Variational hybrid quantum classical algorithms to optimizations are important applications for near-term quantum computing. This paper proposes two quantum algorithms (the second one is variational) for training neural networks. Both of them obtain exponential speedup at the number of samples and polynomial speedup at the dimension of the samples over classical training algorithms. Moreover, the proposed quantum algorithms return the classical information of the training weight so that the outputs can be used directly to solve other problems. For practicality, we draw the quantum circuits to implement the two algorithms. Finally, as an inspiration, we show how to apply the variational algorithm to achieve speedup at the number of constraints in solving convex optimization problems.
Variational hybrid quantum classical algorithms to optimizations are important applications for near-term quantum computing. This paper proposes two quantum algorithms (the second one is variational) for training neural networks. Both of them obtain exponential speedup at the number of samples and polynomial speedup at the dimension of the samples over classical training algorithms. Moreover, the proposed quantum algorithms return the classical information of the training weight so that the outputs can be used directly to solve other problems. For practicality, we draw the quantum circuits to implement the two algorithms. Finally, as an inspiration, we show how to apply the variational algorithm to achieve speedup at the number of constraints in solving convex optimization problems.
The problem of the neural network optimization is local minima and flat areas. According to quantum-inspired methods, quantum mechanics, superposition, entanglement and interference are inspirational in order to possibly surpass these constraints in a more efficient manner. Variational quantum algorithms use parameterized quantum circuits parameterized quantum circuits (ansatz) to optimize weights. Quantum kernel methods also take advantage of quantum embeddings in order to transform features better. Hybrid quantum-classical systems incorporate quantum-inspired processing with already known reliability of optimization. There are, however, serious weaknesses that limit implementation. There are exponentially decreasing gradients due to the barren plateau phenomenon that makes it hard to train at scale. The modern quantum machines have a 0.1-1% error and a microsecond coherence time. The overheads in state preparation, the cost of measurement, circuit depth, and poor qubit connectivity are still impediments. The quantum-inspired classical algorithms, which include the tensor networks and optimization heuristics and run-on standard computers, are the most effective in the near future. These come with small advances in certain areas without involving quantum hardware. Although both quantum kernel methods and variational approaches show theoretical potential, the advantages are still small (relative to the advantages of optimized classical methods). Recent evaluations show that quantum benefits are more constrained than originally claimed them to be. Real quantum applications benefits in neural network optimization may only happen 10-20 years ahead, notwithstanding failure resistant quantum hardware and algorithm developments.
Can near-term gate model based quantum processors offer quantum advantage for practical applications in the pre-fault tolerance noise regime? A class of algorithms which have shown some promise in this regard are the so-called classical-quantum hybrid variational algorithms. Here we develop a low-depth quantum algorithm to generative neural networks using variational quantum circuits. We introduce a method which employs the quantum approximate optimization algorithm as a subroutine in order produce then sample low-energy distributions of Ising Hamiltonians. We sample these states to train neural networks and demonstrate training convergence for numerically simulated noisy circuits with depolarizing errors of rates of up to 4%.
I present a novel approach to training Binary Neural Networks (BiNNs) by leveraging Variational Quantum Algorithms (VQAs). My method employs the hybrid quantum-classical framework of VQAs to train simulated quantum circuits as hypernetworks that generate effective BiNN configurations. By developing approximate methods for calculating the VQA objective and gradient, I significantly scale quantum hypernetworks—over two orders of magnitude larger than previous capabilities, overcoming the constraints of exact computations. Quantum circuits are trained to produce effective BiNNs for binary classification tasks on resized and native resolution MNIST datasets. My empirical investigation extends to various VQA hyperparameters, including sample and batch sizes, circuit depth, and Multi-Basis Encoding (MBE). The results reveal that my optimized circuits consistently generate BiNN configurations with competitive cross-entropy losses. This work showcases the practical scalability of quantum hypernetworks and demonstrates significant interdisciplinary contribution in quantum machine learning, inviting further exploration into hybrid quantum-classical computational techniques.
Variational Quantum Algorithms have emerged as a leading paradigm for near-term quantum computation. In such algorithms, a parameterized quantum circuit is controlled via a classical optimization method that seeks to minimize a problem-dependent cost function. Although such algorithms are powerful in principle, the non-convexity of the associated cost landscapes and the prevalence of local minima means that local optimization methods such as gradient descent typically fail to reach good solutions. In this work we suggest a method to improve gradient-based approaches to variational quantum circuit optimization, which involves coupling the output of the quantum circuit to a classical neural network. The effect of this neural network is to peturb the cost landscape as a function of its parameters, so that local minima can be escaped or avoided via a modification to the cost landscape itself. We present two algorithms within this framework and numerically benchmark them on small instances of the Max-Cut optimization problem. We show that the method is able to reach deeper minima and lower cost values than standard gradient descent based approaches. Moreover, our algorithms require essentially the same number of quantum circuit evaluations per optimization step as the standard approach since, unlike the gradient with respect to the circuit, the neural network updates can be estimated in parallel via the backpropagation method. More generally, our approach suggests that relaxing the cost landscape is a fruitful path to improving near-term quantum computing algorithms.
Hybrid quantum-classical computing in the noisy intermediate-scale quantum (NISQ) era with variational algorithms can exhibit barren plateau issues, causing difficult convergence of gradient-based optimization techniques. In this paper, we discuss "post-variational strategies", which shift tunable parameters from the quantum computer to the classical computer, opting for ensemble strategies when optimizing quantum models. We discuss various strategies and design principles for constructing individual quantum circuits, where the resulting ensembles can be optimized with convex programming. Further, we discuss architectural designs of post-variational quantum neural networks and analyze the propagation of estimation errors throughout such neural networks. Finally, we show that empirically, post-variational quantum neural networks using our architectural designs can potentially provide better results than variational algorithms and performance comparable to that of two-layer neural networks.
Quantum operations on pure states can be fully represented by unitary matrices. Variational quantum circuits, also known as quantum neural networks, embed data and trainable parameters into gate-based operations and optimize the parameters via gradient descent. The high cost of training and low fidelity of current quantum devices, however, restricts much of quantum machine learning to classical simulation. For few-qubit problems with large datasets, training the matrix elements directly, as is done with weight matrices in classical neural networks, can be faster than decomposing data and parameters into gates. We propose a method that trains matrices directly while maintaining unitarity through a single regularization term added to the loss function. A second training step, circuit alignment, then recovers a gate-based architecture from the resulting soft-unitary. On a five-qubit supervised classification task with 1000 datapoints, this two-step process produces a trained variational circuit in under four minutes, compared to over two hours for direct circuit training, while achieving lower binary cross-entropy loss. In a second experiment, soft-unitaries are embedded in a hybrid quantum-classical network for a reinforcement learning cartpole task, where the hybrid agent outperforms a purely classical baseline of comparable size.