Tight and Efficient Gradient Bounds for Parameterized Quantum Circuits

Abstract

The training of a parameterized model largely depends on the landscape of the underlying loss function. In particular, vanishing gradients are a central bottleneck in the scalability of variational quantum algorithms (VQAs), and are known to arise in various ways. However, a caveat of most existing gradient bound results is the requirement of t-design circuit assumptions that are typically not satisfied in practice. In this work, we loosen these assumptions altogether and derive tight upper and lower bounds on loss and gradient concentration for a large class of parameterized quantum circuits and arbitrary observables, which are significantly stronger than prior work. Moreover, we show that these bounds, as well as the variance of the loss itself, can be estimated efficiently and classically-providing practical tools to study the loss landscapes of VQA models, including verifying whether or not a circuit/observable induces barren plateaus. In particular, our results can readily be leveraged to rule out barren plateaus for a realistic class of ansätze and mixed observables, namely, observables containing a non-vanishing local term. This insight has direct implications for hybrid Quantum Generative Adversarial Networks (qGANs). We prove that designing the discriminator appropriately leads to 1-local weights that stay constant in the number of qubits, regardless of discriminator depth. This implies that qGANs with appropriately chosen generators do not suffer from barren plateaus even at scale-making them a promising candidate for applications in generative quantum machine learning. We demonstrate this result by training a qGAN to learn a 2D mixture of Gaussian distributions with up to 16 qubits, and provide numerical evidence that global contributions to the gradient, while initially exponentially small, may kick in substantially over the course of training.

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• Variational quantum simulation: a case study for understanding warm starts
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• Classically estimating observables of noiseless quantum circuits
• Provable bounds for noise-free expectation values computed from noisy samples
• Quantum Convolutional Neural Networks are Effectively Classically Simulable
• Computing exact moments of local random quantum circuits via tensor networks
• Symmetry-invariant quantum machine learning force fields
• Efficient quantum-enhanced classical simulation for patches of quantum landscapes
• Hamiltonian-based graph-state ansatz for variational quantum algorithms
• Architectures and random properties of symplectic quantum circuits
• Pitfalls when tackling the exponential concentration of parameterized quantum models
• Cross-platform hardware benchmark of style-based quantum GANs for data augmentation on superconducting and trapped-ion processors
• Moments of Quantum Channel Ensembles
• Direct Gradient Computation for Barren Plateaus in Parameterized Quantum Circuits
• Quantum Machine Learning for State Tomography Using Classical Data
• Gradient Scalability and Taylor Surrogation of Quantum Cost Landscapes