Quantum Neural Networks (QNNs), or the so-called variational quantum circuits, are important quantum applications both because of their similar promises as classical neural networks and because of the feasibility of their implementation on near-term intermediate-size noisy quantum machines (NISQ). However, the training task of QNNs is challenging and much less understood. We conduct a quantitative investigation on the landscape of loss functions of QNNs and identify a class of simple yet extremely hard QNN instances for training. Specifically, we show for typical under-parameterized QNNs, there exists a dataset that induces a loss function with the number of spurious local minima depending exponentially on the number of parameters. Moreover, we show the optimality of our construction by providing an almost matching upper bound on such dependence. While local minima in classical neural networks are due to non-linear activations, in quantum neural networks local minima appear as a result of the quantum interference phenomenon. Finally, we empirically confirm that our constructions can indeed be hard instances in practice with typical gradient-based optimizers, which demonstrates the practical value of our findings.
Quantum Neural Networks (QNNs), or the so-called variational quantum circuits, are important quantum applications both because of their similar promises as classical neural networks and because of the feasibility of their implementation on near-term intermediate-size noisy quantum machines (NISQ). However, the training task of QNNs is challenging and much less understood. We conduct a quantitative investigation on the landscape of loss functions of QNNs and identify a class of simple yet extremely hard QNN instances for training. Specifically, we show for typical under-parameterized QNNs, there exists a dataset that induces a loss function with the number of spurious local minima depending exponentially on the number of parameters. Moreover, we show the optimality of our construction by providing an almost matching upper bound on such dependence. While local minima in classical neural networks are due to non-linear activations, in quantum neural networks local minima appear as a result of the quantum interference phenomenon. Finally, we empirically confirm that our constructions can indeed be hard instances in practice with typical gradient-based optimizers, which demonstrates the practical value of our findings.
Quantum Neural Networks (QNNs) are a promising variational learning paradigm with applications to near-term quantum processors, however they still face some significant challenges. One such challenge is finding good parameter initialization heuristics that ensure rapid and consistent convergence to local minima of the parameterized quantum circuit landscape. In this work, we train classical neural networks to assist in the quantum learning process, also know as meta-learning, to rapidly find approximate optima in the parameter landscape for several classes of quantum variational algorithms. Specifically, we train classical recurrent neural networks to find approximately optimal parameters within a small number of queries of the cost function for the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut, QAOA for Sherrington-Kirkpatrick Ising model, and for a Variational Quantum Eigensolver for the Hubbard model. By initializing other optimizers at parameter values suggested by the classical neural network, we demonstrate a significant improvement in the total number of optimization iterations required to reach a given accuracy. We further demonstrate that the optimization strategies learned by the neural network generalize well across a range of problem instance sizes. This opens up the possibility of training on small, classically simulatable problem instances, in order to initialize larger, classically intractably simulatable problem instances on quantum devices, thereby significantly reducing the number of required quantum-classical optimization iterations.
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.
Quantum neural networks (QNNs) have become a leading paradigm for establishing near-term quantum applications in recent years. The trainability issue of QNNs has garnered extensive attention, spurring demand for a comprehensive analysis of QNNs in order to identify viable solutions. In this work, we propose a perspective that characterizes the trainability of QNNs based on their locality. We prove that the entire variation range of the loss function via adjusting any local quantum gate vanishes exponentially in the number of qubits with a high probability for a broad class of QNNs. This result reveals extra harsh constraints independent of gradients and unifies the restrictions on gradient-based and gradient-free optimizations naturally. We showcase the validity of our results with numerical simulations of representative models and examples. Our findings, as a fundamental property of random quantum circuits, deepen the understanding of the role of locality in QNNs and serve as a guideline for assessing the effectiveness of diverse training strategies for quantum neural networks.
One of the most important properties of classical neural networks is how surprisingly trainable they are, though their training algorithms typically rely on optimizing complicated, nonconvex loss functions. Previous results have shown that unlike the case in classical neural networks, variational quantum models are often not trainable. The most studied phenomenon is the onset of barren plateaus in the training landscape of these quantum models, typically when the models are very deep. This focus on barren plateaus has made the phenomenon almost synonymous with the trainability of quantum models. Here, we show that barren plateaus are only a part of the story. We prove that a wide class of variational quantum models-which are shallow, and exhibit no barren plateaus-have only a superpolynomially small fraction of local minima within any constant energy from the global minimum, rendering these models untrainable if no good initial guess of the optimal parameters is known. We also study the trainability of variational quantum algorithms from a statistical query framework, and show that noisy optimization of a wide variety of quantum models is impossible with a sub-exponential number of queries. Finally, we numerically confirm our results on a variety of problem instances. Though we exclude a wide variety of quantum algorithms here, we give reason for optimism for certain classes of variational algorithms and discuss potential ways forward in showing the practical utility of such algorithms.
Abstract Machine learning techniques enhanced by noisy intermediate-scale quantum (NISQ) devices and especially variational quantum circuits (VQC) have recently attracted much interest and have already been benchmarked for certain problems. Inspired by classical deep learning, VQCs are trained by gradient descent methods which allow for efficient training over big parameter spaces. For NISQ sized circuits, such methods show good convergence. There are however still many open questions related to the convergence of the loss function and to the trainability of these circuits in situations of vanishing gradients. Furthermore, it is not clear how ‘good’ the minima are in terms of generalization and stability against perturbations of the data and there is, therefore, a need for tools to quantitatively study the convergence of the VQCs. In this work, we introduce a way to compute the Hessian of the loss function of VQCs and show how to characterize the loss landscape with it. The eigenvalues of the Hessian give information on the local curvature and we discuss how this information can be interpreted and compared to classical neural networks. We benchmark our results on several examples, starting with a simple analytic toy model to provide some intuition about the behaviour of the Hessian, then going to bigger circuits, and also train VQCs on data. Finally, we show how the Hessian can be used to adjust the learning rate for faster convergence during the training of variational circuits.
One of the most important properties of neural networks is the clustering of local minima of the loss function near the global minimum, enabling efficient training. Though generative models implemented on quantum computers are known to be more expressive than their traditional counterparts, it has empirically been observed that these models experience a transition in the quality of their local minima. Namely, below some critical number of parameters, all local minima are far from the global minimum in function value; above this critical parameter count, all local minima are good approximators of the global minimum. Furthermore, for a certain class of quantum generative models, this transition has empirically been observed to occur at parameter counts exponentially large in the problem size, meaning practical training of these models is out of reach. Here, we give the first proof of this transition in trainability, specializing to this latter class of quantum generative model. We use techniques inspired by those used to study the loss landscapes of classical neural networks. We also verify that our analytic results hold experimentally even at modest model sizes.