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IEEE International Conference on Quantum Computing and Engineering (QCE)

Neural Quantum Annealing for Real-World Quadratic Unconstrained Binary Optimization

Pietro Torta, Luca Leone, Rebecca Casati, Enrico Prati

Published Sep 15, 2024• 3 citations
Publisher

Abstract

We simulate Quantum Annealing on a variational manifold defined by a parametric family of wavefunctions represented by a Restricted Boltzmann Machine architecture. By iteratively lowering the transverse field and optimizing the neural network parameters, we prove the effectiveness of our methods for a challenging class of real-world Quadratic Unconstrained Binary Optimization (QUBO) instances, namely the Job Shop Scheduling Problem. Our methods outperform a straightforward optimization of the same problem without simulated quantum fluctuations. Despite the QUBO model being equivalent to a frustrated Ising spin glass, we obtain exact solutions up to 50-100 binary variables and empirically outperform results obtained with D-Wave quantum annealers.

DOI
10.1109/qce60285.2024.10382

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We simulate Quantum Annealing on a variational manifold defined by a parametric family of wavefunctions represented by a Restricted Boltzmann Machine architecture. By iteratively lowering the transverse field and optimizing the neural network parameters, we prove the effectiveness of our methods for a challenging class of real-world Quadratic Unconstrained Binary Optimization (QUBO) instances, namely the Job Shop Scheduling Problem. Our methods outperform a straightforward optimization of the same problem without simulated quantum fluctuations. Despite the QUBO model being equivalent to a frustrated Ising spin glass, we obtain exact solutions up to 50-100 binary variables and empirically outperform results obtained with D-Wave quantum annealers.
Quantum optimization which is the use of quantum computers and algorithms for solving complex optimization problems, is one of the topics in quantum computing with the highest potential. Two types of approaches are generally used to solve combinatorial optimization problems with quantum computers: exact methods such as the Grover’s search algorithm, and meta-heuristics such as the Quantum Annealing and the Quantum Approximate Optimization Algorithm (QAOA). Exact methods and variational methods like QAOA make use of universal gate-based quantum computers, such as the ones developed by the IBM company.Quantum annealing is designed for analogic quantum computers developed by D-Wave. Solving a combinatorial optimization problem with heuristics generally requires a transformation of the problem to a format suitable for the quantum computer. While some existing frameworks can cope with Constrained Quadratic Models (CQM), Quadratic Unconstrained Binary Optimization (QUBO) is the best option to map an optimization problem to a quantum computer or simulator. We consider in this paper the job shop scheduling problem.
Variational neural network models have achieved remarkable success in solving ground-state problems of quantum many-body systems. However, addressing classical and quantum spin glasses remains challenging, as disorder and energy frustration give rise to an exponentially large number of local energy minima separated by high-energy barriers, hindering the efficiency of conventional Metropolis-based Monte Carlo methods. To bridge this gap, we introduce Deep Boltzmann Quantum States, a class of neural quantum states inspired by deep Boltzmann machines that inherit efficient block Gibbs sampling. We also propose two key advances in the training algorithm. Firstly, we combine natural-gradient updates with state-of-the-art stochastic optimizers. Secondly, we gradually tune the hardness of the problem Hamiltonian by interpolating from an easy to a hard regime, without the need to closely approximate the instantaneous adiabatic state at intermediate times. We match the exact solution or the best available estimate for several instances of classical and quantum Ising spin-glass models with infinite-range interactions and hundreds of spins. We also solve instances of the NP-hard Job Shop Scheduling Problem exceeding the current limitations of quantum annealing hardware. To summarize, deep neural architectures with efficient global update rules and trained within an annealing-like scheme, provide a powerful framework for solving real-world hard combinatorial optimization and for investigating disordered quantum many-body systems.
A quantum annealing solver for the renowned job-shop scheduling problem (JSP) is presented in detail. After formulating the problem as a time-indexed quadratic unconstrained binary optimization problem, several pre-processing and graph embedding strategies are employed to compile optimally parametrized families of the JSP for scheduling instances of up to six jobs and six machines on the D-Wave Systems Vesuvius processor. Problem simplifications and partitioning algorithms, including variable pruning and running strategies that consider tailored binary searches, are discussed and the results from the processor are compared against state-of-the-art global-optimum solvers.
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Quantum annealing has the potential to outperform classical transistor-based computer technologies in tackling intricate combinatorial optimization problems. However, ongoing scientific debates cast doubts on whether quantum annealing devices (or quantum annealers) can genuinely provide better problem-solving capabilities than classical computers. The question of whether quantum annealing algorithms (QAAs) running on quantum annealers have computational advantages over classical algorithms (CAs) running on classical computers still remains unclear. This paper aims to clarify the question by classifying and benchmarking QAAs that utilize quadratic unconstrained binary optimization (QUBO) formulas to solve NP-hard problems. It proposes a four-class classification of QUBO formulas and exemplifies each class by QUBO formulas used by QAAs for solving specific NP-hard problems, such as the subset sum, maximum cut, vertex cover, 0/1 knapsack, graph coloring, Hamiltonian cycle, traveling salesperson, and job shop scheduling problems. The classification is based on the following two criteria: (i) Does the number of QUBO variables scale linearly with the problem input size? (ii) Does the QUBO formula have both the constraint term and the optimization term? QAAs are implemented and run on a D-Wave quantum annealer for benchmarking. They are benchmarked against related classical algorithms (CAs) running on classical computers in terms of the quality of the solution and the time to the solution. The benchmarking results reveal which classes of QUBO formulas are likely to provide advantages to QAAs over CAs. Furthermore, based on the benchmarking results, observations and suggestions are given for each class of QUBO formulas, facilitating the adoption of appropriate actions to improve the performance of QAAs.
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.