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1. 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. 

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2. Analyzing convergence in quantum neural networks: deviations from neural tangent kernels

   You, Xuchen; Chakrabarti, Shouvanik; Chen, Boyang; Wu, Xiaodi (July 2023, ICML'23: Proceedings of the 40th International Conference on Machine Learning)

   A quantum neural network (QNN) is a parameterized mapping efficiently implementable on near-term Noisy Intermediate-Scale Quantum (NISQ) computers. It can be used for supervised learning when combined with classical gradient-based optimizers. Despite the existing empirical and theoretical investigations, the convergence of QNN training is not fully understood. Inspired by the success of the neural tangent kernels (NTKs) in probing into the dynamics of classical neural networks, a recent line of works proposes to study over-parameterized QNNs by examining a quantum version of tangent kernels. In this work, we study the dynamics of QNNs and show that contrary to popular belief it is qualitatively different from that of any kernel regression: due to the unitarity of quantum operations, there is a nonnegligible deviation from the tangent kernel regression derived at the random initialization. As a result of the deviation, we prove the at-most sublinear convergence for QNNs with Pauli measurements, which is beyond the explanatory power of any kernel regression dynamics. We then present the actual dynamics of QNNs in the limit of over-parameterization. The new dynamics capture the change of convergence rate during training, and implies that the range of measurements is crucial to the fast QNN convergence. 

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3. Analyzing Convergence in Quantum Neural Networks: Deviations from Neural Tangent Kernels

   Xuchen You; Shouvanik Chakrabarti; Boyang Chen; Xiaodi Wu (January 2023, Proceedings of the 40th International Conference on Machine Learning)

   A quantum neural network (QNN) is a parameterized mapping efficiently implementable on near- term Noisy Intermediate-Scale Quantum (NISQ) computers. It can be used for supervised learn- ing when combined with classical gradient-based optimizers. Despite the existing empirical and theoretical investigations, the convergence of QNN training is not fully understood. Inspired by the success of the neural tangent kernels (NTKs) in probing into the dynamics of classical neural net- works, a recent line of works proposes to study over-parameterized QNNs by examining a quantum version of tangent kernels. In this work, we study the dynamics of QNNs and show that contrary to popular belief it is qualitatively different from that of any kernel regression: due to the unitarity of quantum operations, there is a non- negligible deviation from the tangent kernel regression derived at the random initialization. As a result of the deviation, we prove the at-most sub- linear convergence for QNNs with Pauli measurements, which is beyond the explanatory power of any kernel regression dynamics. We then present the actual dynamics of QNNs in the limit of over- parameterization. The new dynamics capture the change of convergence rate during training, and implies that the range of measurements is crucial to the fast QNN convergence. 

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4. Skyrmion-mediated nonvolatile ternary memory

   https://doi.org/10.1038/s41598-024-66853-w  

   Rajib, Md Mahadi; Bindal, Namita; Raj, Ravish Kumar; Kaushik, Brajesh Kumar; Atulasimha, Jayasimha (July 2024, Scientific Reports)

   Multistate memory systems have the ability to store and process more data in the same physical space as binary memory systems, making them a potential alternative to existing binary memory systems. In the past, it has been demonstrated that voltage-controlled magnetic anisotropy (VCMA) based writing is highly energy-efficient compared to other writing methods used in non-volatile nano-magnetic binary memory systems. In this study, we introduce a new, VCMA-based and skyrmion-mediated non-volatile ternary memory system using a perpendicular magnetic tunnel junction (p-MTJ) in the presence of room temperature thermal perturbation. We have also shown that ternary states {− 1, 0, + 1} can be implemented with three magnetoresistance values obtained from a p-MTJ corresponding to ferromagnetic up, down, and skyrmion state, with 99% switching probability in the presence of room temperature thermal noise in an energy-efficient way, requiring ~ 2 fJ energy on an average for each switching operation. Additionally, we show that our proposed ternary memory demonstrates an improvement in area and energy by at least 2X and ~ 10e4 X respectively, compared to state-of-the-art spin-transfer torque (STT)-based non-volatile magnetic multistate memories. Furthermore, these three states can be potentially utilized for energy-efficient, high-density in-memory quantized deep neural network implementation. 

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5. QTROJAN: A Circuit Backdoor Against Quantum Neural Networks

   https://doi.org/10.1109/ICASSP49357.2023.10096293  

   Chu, Cheng; Jiang, Lei; Swany, Martin; Chen, Fan (June 2023, IEEE International Conference on Acoustics, Speech and Signal Processing)

   We propose a circuit-level backdoor attack, QTrojan, against Quantum Neural Networks (QNNs) in this paper. QTrojan is implemented by a few quantum gates inserted into the variational quantum circuit of the victim QNN. QTrojan is much stealthier than a prior Data-Poisoning-based Backdoor Attack (DPBA) since it does not embed any trigger in the inputs of the victim QNN or require access to original training datasets. Compared to a DPBA, QTrojan improves the clean data accuracy by 21% and the attack success rate by 19.9%. 

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