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Expanding the scope of quantum annealing applicability

Hiroshi Isshiki, Koki Asari

Published Oct 1, 2023• 1 citation
Publisher

Abstract

With the emergence of D-wave’s quantum annealing machine, the power of quantum computing has become tangible, and the development of quantum computers is accelerating rapidly. Unfortunately, at present, the minimum value search that D-wave’s quantum annealing machine can handle is limited to quadratic functions of binary integer variables. This constraint is due to both hardware and software limitations. If this constraint was to be lifted, it would undoubtedly bring about a revolution in the world. Therefore, in this study, we obtained the basic mathematics of using quantum annealing, which enables hyper-parallel computing using qubits, to find the global minimum of any minimum value problem. Furthermore, we conducted simple simulations using classical computers to demonstrate the effectiveness of the ideas and algorithms. Furthermore, we conducted simple simulations using classical computers to demonstrate the effectiveness of the ideas and algorithms. We obtained also the mathematical method of using quantum annealing in learning neural networks and solving linear equations.

DOI
10.1063/5.0156410

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With the emergence of D-wave’s quantum annealing machine, the power of quantum computing has become tangible, and the development of quantum computers is accelerating rapidly. Unfortunately, at present, the minimum value search that D-wave’s quantum annealing machine can handle is limited to quadratic functions of binary integer variables. This constraint is due to both hardware and software limitations. If this constraint was to be lifted, it would undoubtedly bring about a revolution in the world. Therefore, in this study, we obtained the basic mathematics of using quantum annealing, which enables hyper-parallel computing using qubits, to find the global minimum of any minimum value problem. Furthermore, we conducted simple simulations using classical computers to demonstrate the effectiveness of the ideas and algorithms. Furthermore, we conducted simple simulations using classical computers to demonstrate the effectiveness of the ideas and algorithms. We obtained also the mathematical method of using quantum annealing in learning neural networks and solving linear equations.
Quantum annealing enables a global minimum search for quadratic functions in binary integer variables. It should be applied to more general problems. Therefore, in this research, we investigate the extension of the method to solve global minimum search by the quantum annealing method. There are three issues that need to be addressed: (1) conversion of decimal integer variables to binary integer variables; (2) binary discrete variable approximation of decimal continuous variables; (3) extension to general non-quadratic functions. This paper aims to develop the basic mathematics necessary for expanding the scope of application of the quantum annealing method, which was created as a solution to the minimum problem of quadratic functions of binary integer variables.
Quantum annealing is a method based on simulated annealing where temperature variations are replaced by quantum fluctuations that cause qubit state transitions. ISING problems and Quadratic Unconstrained Binary Optimization problems (QUBO) can be tackled by quantum annealing-based machines. We can link the interconnected qubits of the machine and the binaries of our model: qubits are binary variables and each pair of them linked by a coupler has a strong impact on the equality or the non-equality of the two binaries of the associated pair. Quantum annealing is based on the fact that any system tends to seek its minimum energy state. Starting from qubits in a state of superposition where all the solutions to the problem are fairly represented, the machine will apply a magnetic field by targeting the qubits and couplers in such a way as to make their value energetically favorable in the direction of optimization (minimization). For the coupled qubits, it is the quadratic products of the binary variables that are considered here and the physical system will make it energetically favorable for them to take (or not) the same values. To work on these machines based on quantum annealing in order to solve optimization problems with constraints, the latter must be relaxed while penalizing the Objective function. After presenting the case of the Max-Cut problem, we will look at such constrained problems.
To date, conventional computers have never been able to efficiently handle certain tasks, where the number of computation steps is likely to blow up as the problem size increases. As an emerging technology and new computing paradigm, quantum computing has a great potential to tackle those hard tasks efficiently. Among all the existing quantum computation models, quantum annealing has drawn significant attention in recent years due to the realization of the commercialized quantum annealer, sparking research interests in developing applications to solve problems that are intractable for classical computers.However, designing and implementing algorithms that manage to harness the enormous computation power from quantum annealer remains a challenging task. Generally, it requires mapping of the given optimization problem into quadratic unconstrained binary optimization(QUBO) problem and embedding the subsequent QUBO onto the physical architecture of quantum annealer. Additionally, practical quantum annealers are susceptible to errors leading to low probability of the correct solution.In this study, we focus on solving Boolean satisfiability (SAT) problem using quantum annealer while addressing practical limitations. We have proposed a mapping technique that maps SAT problem to QUBO, and we have further devised a tool flow that embeds the QUBO onto the architecture of a quantum annealing device. Additionally, We have optimized the proposed embedding flow to reduce run-time in addition to shortening the qubit chain length, leading to robust quantum annealing. To further improve the reliability of quantum annealing, we have also developed a post processing embedding technique that enlarges the energy gap between ground state and the first excited state. To demonstrate the effectiveness of proposed methods, we have conducted experiments on real quantum annealing devices manufactured by D-Wave Systems, showing compelling result of using quantum annealer to solve SAT problem.
The quantum annealing devices, which encode the solution to a computational problem in the ground state of a quantum Hamiltonian, are implemented in D-Wave systems with more than 2,000 qubits. However, quantum annealing can solve only a classical combinatorial optimization problem such as an Ising model, or equivalently, a quadratic unconstrained binary optimization (QUBO) problem. In this paper, we formulate the QUBO model to solve elliptic problems with Dirichlet and Neumann boundary conditions using the finite element method. In this formulation, we develop the objective function of quadratic binary variables represented by qubits and the system finds the binary string combination minimizing the objective function globally. Based on the QUBO formulation, we introduce an iterative algorithm to solve the elliptic problems. We discuss the validation of the modeling on the D-Wave quantum annealing system.
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.
Natural Language Processing and Semantic Web include several NP complete/hard problems that are intractable for classical computing machines. Even though distributed computing has provided remarkable advances (more precisely in dealing with big data), non-decomposable NP problems are still intractable in many real-world applications. And, from quantum computing perspective, solving complex problems with universal quantum gates requires developing of quantum algorithms. Considering commercializing quantum annealing machines by D-Wave, achieving global optimum for discrete optimization problems has been realized. In this study, a novel approach has been introduced to convert symbolic AI problems into quadratic unconstrained binary optimization (QUBO) form. More narrowly, this method represents classification of text documents (fragments) as optimizing a QUBO function. After embedding the train corpus into a QUBO function, D-Wave quantum annealer is used to classify new observations with finding the minimum energy level of the system.