This paper explores the application of Quadratic Unconstrained Binary Optimization (QUBO) models in solving the Travelling Salesman Problem (TSP) through Quantum Annealing algorithms and Graph Neural Networks. Quantum Annealing (QA), a quantum-inspired optimization method that exploits quantum tunneling to escape local minima, is used to solve QUBO formulations of TSP instances on Coherent Ising Machines (CIMs). The paper also presents a novel approach where QUBO is employed as a loss function within a GNN architecture tailored for solving TSP efficiently. By leveraging GNN's capability to learn graph representations, this method finds approximate solutions to TSP with improved computational time compared to traditional exact solvers. The paper details how to construct a QUBO model for TSP by encoding city visits into binary variables and formulating constraints that guarantee valid tours. It further discusses the implementation of QUBO-based Quantum Annealing algorithm for TSP (QQA-TSP) and its feasibility demonstration using quantum simulation platforms. In addition, it introduces a Graph Neural Network solution for TSP (QGNN-TSP), which learns the underlying structure of the problem and produces competitive solutions via gradient descent over a QUBO-based loss function. The experimental results compare the performance of QQA-TSP against state-of-the-art classical solvers such as dynamic programming, Concorde, and Gurobi, while also presenting empirical outcomes from training and evaluating QGNN-TSP on various TSP datasets. The study highlights the promise of combining deep learning techniques with quantum-inspired optimization methods for solving NP-hard problems like TSP, suggesting future directions for enhancing GNN architectures and applying QUBO frameworks to more complex combinatorial optimization tasks.
We propose a new Quadratic Unconstrained Binary Optimization (QUBO) formulation of the Travelling Salesman Problem (TSP), with which we overcame the best formulation of the Vehicle Routing Problem (VRP) in terms of the minimum number of necessary variables. After, we present a detailed study of the constraints subject to the new TSP model and benchmark it with MTZ and native formulations. Finally, we tested whether the correctness of the formulation by entering it into a QUBO problem solver. The solver chosen is a D-Wave\_2000Q6 quantum computer simulator due to the connection between Quantum …
Current hardware limitations restrict the potential when solving quadratic unconstrained binary optimization (QUBO) problems via the quantum approximate optimization algorithm (QAOA) or quantum annealing (QA). Thus, we consider training neural networks in this context. We first discuss QUBO problems that originate from translated instances of the traveling salesman problem (TSP): Analyzing this representation via autoencoders shows that there is way more information included than necessary to solve the original TSP. Then we show that neural networks can be used to solve TSP instances from both…
In this paper we present a novel strategy to solve optimization problems within a hybrid quantum-classical scheme based on quantum annealing, with a particular focus on QUBO problems. The proposed algorithm is based on an iterative structure where the representation of an objective function into the annealer architecture is learned and already visited solutions are penalized by a tabu-inspired search. The result is a heuristic search equipped with a learning mechanism to improve the encoding of the problem into the quantum architecture. We prove the convergence of the algorithm to a global opt…
Quantum annealing promises to be an effective heuristic for complex NP-hard problems. However, clear demonstrations of quantum advantage are wanting, primarily constrained by the difficulty of embedding the problem into the quantum hardware. Community detection methods such as the Girvin--Newman algorithm can provide a divide-and-conquer approach to large problems. Here, we propose a problem-focused division for embedding, deliberately worsening typical measures of embedding quality to improve the partial solutions we obtain. We apply this approach first to the highly irregular graph of an int…
We present PyQUBO, an open-source, Python library for constructing quadratic unconstrained binary optimizations (QUBOs) from the objective functions and the constraints of optimization problems. PyQUBO enables users to prepare QUBOs or Ising models for various combinatorial optimization problems with ease thanks to the abstraction of expressions and the extensibility of the program. QUBOs and Ising models formulated using PyQUBO are solvable by Ising machines, including quantum annealing machines. We introduce the features of PyQUBO with applications in the number partitioning problem, knapsac…
We propose a quadratic unconstrained binary optimization (QUBO) formulation of the l1-norm, which enables us to perform sparse estimation of Ising-type annealing methods such as quantum annealing. The QUBO formulation is derived using the Legendre transformation and the Wolfe theorem, which have recently been employed to derive the QUBO formulations of ReLU-type functions. It is shown that a simple application of the derivation method to the l1-norm case results in a redundant variable. Finally a simplified QUBO formulation is obtained by removing the redundant variable.
We propose a quadratic unconstrained binary optimization (QUBO) formulation of rectified linear unit (ReLU) type functions. Different from the q-loss function proposed by Denchev et al. (2012), a simple discussion based on the Legendre duality is not sufficient to obtain the QUBO formulation of the ReLU-type functions. In addition to the Legendre duality, we employ the Wolfe duality, and the QUBO formulation of the ReLU-type is derived. The QUBO formulation is available in Ising-type annealing methods, including quantum annealing machines.
The Quadratic Unconstrained Binary Optimization (QUBO) problems are NP hard; thus, so far, there are no algorithms to solve them efficiently. There are exact methods like the Branch-and-Bound algorithm for smaller problems, and for larger ones, many good approximations like stochastic simulated annealing for discrete variables or the mean field annealing for continuous variables. This paper will show that the statistical physics approach and the quantum mechanical approach in the mean field annealing give the same result. We examined the Ising problem, which is an alternative formulation of th…
Quadratic unconstrained binary optimization (QUBO) is the mathematical formalism for phrasing and solving a class of optimization problems that are combinatorial in nature. Due to their natural equivalence with the two dimensional Ising model for ferromagnetism in statistical mechanics, problems from the QUBO class can be solved on quantum annealing hardware. In this paper, we report a QUBO formatting of the problem of optimal control of time-dependent traffic signals on an artificial grid-structured road network so as to ease the flow of traffic, and the use of D-Wave Systems' quantum anneale…
We present a map from the travelling salesman problem (TSP), a prototypical NP-complete combinatorial optimisation task, to the ground state associated with a system of many-qudits. Conventionally, the TSP is cast into a quadratic unconstrained binary optimisation (QUBO) problem, that can be solved on an Ising machine. The size of the corresponding physical system's Hilbert space is $2^{N^2}$, where $N$ is the number of cities considered in the TSP. Our proposal provides a many-qudit system with a Hilbert space of dimension $2^{N\log_2N}$, which is considerably smaller than the dimension of th…
Comparing Quantum Annealing and Spiking Neuromorphic Computing for Sampling Binary Sparse Coding QUBO Problems
Los Alamos National Laboratory, CCS-3 Information Sciences（洛斯阿拉莫斯国家实验室，CCS-3信息科学部门）；Harvard University, T.H. Chan School of Public Health（哈佛大学，T.H. Chan公共卫生学院）
Quantum annealing technologies aim to solve computational optimization and sampling problems. QPU (Quantum Processing Unit) machines such as the D-Wave system use the QUBO (Quadratic Unconstrained Binary Optimization) formula to define model optimization problems for quantum annealing. This machine uses quantum effects to speed up computing time better than classical computers. We propose a vehicle routing problem that can be formulated in the QUBO model as a combinatorial problem, which gives the possible route solutions increases exponentially. The solution aims to optimize the vehicle's jou…
In this note, we describe an experiment on portfolio optimization using the Quadratic Unconstrained Binary Optimization (QUBO) formulation. The dataset we use is taken from a real-world problem for which a classical solution is currently deployed and used in production. In this work, carried out in a collaboration between the Raiffeisen Bank International (RBI) and Reply, we derive a QUBO formulation, which we solve using various methods: two D-Wave hybrid solvers, that combine the employment of a quantum annealer together with classical methods, and a purely classical algorithm. Particular fo…
Annealing machines specialized for combinatorial optimization problems have been developed, and some companies offer services to use those machines. Such specialized machines can only handle binary variables, and their input format is the quadratic unconstrained binary optimization (QUBO) formulation. Therefore, discretization is necessary to solve problems with continuous variables. However, there is a severe constraint on the number of binary variables with such machines. Although the simple binary expansion in the previous research requires many binary variables, we need to reduce the numbe…
Recent studies on quantum computing algorithms focus on excavating features of quantum computers which have potential for contributing to computational model enhancements. Among various approaches, quantum annealing methods effectively parallelize quadratic unconstrained binary optimization (QUBO) formulations of systems of linear equations. In this paper, we simplify these formulations by exploiting congruence of real symmetric matrices to diagonal matrices. We further exhibit computational merits of the proposed QUBO models, which can outperform classical algorithms such as QR and SVD decomp…
Quantum Image Denoising: A Framework via Boltzmann Machines, QUBO, and Quantum Annealing
Johns Hopkins University（约翰斯·霍普金斯大学）；NASA Ames Research Center（美国国家航空航天局艾姆斯研究中心）；Research Institute of Advanced Computer Science, USRA（大学空间研究协会高级计算机科学研究所）；NEC Corporation（日本电气股份有限公司）；National Institute of Advanced Industrial Science and Technology (AIST)（日本产业技术综合研究所）
We propose a new kernel that quantifies success for the task of computing a core-periphery partition for an undirected network. Finding the associated optimal partitioning may be expressed in the form of a quadratic unconstrained binary optimization (QUBO) problem, to which a state-of-the-art quantum annealer may be applied. We therefore make use of the new objective function to (a) judge the performance of a quantum annealer, and (b) compare this approach with existing heuristic core-periphery partitioning methods. The quantum annealing is performed on the commercially available D-Wave machin…
Universität Siegen（锡根大学）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；Aalto University（阿尔托大学）
Modern quantum annealers can find high-quality solutions to combinatorial optimisation objectives given as quadratic unconstrained binary optimisation (QUBO) problems. Unfortunately, obtaining suitable QUBO forms in computer vision remains challenging and currently requires problem-specific analytical derivations. Moreover, such explicit formulations impose tangible constraints on solution encodings. In stark contrast to prior work, this paper proposes to learn QUBO forms from data through gradient backpropagation instead of deriving them. As a result, the solution encodings can be chosen flex…
QUBO.jl: A Julia Ecosystem for Quadratic Unconstrained Binary Optimization
Federal University of Rio de Janeiro（里约热内卢联邦大学）；PSR；Davidson School of Chemical Engineering, Purdue University（普渡大学戴维森化学工程学院）；Pontifical Catholic University of Rio de Janeiro（里约热内卢天主教大学）；Research Institute of Advanced Computer Science, Universities Space Research Association（美国大学航天协会高级计算机科学研究所）；Quantum Artificial Intelligence Laboratory, NASA Ames Research Center（NASA艾姆斯研究中心量子人工智能实验室）
Previous research on quantum annealing methods focused on effectively modeling systems of linear equations by utilizing quadratic unconstrained binary optimization (QUBO) formulations. These studies take part in enhancing quantum computing algorithms, which extract properties of quantum computers suitable for improving classical computational models. In this paper, we further develop the QUBO formulations of systems of linear equations by applying Sylvester's law of inertia, which explores matrix congruence of any real symmetric matrix to a diagonal matrix. We expect that the proposed algorith…
In this paper, we propose a hybrid framework to solve large-scale permutation-based combinatorial problems effectively using a high-performance quadratic unconstrained binary optimization (QUBO) solver. To do so, transformations are required to change a constrained optimization model to an unconstrained model that involves parameter tuning. We propose techniques to overcome the challenges in using a QUBO solver that typically comes with limited numbers of bits. First, to smooth the energy landscape, we reduce the magnitudes of the input without compromising optimality. We propose a machine lea…
To run an algorithm on a quantum computer, one must choose an assignment from logical qubits in a circuit to physical qubits on quantum hardware. This task of initial qubit placement, or qubit allocation, is especially important on present-day quantum computers which have a limited number of qubits, connectivity constraints, and varying gate fidelities. In this work we formulate and implement the qubit placement problem as a quadratic, unconstrained binary optimization (QUBO) problem and solve it using simulated annealing to obtain a spectrum of initial placements. Compared to contemporary all…
QUBO-Based Optimization of Social Indicator Configurations for Working-Age Population Growth
Graduate School of Engineering, Tohoku University（东北大学研究生院工学研究科）；Graduate School of Information Sciences, Tohoku University（东北大学研究生院信息科学研究科）；Department of Physics, Institute of Science Tokyo（东京科学大学物理系）；Research and Education Institute for Semiconductors and Informatics, Kumamoto University（熊本大学半导体与信息学教育研究所）；Sigma-i Co., Ltd.（Sigma-i有限公司）
Quantum devices can be used to solve constrained combinatorial optimization (COPT) problems thanks to the use of penalization methods to embed the COPT problem's constraints in its objective to obtain a quadratic unconstrained binary optimization (QUBO) reformulation of the COPT. However, the particular way in which this penalization is carried out, affects the value of the penalty parameters, as well as the number of additional binary variables that are needed to obtain the desired QUBO reformulation. In turn, these factors substantially affect the ability of quantum computers to efficiently …
Diagnosing the minimal set of faults capable of explaining a set of given observations, e.g., from sensor readouts, is a hard combinatorial optimization problem usually tackled with artificial intelligence techniques. We present the mapping of this combinatorial problem to quadratic unconstrained binary optimization (QUBO), and the experimental results of instances embedded onto a quantum annealing device with 509 quantum bits. Besides being the first time a quantum approach has been proposed for problems in the advanced diagnostics community, to the best of our knowledge this work is also the…
Quadratic unconstrained binary optimization (QUBO) has become the standard format for optimization using quantum computers, i.e., for both the quantum approximate optimization algorithm (QAOA) and quantum annealing (QA). We present a toolkit of methods to transform almost arbitrary problems to QUBO by (i) approximating them as a polynomial and then (ii) translating any polynomial to QUBO. We showcase the usage of our approaches on two example problems (ratio cut and logistic regression).
Thermodynamic significance of QUBO encoding on quantum annealers
University of Massachusetts, Boston（马萨诸塞大学波士顿分校）；University of Maryland, Baltimore County（马里兰大学巴尔的摩县分校）；Quantum Science Institute, University of Maryland, Baltimore County（马里兰大学巴尔的摩县分校量子科学研究所）；Universität Tübingen（蒂宾根大学）；Forschungszentrum Jülich（于利希研究中心）；Polish Academy of Sciences（波兰科学院）
Renewable energy optimisation poses computationally-intensive challenges. Yet, often the continuous nature of the decision space precludes the use of many emerging, non-von-Neumann computing platforms such as quantum annealing, which are limited to discrete problems. We propose Quantum Annealing Continuous Optimisation (QuAnCO), a Trust Region (TR)-based algorithm, where the TR Newton sub-problem is transformed into Quadratic Unconstrained Binary Optimisation (QUBO), thereby allowing the use of Ising solvers such as D-Wave's quantum annealer. This transformation to QUBO is done by 1) using a h…
We propose a brand-new formulation of capacitated vehicle routing problem (CVRP) as quadratic unconstrained binary optimization (QUBO). The formulated CVRP is equipped with time-table which describes time-evolution of each vehicle. Therefore, various constraints associated with time are successfully realized. With a similar method, constraints of capacities are also introduced, where capacitated quantities are allowed to increase and decrease according to the cities which vehicles arrive. As a bonus of capacity-qubits, one also obtains a description of state, which allows us to set a variety o…
We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. The idea is tested by the two models, the transverse Ising model and the traveling salesman problem (TSP). Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations. The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method. TSP can be described by the Ising spin, so that we also apply the same technique to TSP. We directory solve the time-…
Classical and quantum annealing are computing paradigms that have been proposed to solve a wide range of optimization problems. In this paper, we aim to enhance the performance of annealing algorithms by introducing the technique of degeneracy engineering, through which the relative degeneracy of the ground state is increased by modifying a subset of terms in the objective Hamiltonian. We illustrate this novel approach by applying it to the example of $\ell_0$-norm regularization for sparse linear regression, which is in general an NP-hard optimization problem. Specifically, we show how to cas…
We present herein an introduction to implementing 2-color cellular automata on quantum annealing systems, such as the D-Wave quantum computer. We show that implementing nearest-neighbor cellular automata is possible. We present an implementation of Wolfram's cellular automata Rule 110, which has previously been shown to be a universal Turing machine, as a QUBO suitable for use on quantum annealing systems. We demonstrate back-propagation of cellular automata rule sets to determine initial cell states for a desired later system state. We show 2-D 2-color cellular automata, such as Conway's Game…
