Characterizing the loss landscape of variational quantum circuits

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.

Quantum generative adversarial networks with multiple superconducting qubits

“…Moreover, another possible challenge for larger circuits is the so called "barren plateaus" problem, namely that the gradient might be vanishing for most of the parameter regions (see ref. [20][21][22][23] for more details). In fact, these challenges are the common ones facing most of the current experiments on variational quantum circuits.…”

“…Superconducting qubits are a promising platform for realizing QGANs, owing to their flexible design, excellent scalability, and remarkable controllability. In our implementation, both the generator and discriminator are composed of multiqubit parameterized quantum circuits, also referred to as quantum neural networks in some contexts [20][21][22][23] . Here, we benchmark the functionality of the quantum gradient method by learning an arbitrary mixed state, where the state is replicated with a fidelity up to 0.999.…”

Survey of NISQ Era Hybrid Quantum-Classical Machine Learning Research

“…To explore the loss function of PQCs, the Hessian is employed. The Hessian allows detection of local minima, maxima, and saddle points, akin to the second derivative test [14]. One of the issues with implementing the Hessian, therefore, is the calculation of the second derivative of the loss function.…”

“…Using the limit definition of the derivative to calculate the gradient amplifies the inherent measurement noise. Instead, the authors use parameter shift rules and the chain rule to calculate the Hessian of a quantum circuit [14].…”

“…Thus, QNG has poor performance when the circuit is initialized in a region with small gradients. Although the Hessian methods can help avoid barren plateaus, they struggle with local minima and the Hessian is costlier to calculate than the quantum metric tensor used for QNG [14]. One potential solution to this issue is to use a Hessian approach to escape flat regions of the loss landscape and QNG when the gradient is larger.…”