Comparing Three Generations of D-Wave Quantum Annealers for Minor Embedded Combinatorial Optimization Problems

Quantum annealing is a novel type of analog computation that aims to use quantum mechanical fluctuations to search for optimal solutions of Ising problems. Quantum annealing in the Transverse Ising model, implemented on D-Wave QPUs, are available as cloud computing resources. In this article we report concise benchmarks across three generations of D-Wave quantum annealers, consisting of four different devices, for the NP-Hard combinatorial optimization problems unweighted maximum clique and unweighted maximum cut on random graphs. The Ising, or equivalently QUBO, formulation of these problems do not require auxiliary variables for order reduction, and their overall structure and weights are not highly complex, which makes these problems simple test cases to understand the sampling capability of current D-Wave quantum annealers. All-to-all minor embeddings of size $52$, with relatively uniform chain lengths, are used for a direct comparison across the Chimera, Pegasus, and Zephyr device topologies. A grid search over annealing times and the minor embedding chain strengths is performed in order to determine the level of reasonable performance for each device and problem type. Experiment metrics that are reported are approximation ratios for non-broken chain samples and chain break proportions. How fairly the quantum annealers sample optimal maximum cliques, for instances which contain multiple maximum cliques, is also quantified using entropy of the measured ground state distributions. The newest generation of quantum annealing hardware, which has a Zephyr hardware connectivity, performed the best overall with respect to approximation ratios and chain break frequencies

4-clique network minor embedding for quantum annealers

Quantum annealing is a proposed algorithm for computing solutions to combinatorial optimization problems. Current quantum annealing hardware is relatively sparse and therefore requires graph minor embedding in order to map an arbitrarily structured problem onto the sparse, and relatively small, quantum annealing processor. This paper proposes a new minor embedding method called 4-clique minor embedding. This is in contrast to the standard minor embedding technique of using a path of linearly connected qubits in order to represent a logical variable state. The 4-clique minor embedding is possible because of Pegasus graph connectivity, which is the native hardware graph for some of the current D-Wave quantum annealers. The Pegasus hardware graph has many 4-cliques, and it is possible to form a graph composed entirely of paths of connected 4-cliques, on which a problem can be minor embedded. The 4-clique chains come at the cost of additional qubit usage on the hardware graph, but they allow for stronger coupling within each chain thereby increasing chain integrity and reducing chain breaks. This 4-clique minor embedding technique is described in detail, and is compared against the standard linear path minor embedding with some experiments on two D-Wave quantum annealing processors with Pegasus hardware graphs. 4-clique minor embeddings can use weak chain strengths while successfully carrying out the computation of minimizing random all-to-all spin glass problem instances, in contrast to the linear path minor embeddings which have high chain break frequencies for weak chain strengths. This work shows that non standard minor embedding methods could be useful. For future quantum annealing architectures, distributing minor embeddings over more densely connected regions of hardware instead of linear paths may provide more robust computations for minor embedding problems

Inferring the Dynamics of the State Evolution During Quantum Annealing

To solve an optimization problem using a commercial quantum annealer, one has to represent the problem of interest as an Ising or a quadratic unconstrained binary optimization (QUBO) problem and submit its coefficients to the annealer, which then returns a user-specified number of low-energy solutions. It would be useful to know what happens in the quantum processor during the anneal process so that one could design better algorithms or suggest improvements to the hardware. However, existing quantum annealers are not able to directly extract such information from the processor. Hence, in this work we propose to use advanced features of D-Wave 2000Q to indirectly infer information about the dynamics of the state evolution during the anneal process. Specifically, D-Wave 2000Q allows the user to customize the anneal schedule, that is, the schedule with which the anneal fraction is changed from the start to the end of the anneal. Using this feature, we design a set of modified anneal schedules whose outputs can be used to generate information about the states of the system at user-defined time points during a standard anneal. With this process, called "slicing", we obtain approximate distributions of lowest-energy anneal solutions as the anneal time evolves. We use our technique to obtain a variety of insights into the annealer, such as the state evolution during annealing, when individual bits in an evolving solution flip during the anneal process and when they stabilize, and we introduce a technique to estimate the freeze-out point of both the system as well as of individual qubits

Advanced anneal paths for improved quantum annealing

Advances in quantum annealing technology make it possible to obtain high quality approximate solutions of important NP-hard problems. With the newer generations of the D-Wave annealer, more advanced features are available which allow the user to have greater control of the anneal process. In this contribution, we study how such features can help in improving the quality of the solutions returned by the annealer. Specifically, we focus on two of these features: reverse annealing and h-gain. Reverse annealing (RA) was designed to allow refining a known solution by backward annealing from a classical state representing the solution to a mid-anneal point where a transverse field is present, followed by an ordinary forward anneal, which is hoped to improve on the previous solution. The h-gain (HG) feature stands for time-dependent gain in Hamiltonian linear ($h$) biases and was originally developed to help study freezeout times and phase transitions in spin glasses. Here we apply HG to bias the quantum state in the beginning of the annealing process towards the known solution as in the RA case, but using a different apparatus. We also investigate a hybrid reverse annealing/h-gain schedule, which has a backward phase resembling an RA step and whose forward phase uses the HG idea. To optimize the parameters of the schedules, we employ a Bayesian optimization framework. We test all techniques on a variety of input problems including the weighted Maximum Cut problem and the weighted Maximum Clique problem. Our results show that each technique may dominate the others depending on the input instance, and that the HG technique is a viable alternative to RA for some problems

Peering into the Anneal Process of a Quantum Annealer

Commercial adiabatic quantum annealers have the potential to solve important NP-hard optimization problems efficiently. The newest generation of those machines additionally allows the user to customize the anneal schedule, that is, the schedule with which the anneal fraction is changed from the start to the end of the annealing. In this work we use the aforementioned feature of the D-Wave 2000Q to attempt to monitor how the anneal solution evolves during the anneal process. This process we call slicing: at each time slice during the anneal, we are able to obtain an approximate distribution of anneal solutions. We use our technique to obtain a variety of insights into the D-Wave 2000Q. For example, we observe when individual bits flip during the anneal process and when they stabilize, which allows us to determine the freeze-out point for each qubit individually. We highlight our results using both random QUBO (quadratic unconstrained binary optimization) instances and, for better visualization, instances which we specifically optimize (using our own genetic algorithm) to exhibit a pronounced evolution of its solution during the anneal

Advanced unembedding techniques for quantum annealers

The D-Wave quantum annealers make it possible to obtain high quality solutions of NP-hard problems by mapping a problem in a QUBO (quadratic unconstrained binary optimization) or Ising form to the physical qubit connectivity structure on the D-Wave chip. However, the latter is restricted in that only a fraction of all pairwise couplers between physical qubits exists. Modeling the connectivity structure of a given problem instance thus necessitates the computation of a minor embedding of the variables in the problem specification onto the logical qubits, which consist of several physical qubits "chained" together to act as a logical one. After annealing, it is however not guaranteed that all chained qubits get the same value (-1 or +1 for an Ising model, and 0 or 1 for a QUBO), and several approaches exist to assign a final value to each logical qubit (a process called "unembedding"). In this work, we present tailored unembedding techniques for four important NP-hard problems: the Maximum Clique, Maximum Cut, Minimum Vertex Cover, and Graph Partitioning problems. Our techniques are simple and yet make use of structural properties of the problem being solved. Using Erd\H{o}s-R\'enyi random graphs as inputs, we compare our unembedding techniques to three popular ones (majority vote, random weighting, and minimize energy). We demonstrate that our proposed algorithms outperform the currently available ones in that they yield solutions of better quality, while being computationally equally efficient

Initial state encoding via reverse quantum annealing and h-gain features

Quantum annealing is a specialized type of quantum computation that aims to use quantum fluctuations in order to obtain global minimum solutions of combinatorial optimization problems. D-Wave Systems, Inc., manufactures quantum annealers, which are available as cloud computing resources, and allow users to program the anneal schedules used in the annealing computation. In this paper, we are interested in improving the quality of the solutions returned by a quantum annealer by encoding an initial state. We explore two D-Wave features allowing one to encode such an initial state: the reverse annealing and the h-gain features. Reverse annealing (RA) aims to refine a known solution following an anneal path starting with a classical state representing a good solution, going backwards to a point where a transverse field is present, and then finishing the annealing process with a forward anneal. The h-gain (HG) feature allows one to put a time-dependent weighting scheme on linear ($h$) biases of the Hamiltonian, and we demonstrate that this feature likewise can be used to bias the annealing to start from an initial state. We also consider a hybrid method consisting of a backward phase resembling RA, and a forward phase using the HG initial state encoding. Importantly, we investigate the idea of iteratively applying RA and HG to a problem, with the goal of monotonically improving on an initial state that is not optimal. The HG encoding technique is evaluated on a variety of input problems including the weighted Maximum Cut problem and the weighted Maximum Clique problem, demonstrating that the HG technique is a viable alternative to RA for some problems. We also investigate how the iterative procedures perform for both RA and HG initial state encoding on random spin glasses with the native connectivity of the D-Wave Chimera and Pegasus chips.Comment: arXiv admin note: substantial text overlap with arXiv:2009.0500

Parallel Quantum Annealing

Quantum annealers of D-Wave Systems, Inc., offer an efficient way to compute high quality solutions of NP-hard problems. This is done by mapping a problem onto the physical qubits of the quantum chip, from which a solution is obtained after quantum annealing. However, since the connectivity of the physical qubits on the chip is limited, a minor embedding of the problem structure onto the chip is required. In this process, and especially for smaller problems, many qubits will stay unused. We propose a novel method, called parallel quantum annealing, to make better use of available qubits, wherein either the same or several independent problems are solved in the same annealing cycle of a quantum annealer, assuming enough physical qubits are available to embed more than one problem. Although the individual solution quality may be slightly decreased when solving several problems in parallel (as opposed to solving each problem separately), we demonstrate that our method may give dramatic speed-ups in terms of Time-to-Solution (TTS) for solving instances of the Maximum Clique problem when compared to solving each problem sequentially on the quantum annealer. Additionally, we show that solving a single Maximum Clique problem using parallel quantum annealing reduces the TTS significantly.Comment: 13 pages. v4: format improvement

A Robust Cybersecurity Topic Classification Tool

In this research, we use user defined labels from three internet text sources (Reddit, Stackexchange, Arxiv) to train 21 different machine learning models for the topic classification task of detecting cybersecurity discussions in natural text. We analyze the false positive and false negative rates of each of the 21 model's in a cross validation experiment. Then we present a Cybersecurity Topic Classification (CTC) tool, which takes the majority vote of the 21 trained machine learning models as the decision mechanism for detecting cybersecurity related text. We also show that the majority vote mechanism of the CTC tool provides lower false negative and false positive rates on average than any of the 21 individual models. We show that the CTC tool is scalable to the hundreds of thousands of documents with a wall clock time on the order of hours.Comment: Improved formattin