasingly diverse set of applications driven by commercial and academic interest. In this work, 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 processor can be used to sample from the model to train the Boltzmann machine. This novel hybrid quantum-classical algorithm joins a growing family of algorithms that use a quantum processor sampling subroutine in deep learning, and provides a scalable framework to test the advantages of quantum-assisted learning. For the latent space model, fully connected, symmetric bipartite and Chimera graph topologies are compared on a reduced stochastically binarized MNIST dataset, for both classical and quantum sampling methods. The quantum-assisted associative adversarial network successfully learns a generative model of the MNIST dataset for all topologies. Evaluated using the Fréchet inception distance and inception score, the quantum and classical versions of the algorithm are found to have equivalent performance for learning an implicit generative model of the MNIST dataset. Classical sampling is used to demonstrate the algorithm on the LSUN bedrooms dataset, indicating scalability to larger and color datasets. Though the quantum processor used here is a quantum annealer, the algorithm is general enough such that any quantum processor, such as gate model quantum computers, may be substituted as a sampler.

Tropical Tensor Network for Ground States of Spin Glasses.

We present a unified exact tensor network approach to compute the ground state energy, identify the optimal configuration, and count the number of solutions for spin glasses. The method is based on tensor networks with the tropical algebra defined on the semiring of (R∪{-∞},⊕,⊙). Contracting the tropical tensor network gives the ground state energy; differentiating through the tensor network contraction gives the ground state configuration; mixing the tropical algebra and the ordinary algebra counts the ground state degeneracy. The approach brings together the concepts from graphical models, tensor networks, differentiable programming, and quantum circuit simulation, and easily utilizes the computational power of graphical processing units (GPUs). For applications, we compute the exact ground state energy of Ising spin glasses on square lattice up to 1024 spins, on cubic lattice up to 216 spins, and on three regular random graphs up to 220 spins, on a single GPU; we obtain exact ground state energy of ±J Ising spin glass on the chimera graph of D-Wave quantum annealer of 512 qubits in less than 100s and investigate the exact value of the residual entropy of ±J spin glasses on the chimera graph; finally, we investigate ground-state energy and entropy of three-state Potts glasses on square lattices up to size 18×18. Our approach provides baselines and benchmarks for exact algorithms for spin glasses and combinatorial optimization problems, and for evaluating heuristic algorithms and mean-field theories.

Investigating the Chinese postman problem on a quantum annealer

The recent availability of quantum annealers has fueled a new area of information technology where such devices are applied to address practically motivated and computationally difficult problems with hardware that exploits quantum mechanical phenomena. D-Wave annealers are promising platforms to solve these problems in the form of quadratic unconstrained binary optimization. Here we provide a formulation of the Chinese postman problem that can be used as a tool for probing the local connectivity of graphs and networks. We treat the problem classically with a tabu algorithm and simulated annealing, and using a D-Wave device. The efficiency of quantum annealing with respect to the simulated annealing has been demonstrated using the optimal time to solution metric. We systematically analyze computational parameters associated with the specific hardware. Our results clarify how the interplay between the embedding due to limited connectivity of the Chimera graph, the definition of logical qubits, and the role of spin-reversal controls the probability of reaching the expected solution.

Griffiths-McCoy singularity on the diluted Chimera graph: Monte Carlo simulations and experiments on quantum hardware

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 is found for the existence of the Griffiths-McCoy singularity in the paramagnetic phase. The experimental approach on the quantum hardware produces results that are less clear-cut due to the intrinsic noise and errors in the analog quantum device but can nonetheless be interpreted to be consistent with the existence of the Griffiths-McCoy singularity as in the Monte Carlo case. This is the first experimental approach based on an analog quantum simulator to study the subtle phenomenon of Griffiths-McCoy singularities in a disordered quantum spin system, through which we have clarified the capabilities and limitations of the D-Wave quantum annealer as a quantum simulator.

Quantum Advantage in Cryptography with a Low-Connectivity Quantum Annealer

The application in cryptography of quantum algorithms for prime factorization fostered the interest in quantum computing. However, quantum computers, and particularly quantum annealers, can also be helpful to construct secure cryptographic keys. Indeed, finding robust Boolean functions for cryptography is an important problem in sequence ciphers, block ciphers, and hash functions, among others. Due to the super-exponential size $\mathcal{O}(2^{2^n})$ of the associated space, finding $n$-variable Boolean functions with global cryptographic constraints is computationally hard. This problem has already been addressed employing generic low-connected incoherent D-Wave quantum annealers. However, the limited connectivity of the Chimera graph, together with the exponential growth in the complexity of the Boolean function design problem, limit the problem scalability. Here, we propose a special-purpose coherent quantum annealing architecture with three couplers per qubit, designed to optimally encode the bent function design problem. A coherent quantum annealer with this tree-type architecture has the potential to solve the $8$-variable bent function design problem, which is classically unsolved, with only $127$ physical qubits and $126$ couplers. This paves the way to reach useful quantum supremacy within the framework of quantum annealing for cryptographic purposes.

Breaking limitation of quantum annealer in solving optimization problems under constraints

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 problem including fully or even partly connected spins suffers from low embeddable size on the chimera graph. In the present study, we propose an alternative approach to solve a large-scale optimization problem on the chimera graph via a well-known method in statistical mechanics called the Hubbard-Stratonovich transformation or its variants. The proposed method can be used to deal with a fully connected Ising model without embedding on the chimera graph and leads to nontrivial results of the optimization problem. We tested the proposed method with a number of partition problems involving solving linear equations and the traffic flow optimization problem in Sendai and Kyoto cities in Japan.

Disorder-assisted graph coloring on quantum annealers

We are at the verge of a new era, which will be dominated by Noisy Intermediate-Scale Quantum Devices. Prototypical examples for these new technologies are present-day quantum annealers. In the present work, we investigate to what extent static disorder generated by an external source of noise does not have to be detrimental, but can actually assist quantum annealers in achieving better performance. In particular, we analyze the graph coloring problem that can be solved on a sparse topology (i.e. chimera graph) via suitable embedding. We show that specifically tailored disorder can enhance the fidelity of the annealing process and thus increase the overall performance of the annealer.

Domain wall encoding of discrete variables for quantum annealing and QAOA

In this paper I propose a new method of encoding discrete variables into Ising model qubits for quantum optimisation. The new method is based on the physics of domain walls in one-dimensional Ising spin chains. I find that these encodings and the encoding of arbitrary two variable interactions is possible with only two body Ising terms Following on from similar results for the ‘one hot’ method of encoding discrete variables (Hadfield et al 2019 Algorithms 12 34) I also demonstrate that it is possible to construct two body mixer terms which do not leave the logical subspace, an important consideration for optimising using the quantum alternating operator ansatz. I additionally discuss how, since the couplings in the domain wall encoding only need to be ferromagnetic and therefore could in principle be much stronger than anti-ferromagnetic couplers, application specific quantum annealers for discrete problems based on this construction may be beneficial. Finally, I compare embedding for synthetic scheduling and colouring problems with the domain wall and one hot encodings on two graphs which are relevant for quantum annealing, the chimera graph and the Pegasus graph. For every case I examine I find a similar or better performance from the domain wall encoding as compared to one hot, but this advantage is highly dependent on the structure of the problem. For encoding some problems, I find an advantage similar to the one found by embedding in a Pegasus graph compared to embedding in a chimera graph.

Impact of global and local interaction on quantum spatial search on chimera graph

In this 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 the 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 turn would imply that poorly connected vertices are hard to be found.

Efficiently embedding QUBO problems on adiabatic quantum computers

Adiabatic quantum computers like the D-Wave 2000Q can approximately solve the QUBO problem, which is an NP-hard problem, and have been shown to outperform classical computers on several instances. Solving the QUBO problem literally means solving virtually any NP-hard problem like the traveling salesman problem, airline scheduling problem, protein folding problem, genotype imputation problem, thereby enabling significant scientific progress, and potentially saving millions/billions of dollars in logistics, airlines, healthcare and many other industries. However, before QUBO problems are solved on quantum computers, they must be embedded (or compiled) onto the hardware of quantum computers, which in itself is a very hard problem. In this work, we propose an efficient embedding algorithm, that lets us embed QUBO problems fast, uses less qubits and gets the objective function value close to the global minimum value. We then compare the performance of our embedding algorithm to that of D-Wave’s embedding algorithm, which is the current state of the art, and show that our embedding algorithm convincingly outperforms D-Wave’s embedding algorithm. Our embedding approach works with perfect Chimera graphs, i.e., Chimera graphs with no missing qubits.