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One of the main bottlenecks in solving combinatorial optimization problems with quantum annealers is the qubit connectivity in the hardware. A possible solution for larger connectivty is minor embedding. This techniques makes the geometrical properties of the combinatorial optimization problem, encoded as a Hamiltonian, match the properties of the quantum annealing hardware. The embedding itself is a hard computational problem and therefore heuristic algorithms are required. In this work, we present fixed, modular and scalable embeddings that can be used to embed any combinatorial optimization problem described as an Ising Hamiltonian. These embeddings are the result of an extension of the well-known parity mapping, which has been used in the past to map higher-order Ising Hamiltonians to quadratic Hamiltonians, which are suitable for existing quantum hardware. We show how our new embeddings can be mapped to existing quantum annealers and that the embedded Hamiltonian physical properties match the original Hamiltonian properties.

Recent advances in quantum computers are demonstrating the ability to solve problems at a scale beyond brute force classical simulation. As such, a widespread interest in quantum algorithms has developed in many areas, with optimization being one of the most pronounced domains. Across computer science and physics, there are a number of different approaches for major classes of optimization problems, such as combinatorial optimization, convex optimization, non-convex optimization, and stochastic extensions. This work draws on multiple approaches to study quantum optimization. Provably exact versus heuristic settings are first explained using computational complexity theory - highlighting where quantum advantage is possible in each context. Then, the core building blocks for quantum optimization algorithms are outlined to subsequently define prominent problem classes and identify key open questions that, if answered, will advance the field. The effects of scaling relevant problems on noisy quantum devices are also outlined in detail, alongside meaningful benchmarking problems. We underscore the importance of benchmarking by proposing clear metrics to conduct appropriate comparisons with classical optimization techniques. Lastly, we highlight two domains - finance and sustainability - as rich sources of optimization problems that could be used to benchmark, and eventually validate, the potential real-world impact of quantum optimization., Comment: 72 pages, 9 Figures, 4 Tables. Fixed typos and added references

We present a method to split quantum circuits of variational quantum algorithms (VQAs) to allow for parallel training and execution, that maximally exploits the limited number of qubits in hardware to solve large problem instances. We apply this specifically to combinatorial optimization problems, where inherent structures from the problem can be identified, thus directly informing how to create these parallelized quantum circuits, which we call slices. We test our method by creating a parallelized version of the Quantum Approximate Optimization Algorithm, which we call pQAOA, and explain how our methods apply to other quantum algorithms like the Variational Quantum Eigensolver and quantum annealing. We show that not only can our method address larger problems, but that it is also possible to run full VQA models while training parameters using only one slice. These results show that the loss of information induced by splitting does not necessarily affect the training of parameters in quantum circuits for optimization. This implies that combinatorial optimization problems are encoded with redundant information in quantum circuits of current VQAs. Therefore, to attain quantum advantage for combinatorial optimization, future quantum algorithms should be designed to incorporate information that is free of such redundancies.

Optimization problems are ubiquitous in various industrial settings, and multi-knapsack optimization is one recurrent task faced daily by several industries. The advent of quantum computing has opened a new paradigm for computationally intensive tasks, with promises of delivering better and faster solutions for specific classes of problems. This work presents a comprehensive study of quantum computing approaches for multi-knapsack problems, by investigating some of the most prominent and state-of-the-art quantum algorithms using different quantum software and hardware tools. The performance of the quantum approaches is compared for varying hyperparameters. We consider several gate-based quantum algorithms, such as QAOA and VQE, as well as quantum annealing, and present an exhaustive study of the solutions and the estimation of runtimes. Additionally, we analyze the impact of warm-starting QAOA to understand the reasons for the better performance of this approach. We discuss the implications of our results in view of utilizing quantum optimization for industrial applications in the future. In addition to the high demand for better quantum hardware, our results also emphasize the necessity of more and better quantum optimization algorithms, especially for multi-knapsack problems., Comment: 20 pages

Many emerging commercial services are based on the sharing or pooling of resources for common use with the aim of reducing costs. Businesses such as delivery-, mobility-, or transport-as-a-service have become standard in many parts of the world, fulfilling on-demand requests for customers in live settings. However, it is known that many of these problems are NP-hard, and therefore both modeling and solving them accurately is a challenge. Here we focus on one such routing problem, the Ride Pooling Problem (RPP), where multiple customers can request on-demand pickups and drop-offs from shared vehicles within a fleet. The combinatorial optimization task is to optimally pool customer requests using the limited set of vehicles, akin to a small-scale flexible bus route. In this work, we propose a quadratic unconstrained binary optimization (QUBO) program and introduce efficient formulation methods for the RPP to be solved using metaheuristics, and specifically emerging quantum optimization algorithms.

Genetic algorithms have unique properties which are useful when applied to black box optimization. Using selection, crossover, and mutation operators, candidate solutions may be obtained without the need to calculate a gradient. In this work, we study results obtained from using quantum-enhanced operators within the selection mechanism of a genetic algorithm. Our approach frames the selection process as a minimization of a binary quadratic model with which we encode fitness and distance between members of a population, and we leverage a quantum annealing system to sample low energy solutions for the selection mechanism. We benchmark these quantum-enhanced algorithms against classical algorithms over various black-box objective functions, including the OneMax function, and functions from the IOHProfiler library for black-box optimization. We observe a performance gain in average number of generations to convergence for the quantum-enhanced elitist selection operator in comparison to classical on the OneMax function. We also find that the quantum-enhanced selection operator with non-elitist selection outperform benchmarks on functions with fitness perturbation from the IOHProfiler library. Additionally, we find that in the case of elitist selection, the quantum-enhanced operators outperform classical benchmarks on functions with varying degrees of dummy variables and neutrality.

Variational quantum algorithms are the leading candidate for advantage on near-term quantum hardware. When training a parametrized quantum circuit in this setting to solve a specific problem, the choice of ansatz is one of the most important factors that determines the trainability and performance of the algorithm. In quantum machine learning (QML), however, the literature on ansatzes that are motivated by the training data structure is scarce. In this work, we introduce an ansatz for learning tasks on weighted graphs that respects an important graph symmetry, namely equivariance under node permutations. We evaluate the performance of this ansatz on a complex learning task, namely neural combinatorial optimization, where a machine learning model is used to learn a heuristic for a combinatorial optimization problem. We analytically and numerically study the performance of our model, and our results strengthen the notion that symmetry-preserving ansatzes are a key to success in QML., Comment: 17+3 pages, 10 figures, version accepted at journal

Quantum annealing is a heuristic quantum optimization algorithm that can be used to solve combinatorial optimization problems. In recent years, advances in quantum technologies have enabled the development of small- and intermediate-scale quantum processors that implement the quantum annealing algorithm for programmable use. Specifically, quantum annealing processors produced by D-Wave Systems have been studied and tested extensively in both research and industrial settings across different disciplines. In this paper we provide a literature review of the theoretical motivations for quantum annealing as a heuristic quantum optimization algorithm, the software and hardware that is required to use such quantum processors, and the state-of-the-art applications and proofs-of-concepts that have been demonstrated using them. The goal of our review is to provide a centralized and condensed source regarding applications of quantum annealing technology. We identify the advantages, limitations, and potential of quantum annealing for both researchers and practitioners from various fields., Comment: major revision with extended noise section and discussion of alternative platforms

We present a generalization of the binary paint shop problem (BPSP) to tackle an automotive industry application, the multi-car paint shop (MCPS) problem. The objective of the optimization is to minimize the number of color switches between cars in a paint shop queue during manufacturing, a known NP-hard problem. We distinguish between different sub-classes of paint shop problems, and show how to formulate the basic MCPS problem as an Ising model. The problem instances used in this study are generated using real-world data from a factory in Wolfsburg, Germany. We compare the performance of the D-Wave 2000Q and Advantage quantum processors to other classical solvers and a hybrid quantum-classical algorithm offered by D-Wave Systems. We observe that the quantum processors are well-suited for smaller problems, and the hybrid algorithm for intermediate sizes. However, we find that the performance of these algorithms quickly approaches that of a simple greedy algorithm in the large size limit., Comment: Accepted to IEEE International Conference on Quantum Computing and Engineering (QCE21)

The binary paint shop problem (BPSP) is an APX-hard optimization problem of the automotive industry. In this work, we show how to use the Quantum Approximate Optimization Algorithm (QAOA) to find solutions of the BPSP and demonstrate that QAOA with constant depth is able to beat classical heuristics on average in the infinite size limit $n\rightarrow\infty$. For the BPSP, it is known that no classical algorithm can exist which approximates the problem in polynomial runtime. We introduce a BPSP instance which is hard to solve with QAOA, and numerically investigate its performance and discuss QAOA's ability to generate approximate solutions. We complete our studies by running first experiments of small-sized instances on a trapped-ion quantum computer through AWS Braket.

The Web Summit conference in Lisbon, Portugal, is one of the biggest technology conferences in Europe, attended by tens of thousands of people every year. The high influx of people into Lisbon causes significant stress on the city's transit services for the duration of the conference. For the Web Summit 2019, Volkswagen AG partnered with the city of Lisbon for a pilot project to provide quantum computing-based traffic optimization. A two-phase solution was implemented: the first phase used data science techniques to analyze the movement of people from previous conferences to build temporary new bus routes throughout the city. The second phase used a custom Android navigation app installed in the buses operated by Carris, powered by a quantum optimization service provided by Volkswagen that connected to live traffic data and a D-Wave quantum processing unit to optimize the buses' routes in real-time. To our knowledge, this is the first commercial application that depends on a quantum processor to perform a critical live task.

In this paper we develop methods to solve two problems related to time series (TS) analysis using quantum computing: reconstruction and classification. We formulate the task of reconstructing a given TS from a training set of data as an unconstrained binary optimization (QUBO) problem, which can be solved by both quantum annealers and gate-model quantum processors. We accomplish this by discretizing the TS and converting the reconstruction to a set cover problem, allowing us to perform a one-versus-all method of reconstruction. Using the solution to the reconstruction problem, we show how to extend this method to perform semi-supervised classification of TS data. We present results indicating our method is competitive with current semi- and unsupervised classification techniques, but using less data than classical techniques.

Quantum annealing algorithms belong to the class of meta-heuristic tools, applicable for solving binary optimization problems. Hardware implementations of quantum annealing, such as the quantum processing units (QPUs) produced by D-Wave Systems, have been subject to multiple analyses in research, with the aim of characterizing the technology's usefulness for optimization and sampling tasks. In this paper, we present a real-world application that uses quantum technologies. Specifically, we show how to map certain parts of the real-world traffic flow optimization problem to be suitable for quantum annealing. We show that time-critical optimization tasks, such as continuous redistribution of position data for cars in dense road networks, are suitable candidates for quantum applications. Due to the limited size and connectivity of current-generation D-Wave QPUs, we use a hybrid quantum and classical approach to solve the traffic flow problem., Comment: 17 pages, 6 figures

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.

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., Comment: 28 pages, 14 figures ; supplementary materials 18 pages, 5 figures [v4: updated supplementary material file correcting broken references]

In the evaluation of quantum annealers, metrics based on ground state success rates have two major drawbacks. First, evaluation requires computation time for both quantum and classical processors that grows exponentially with problem size. This makes evaluation itself computationally prohibitive. Second, results are heavily dependent on the effects of analog noise on the quantum processors, which is an engineering issue that complicates the study of the underlying quantum annealing algorithm. We introduce a novel "time-to-target" metric which avoids these two issues by challenging software solvers to match the results obtained by a quantum annealer in a short amount of time. We evaluate D-Wave's latest quantum annealer, the D-Wave 2X system, on an array of problem classes and find that it performs well on several input classes relative to state of the art software solvers running single-threaded on a CPU., Comment: 29 pages