Quantum annealing is a promising paradigm for building practical quantum computers. Compared to other approaches, quantum annealing technology has been scaled up to a larger number of qubits. On the other hand, deep learning has been profoundly successful in pushing the boundaries of AI. It is thus natural to investigate potentially game changing technologies such as quantum annealers to augment the capabilities of deep learning. In this work, we explore binary neural networks, which are lightweight yet powerful models typically intended for resource constrained devices. Departing from current training regimes for binary networks that smooth/approximate the activation functions to make the network differentiable, we devise a quadratic unconstrained binary optimization formulation for the training problem. While the problem is intractable, i.e., the cost to estimate the binary weights scales exponentially with network size, we show how the problem can be optimized directly on a quantum annealer, thereby opening up to the potential gains of quantum computing. We experimentally validated our formulation via simulation and testing on an actual quantum annealer (D-Wave Advantage), the latter to the extent allowable by the capacity of current technology.
Quantum annealing is a promising paradigm for building practical quantum computers. Compared to other approaches, quantum annealing technology has been scaled up to a larger number of qubits. On the other hand, deep learning has been profoundly successful in pushing the boundaries of AI. It is thus natural to investigate potentially game changing technologies such as quantum annealers to augment the capabilities of deep learning. In this work, we explore binary neural networks, which are lightweight yet powerful models typically intended for resource constrained devices. Departing from current training regimes for binary networks that smooth/approximate the activation functions to make the network differentiable, we devise a quadratic unconstrained binary optimization formulation for the training problem. While the problem is intractable, i.e., the cost to estimate the binary weights scales exponentially with network size, we show how the problem can be optimized directly on a quantum annealer, thereby opening up to the potential gains of quantum computing. We experimentally validated our formulation via simulation and testing on an actual quantum annealer (D-Wave Advantage), the latter to the extent allowable by the capacity of current technology.
Artificial neural networks are at the heart of modern deep learning algorithms. We describe how to embed and train a general neural network in a quantum annealer without introducing any classical element in training. To implement the network on a state-of-the-art quantum annealer, we develop three crucial ingredients: binary encoding the free parameters of the network; polynomial approximation of the activation function; and reduction of binary higher-order polynomials into quadratic ones. Together, these ideas allow encoding the loss function as an Ising model Hamiltonian. The quantum annealer then trains the network by finding the ground state. We implement this for an elementary network and illustrate the advantages of quantum training: its consistency in finding the global minimum of the loss function and the fact that the network training converges in a single annealing step, which leads to short training times while maintaining a high classification performance. After training the network using a quantum annealer, one can then use the quantum network weights in a classical network algorithm of identical design for inference. Our approach opens an avenue for the quantum training of general machine learning models.
In recent years, the growing number of devices connected to the internet led network operators to continuously expand their own infrastructures. In order to simplify this scaling process, the research community is currently investigating the opportunity to move the complexity from a hardware to a software domain, through the introduction of a new paradigm, called Network Functions Virtualisation (NFV). It considers standard hardware platforms where many virtual instances are allocated to implement specific network services. However, despite the theoretical benefits, the mapping of the different virtual instances to the available physical resources represents a complex problem, difficult to be solved classically. The present work proposes a Quadratic Unconstrained Binary Optimisation (QUBO) formulation of this embedding process, exploring the implementation possibilities on D-Wave’s Quantum Annealers. Many test cases, with realistic constraints, have been considered to validate and characterise the potential of the model, and the promising results achieved are discussed throughout the document. The technical discussion is enriched with comparisons of the results obtained through heuristic algorithms, highlighting the strengths and the limitations in the resolution of the QUBO formulation proposed on current quantum machines.
We present, QP-SBGD, a novel layer-wise stochastic optimiser tailored towards training neural networks with binary weights, known as binary neural networks (BNNs), on quantum hardware. BNNs reduce the computational requirements and energy consumption of deep learning models with minimal loss in accuracy. However, training them in practice remains to be an open challenge. Most known BNN-optimisers either rely on projected updates or binarise weights post-training. Instead, QP-SBGD approximately maps the gradient onto binary variables, by solving a quadratic constrained binary optimisation. Under practically reasonable assumptions, we show that this update rule converges with a rate of O(1/T). Moreover, we show how the NP-hard projection can be effectively executed on an adiabatic quantum annealer, harnessing recent advancements in quantum computation. We also introduce a projected version of this update rule and prove that if a fixed point exists in the binary variable space, the modified updates will converge to it. Last but not least, our algorithm is implemented layer-wise, making it suitable to train larger networks on resource-limited quantum hardware. Through extensive evaluations, we show that QP-SBGD outperforms or is on par with competitive and well-established baselines such as BinaryConnect, signSGD and ProxQuant when optimising the Rosenbrock function, training BNNs as well as binary graph neural networks.
Abstract Quantum annealing aims at solving optimization problems of practical relevance using quantum-computing hardware. Problems of interest are typically formulated in terms of quadratic unconstrained binary optimization (QUBO) Hamiltonians. However, many optimization problems are much more naturally formulated in terms of polynomial unconstrained binary optimization (PUBO) functions of higher order. As we show with various problem examples, leveraging the PUBO formulation can bring considerable savings in terms of required number of qubits. Moreover, in numerical benchmarks for the paradigmatic 3-SAT problem, we find scenarios where the scaling of the minimum energy gap during the optimization sweep differs significantly, suggesting the possibility of an exponentially faster annealing time when using the PUBO as compared to the QUBO formulation. This advantage persists even when considering the overhead caused by the higher-order interactions necessary for PUBO cost Hamiltonians. As an interesting side effect, the analysis on minimum energy gaps of different 3-SAT instance generators reveals different degrees of hardness, which will be of interest also for classical benchmark calculations. Our findings show a promising path to improving the resource efficiency and sweeping speed of quantum annealing protocols on both analog and digital platforms, which are important prerequisites when aiming at solving larger optimization problems with relevance to industry.
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.
Binary neural networks (BNNs) are increasingly deployed in edge computing applications due to their low hardware complexity and high energy efficiency. However, verifying the robustness of BNNs against input perturbations, including adversarial attacks, remains computationally challenging because the underlying decision problem is inherently combinatorial. In this paper, we propose an Ising- and quantum-inspired framework for BNN robustness verification. We show that, for a broad class of BNN architectures, robustness verification can be formulated as a Quadratic Constrained Boolean Optimization (QCBO) problem and subsequently transformed into a Quadratic Unconstrained Boolean Optimization (QUBO) instance amenable to Ising and quantum-inspired solvers. We demonstrate the feasibility of this formulation on binarized MNIST by solving the resulting QUBOs with a free energy machine (FEM) solver and simulated annealing. We also show the deployment of this framework on quantum annealing and digital annealing platforms. Our results highlight the potential of quantum-inspired computing and Ising computing as a pathway toward trustworthy AI systems.