More Like this

1. Equivalence of cost concentration and gradient vanishing for quantum circuits: an elementary proof in the Riemannian formulation

   https://doi.org/10.1088/2058-9565/ad6fca  

   Miao, Qiang; Barthel, Thomas (September 2024, Quantum Science and Technology)

   Abstract The optimization of quantum circuits can be hampered by a decay of average gradient amplitudes with increasing system size. When the decay is exponential, this is called the barren plateau problem. Considering explicit circuit parametrizations (in terms of rotation angles), it has been shown in Arrasmithet al(2022Quantum Sci. Technol.7045015) that barren plateaus are equivalent to an exponential decay of the variance of cost-function differences. We show that the issue is particularly simple in the (parametrization-free) Riemannian formulation of such optimization problems and obtain a tighter bound for the cost-function variance. An elementary derivation shows that the single-gate variance of the cost function isstrictly equalto half the variance of the Riemannian single-gate gradient, where we sample variable gates according to the uniform Haar measure. The total variances of the cost function and its gradient are then both bounded from above by the sum of single-gate variances and, conversely, bound single-gate variances from above. So, decays of gradients and cost-function variations go hand in hand, and barren plateau problems cannot be resolved by avoiding gradient-based in favor of gradient-free optimization methods. 

   more » « less
2. Random coordinate descent: A simple alternative for optimizing parameterized quantum circuits

   https://doi.org/10.1103/PhysRevResearch.6.033029  

   Ding, Zhiyan; Ko, Taehee; Yao, Jiahao; Lin, Lin; Li, Xiantao (July 2024, Physical Review Research)

   Variational quantum algorithms rely on the optimization of parameterized quantum circuits in noisy settings. The commonly used back-propagation procedure in classical machine learning is not directly applicable in this setting due to the collapse of quantum states after measurements. Thus, gradient estimations constitute a significant overhead in a gradient-based optimization of such quantum circuits. This paper introduces a random coordinate descent algorithm as a practical and easy-to-implement alternative to the full gradient descent algorithm. This algorithm only requires one partial derivative at each iteration. Motivated by the behavior of measurement noise in the practical optimization of parameterized quantum circuits, this paper presents an optimization problem setting that is amenable to analysis. Under this setting, the random coordinate descent algorithm exhibits the same level of stochastic stability as the full gradient approach, making it as resilient to noise. The complexity of the random coordinate descent method is generally no worse than that of the gradient descent and can be much better for various quantum optimization problems with anisotropic Lipschitz constants. Theoretical analysis and extensive numerical experiments validate our findings. Published by the American Physical Society2024 

   more » « less
3. Energy-dependent barren plateau in bosonic variational quantum circuits

   https://doi.org/10.1088/2058-9565/ad80bf  

   Zhang, Bingzhi; Zhuang, Quntao (October 2024, Quantum Science and Technology)

   Abstract Bosonic variational quantum circuits (VQCs) are crucial for information processing in microwave cavities, trapped ions, and optical systems, widely applicable in quantum communication, sensing and error correction. The trainability of such VQCs is less understood, hindered by the lack of theoretical tools such ast-design due to the infinite dimension of the continuous-variable systems involved. We overcome this difficulty to reveal an energy-dependent barren plateau in such VQCs. The variance of the gradient decays as $1/E^{M\nu }$, exponential in the number of modesMbut polynomial in the (per-mode) circuit energyE. The exponentν = 1 for shallow circuits andν = 2 for deep circuits. We prove these results for state preparation of general Gaussian states and number states. We also provide numerical evidence demonstrating that the results extend to general state preparation tasks. As circuit energy is a controllable parameter, we provide a strategy to mitigate the barren plateau in bosonic continuous-variable VQCs. 

   more » « less
4. Estimating the randomness of quantum circuit ensembles up to 50 qubits

   https://doi.org/10.1038/s41534-022-00648-7  

   Liu, Minzhao; Liu, Junyu; Alexeev, Yuri; Jiang, Liang (December 2022, npj Quantum Information)

   Abstract Random quantum circuits have been utilized in the contexts of quantum supremacy demonstrations, variational quantum algorithms for chemistry and machine learning, and blackhole information. The ability of random circuits to approximate any random unitaries has consequences on their complexity, expressibility, and trainability. To study this property of random circuits, we develop numerical protocols for estimating the frame potential, the distance between a given ensemble and the exact randomness. Our tensor-network-based algorithm has polynomial complexity for shallow circuits and is high-performing using CPU and GPU parallelism. We study 1. local and parallel random circuits to verify the linear growth in complexity as stated by the Brown–Susskind conjecture, and; 2. hardware-efficient ansätze to shed light on its expressibility and the barren plateau problem in the context of variational algorithms. Our work shows that large-scale tensor network simulations could provide important hints toward open problems in quantum information science. 

   more » « less
5. Exponentially Many Local Minima in Quantum Neural Networks

   You, Xuchen; Wu, Xiaodi (January 2021, Proceedings of the 38th International Conference on Machine Learning)

   null (Ed.)

   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. 

   more » « less