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arXiv:2111.05995 · Scientific Reports

Parallel quantum annealing

Elijah Pelofske, Georg Hahn, Hristo N. Djidjev

Published Mar 16, 2022• 55 citations
arXivHTMLPDF

Abstract

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 the Time-To-Solution (TTS) metric 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.

DOI
10.1038/s41598-022-08394-8

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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 the Time-To-Solution (TTS) metric 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.
Abstract Quantum annealing has the potential to find low energy solutions of NP-hard problems that can be expressed as quadratic unconstrained binary optimization problems. However, the hardware of the quantum annealer manufactured by D-Wave Systems, which we consider in this work, is sparsely connected and moderately sized (on the order of thousands of qubits), thus necessitating a minor-embedding of a logical problem onto the physical qubit hardware. The combination of relatively small hardware sizes and the necessity of a minor-embedding can mean that solving large optimization problems is not possible on current quantum annealers. In this research, we show that a hybrid approach combining parallel quantum annealing with graph decomposition allows one to solve larger optimization problem accurately. We apply the approach to the Maximum Clique problem on graphs with up to 120 nodes and 6395 edges.
This study introduces an innovative application of Parallel Quantum Annealing (PQA) to concurrently solve multiple NP-hard problems on a quantum annealer, leveraging unused qubits on D-Wave's Pegasus architecture to enhance efficiency and accuracy. While parallel execution of different problems on the same quantum annealing device has been previously explored, our application extends the current PQA method, which typically solves a single problem multiple times or different problems with minor embedded combinatorial optimization. This study primarily focuses on three NP-hard problems relevant to integrated circuit design: the Graph Coloring Problem (GCP), the Minimum Vertex Cover Problem (MVCP), and the Graph Partitioning Problem (GPP). After formulating these problems as Quadratic Unconstrained Binary Optimization (QUBO), they were embedded into the D-Wave quantum annealing machine, and the annealing process was run on the combined QUBO. The results were subsequently decoded into individual problems. Our results, quantified via the Time-To-Target (TTT) metric, showcased enhanced efficiency and accuracy in reaching target solutions, indicating a substantial improvement over traditional Quantum Annealing (QA) and Simulated Annealing (SA) methods. The research illustrates the potential of quantum computing for solving complex real-world problems more efficiently and emphasizes the importance of leveraging all available qubits in quantum annealers. The findings can significantly contribute to the development and optimization of quantum computing methods and have broad application implications, particularly in areas where NP-hard problems are prevalent.
Abstract Quantum annealing is a type of analog computation that aims to use quantum mechanical fluctuations in search of optimal solutions of QUBO (quadratic unconstrained binary optimization) or, equivalently, Ising problems. Since NP-hard problems can in general be mapped to Ising and QUBO formulations, the quantum annealing paradigm has the potential to help solve various NP-hard problems. Current quantum annealers, such as those manufactured by D-Wave Systems, Inc. have various practical limitations including the size (number of qubits) of the problem that can be solved, the qubit connectivity, and error due to the environment or system calibration, which can reduce the quality of the solutions. Typically, for an arbitrary problem instance, the corresponding QUBO (or Ising) structure will not natively embed onto the available qubit architecture on the quantum chip. Thus, in these cases, a minor embedding of the problem structure onto the device is necessary. However, minor embeddings on these devices do not always make use of the full sparse chip hardware graph, and a large portion of the available qubits stay unused during quantum annealing. In this work, we embed a disjoint random QUBO on the unused parts of the chip alongside the QUBO to be solved, which acts as an indicator of the solution quality of the device over time. Using experiments on three different D-Wave quantum annealers, we demonstrate that (i) long term trends in solution quality exist on the D-Wave device, and (ii) the unused qubits can be used to measure the current level of noise of the quantum system.
The recent availability of the first commercial quantum computers has provided a promising tool to tackle NP hard problems which can only be solved heuristically with present techniques. However, it is unclear if the current state of quantum computing already provides a quantum advantage over the current state of the art in classical computing. This article assesses the performance of the D-Wave 2X quantum annealer on two NP hard graph problems, in particular clique finding and graph partitioning. For this, we provide formulations as Qubo and Ising Hamiltonians suitable for the quantum annealer and compare a variety of quantum solvers (Sapi, QBSolv, QSage provided by D-Wave Sys, Inc.) to current classical algorithms (METIS, Simulated Annealing, third-party clique finding and graph splitting heuristics) on certain test sets of graphs. We demonstrate that for small graph instances, classical methods still outperform the quantum annealer in terms of computing time, even though the quality of the best solutions obtained is comparable. Nevertheless, due to the limited problem size which can be embedded on the D-Wave 2X chip, the aforementioned finding applies to most of problems of general nature solvable on the quantum annealer. For instances specifically designed to fit the D-Wave 2X architecture, we observe substantial speed-ups in computing time over classical approaches.
Quantum annealing approximately solves combinatorial optimization problems by leveraging the principles of adiabatic quantum systems. In this approach, the system's Hamiltonian evolves from an initial general state to a problem-specific state. This study introduces multi-tasking quantum annealing (MTQA), a method that enables the parallel processing of multiple optimization problems by embedding them into spatially distinct regions on quantum hardware. MTQA is evaluated using two NP-hard problems: the minimum vertex cover problem (MVCP) and the graph partitioning problem (GPP). This parallel approach optimizes quantum resource utilization by concurrently utilizing idle qubits. The findings demonstrate that MTQA achieves a solution quality comparable to single-problem quantum annealing and classical simulated annealing (SA), while notably reducing the time-to-solution (TTS) metrics. Eigenspectrum analysis further theoretically supports the hypothesis that parallel embedding preserves quantum coherence and does not increase computational complexity by efficiently utilizing available quantum hardware (e.g., qubits and couplers). MTQA enables efficient multitasking in quantum annealing, optimizing hardware utilization and improving throughput for concurrent tasks and demonstrating performance for problems up to 100 nodes in real-world applications.
Parallel Quantum Annealing is a technique to solve multiple optimization problems simultaneously. Parallel quantum annealing aims to optimize the utilization of available qubits on a quantum topology by addressing multiple independent problems in a single annealing cycle. This study provides insights into the potential and the limitations of this parallelization method. The experiments consisting of two different problems are integrated, and various problem dimensions are explored including normalization techniques using specific methods such as DWaveSampler with Default Embedding, DWaveSampler with Custom Embedding and LeapHybridSampler. This method minimizes idle qubits and holds promise for substantial speed-up, as indicated by the Time-to-Solution (TTS) metric, compared to traditional quantum annealing, which solves problems sequentially and may leave qubits unutilized.