Effects of noise on the overparametrization of quantum neural networks

Abstract

Overparametrization is one of the most surprising and notorious phenomena in machine learning. Recently, there have been several efforts to study if, and how, Quantum Neural Networks (QNNs) acting in the absence of hardware noise can be overparametrized. In particular, it has been proposed that a QNN can be defined as overparametrized if it has enough parameters to explore all available directions in state space. That is, if the rank of the Quantum Fisher Information Matrix (QFIM) for the QNN's output state is saturated. Here, we explore how the presence of noise affects the overparametrization phenomenon. Our results show that noise can "turn on" previously-zero eigenvalues of the QFIM. This enables the parametrized state to explore directions that were otherwise inaccessible, thus potentially turning an overparametrized QNN into an underparametrized one. For small noise levels, the QNN is quasi-overparametrized, as large eigenvalues coexists with small ones. Then, we prove that as the magnitude of noise increases all the eigenvalues of the QFIM become exponentially suppressed, indicating that the state becomes insensitive to any change in the parameters. As such, there is a pull-and-tug effect where noise can enable new directions, but also suppress the sensitivity to parameter updates. Finally, our results imply that current QNN capacity measures are ill-defined when hardware noise is present.

arXiv comment14 + 6 pages, 11 figures

References in corpus (47)

• Quantum Computing in the NISQ era and beyond
• Quantum Machine Learning
• Variational Quantum Algorithms
• Barren plateaus in quantum neural network training landscapes
• Noisy intermediate-scale quantum (NISQ) algorithms
• Reconciling modern machine learning practice and the bias-variance trade-off
• Quantum Convolutional Neural Networks
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms
• Quantum Fisher information matrix and multiparameter estimation
• Towards Practical Quantum Variational Algorithms
• Efficient Learning for Deep Quantum Neural Networks
• Challenges and Opportunities in Quantum Machine Learning
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• Quantum advantage in learning from experiments
• Training variational quantum algorithms is NP-hard
• Variational quantum simulation of general processes
• Provably efficient machine learning for quantum many-body problems
• Exploring entanglement and optimization within the Hamiltonian Variational Ansatz
• Diagnosing Barren Plateaus with Tools from Quantum Optimal Control
• Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography
• Noise Resilience of Variational Quantum Compiling
• Theory of overparametrization in quantum neural networks
• Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer
• Qibo: a framework for quantum simulation with hardware acceleration
• Discontinuities of the quantum Fisher information and the Bures metric
• Evaluating the Holevo Cramér-Rao bound for multi-parameter quantum metrology
• Fisher Information in Noisy Intermediate-Scale Quantum Applications
• The jamming transition as a paradigm to understand the loss landscape of deep neural networks
• Quantum Variational Optimization of Ramsey Interferometry and Atomic Clocks
• Variational-State Quantum Metrology
• Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks
• Symmetry Principles in Quantum Systems Theory
• Variational Quantum Algorithm for Estimating the Quantum Fisher Information
• Representation Learning via Quantum Neural Tangent Kernels
• Fidelity susceptibility and quantum Fisher information for density operators with arbitrary ranks
• Quantum Fisher information and symmetric logarithmic derivative via anti-commutators
• Analytic theory for the dynamics of wide quantum neural networks
• Characterizing local noise in QAOA circuits
• On the discontinuity of the quantum Fisher information for quantum statistical models with parameter dependent rank
• Relative Entropy Convergence for Depolarizing Channels
• Progress toward favorable landscapes in quantum combinatorial optimization
• Efficient classical algorithms for simulating symmetric quantum systems
• Exploiting Landscape Geometry to Enhance Quantum Optimal Control
• Quantum simulation with just-in-time compilation
• Inference-Based Quantum Sensing
• Characterization of variational quantum algorithms using free fermions

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