We propose a non-convex training objective for robust binary classification of data sets in which label noise is present. The design is guided by the intention of solving the resulting problem by adiabatic quantum optimization. Two requirements are imposed by the engineering constraints of existing quantum hardware: training problems are formulated as quadratic unconstrained binary optimization; and model parameters are represented as binary expansions of low bit-depth. In the present work we validate this approach by using a heuristic classical solver as a stand-in for quantum hardware. Testing on several popular data sets and comparing with a number of existing losses we find substantial advantages in robustness as measured by test error under increasing label noise. Robustness is enabled by the non-convexity of our hardware-compatible loss function, which we name q-loss.
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.
We study the problem of supervised binary classification from the perspective of deploying adiabatic quantum optimization in training. A vast body of prior academic work consisting of both theoretical and numerical studies has indicated that quantum technology promises to provide computational power that may be fundamentally superior to any classical computing methods. Given the abundance of NP-hard optimization problems that naturally arise in learning, it is clear that machine learning can immensely benefit from such an optimization tool. We describe a series of increasingly complex designs that result in computationally hard training problems of combinatorial nature. In return for accepting classical computational hardness, we retain theoretical properties such as maximal sparsity and robustness to label noise, which are otherwise sacrificed by convex methods for the sake of computational efficiency and sound theoretical footing. In order to be compatible with emerging quantum hardware technology, we formalize the training problem as quadratic unconstrained binary optimization. Our initial investigations focus on a simple training formulation with non-convex regularization that conforms to the architecture of existing quantum hardware and makes frugal use of a limited number of available physical qubits. Next, we extend this baseline formulation to a scalable algorithm, QBoost, which is able to train incrementally large-scale classifiers on data sets of practical interest. Further, we derive another algorithm, TotalQBoost, as a theoretically motivated totally corrective boosting algorithm with cardinality penalization that also makes use of quantum optimization. Both QBoost and TotalQBoost perform explicit cardinality regularization, which is the only known way of achieving maximal sparsity in the trained classifiers. We apply QBoost and TotalQBoost to three different real-world computer vision problems and make use of a quantum processor for solving the sequence of discrete optimization problems generated by one of them. Finally, we study a learning formulation with convex regularization and a non-convex loss function, q-loss, specifically designed for robust supervised learning in the presence of label noise as it occurs in practice. For compatibility with quantum hardware we derive the corresponding quadratic binary problem via variational approximation. For all proposed algorithms we compare results on a variety of popular synthetic and natural data sets against a rich selection of existing rival learning formulations.
Variational quantum circuits for image classification suffer from barren plateaus, while quantum kernel methods scale quadratically with dataset size. We propose an iterative framework based on Quadratic Unconstrained Binary Optimization (QUBO) for training the classifier head of convolutional neural networks (CNNs) via quantum annealing, entirely avoiding gradient-based circuit optimization. Following the Extreme Learning Machine paradigm, convolutional filters are randomly initialized and frozen, and only the fully connected layer is optimized. At each iteration, a convex quadratic surrogate derived from the feature Gram matrix replaces the non-quadratic cross-entropy loss, yielding an iteration-stable curvature proxy. A per-output decomposition splits the
C-class problem into
C independent QUBOs, each with
(d+1)K binary variables, where
d is the feature dimension and
K is the bit precision, so that problem size depends on the image resolution and bit precision, not on the number of training samples. We evaluate the method on six image-classification benchmarks (sklearn digits, MNIST, Fashion-MNIST, CIFAR-10, EMNIST, KMNIST). A precision study shows that accuracy improves monotonically with bit resolution, with 10 bits representing a practical minimum for effective optimization; the 15-bit formulation remains within the qubit and coupler limits of current D-Wave Advantage hardware. The 20-bit formulation matches or exceeds classical stochastic gradient descent on MNIST, Fashion-MNIST, and EMNIST, while remaining competitive on CIFAR-10 and KMNIST. All experiments use simulated annealing, establishing a baseline for direct deployment on quantum annealing hardware.
This paper describes how to make the problem of binary classification amenable to quantum computing. A formulation is employed in which the binary classifier is constructed as a thresholded linear superposition of a set of weak classifiers. The weights in the superposition are optimized in a learning process that strives to minimize the training error as well as the number of weak classifiers used. No efficient solution to this problem is known. To bring it into a format that allows the application of adiabatic quantum computing (AQC), we first show that the bit-precision with which the weights need to be represented only grows logarithmically with the ratio of the number of training examples to the number of weak classifiers. This allows to effectively formulate the training process as a binary optimization problem. Solving it with heuristic solvers such as tabu search, we find that the resulting classifier outperforms a widely used state-of-the-art method, AdaBoost, on a variety of benchmark problems. Moreover, we discovered the interesting fact that bit-constrained learning machines often exhibit lower generalization error rates. Changing the loss function that measures the training error from 0-1 loss to least squares maps the training to quadratic unconstrained binary optimization. This corresponds to the format required by D-Wave's implementation of AQC. Simulations with heuristic solvers again yield results better than those obtained with boosting approaches. Since the resulting quadratic binary program is NP-hard, additional gains can be expected from applying the actual quantum processor.