Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits

Abstract

Variational quantum algorithms (VQAs) optimize the parameters $\vec{\theta }$ of a parametrized quantum circuit $V(\vec{\theta })$ to minimize a cost function $C$. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming $V(\vec{\theta })$ is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining $C$ in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when $V(\vec{\theta })$ is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining $C$ with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of $V(\vec{\theta })$ is $\mathcal{O}(\log n)$. Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.

14 + 25 pages, 8 + 2 figures. Updated to published version. Changed title

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• qLEET: Visualizing Loss Landscapes, Expressibility, Entangling Power and Training Trajectories for Parameterized Quantum Circuits
• Revisiting semiconductor bulk hamiltonians using quantum computers
• A Hybrid Quantum-Classical Particle-in-Cell Method for Plasma Simulations
• Optimal quantum reservoir learning in proximity to universality
• On the Hardness of Measuring Magic
• Performance analysis of a filtering variational quantum algorithm
• Learning quantum symmetries with interactive quantum-classical variational algorithms
• State preparation and evolution in quantum computing: a perspective from Hamiltonian moments
• Fermionic Machine Learning
• Resource-efficient quantum algorithm for linear systems of equations
• Quantum-data-driven dynamical transition in quantum learning
• Estimates of loss function concentration in noisy parametrized quantum circuits
• Encoding strongly-correlated many-boson wavefunctions on a photonic quantum computer: application to the attractive Bose-Hubbard model
• HQNET: Harnessing Quantum Noise for Effective Training of Quantum Neural Networks in NISQ Era
• Batched Line Search Strategy for Navigating through Barren Plateaus in Quantum Circuit Training
• Disentangling quantum neural networks for unified estimation of quantum entropies and distance measures
• Improving Parameter Training for VQEs by Sequential Hamiltonian Assembly
• Unsupervised detection of decoupled subspaces: many-body scars and beyond
• Investigating Techniques to Optimise the Layout of Turbines in a Windfarm using a Quantum Computer
• Efficient Online Quantum Circuit Learning with No Upfront Training
• Vanishing performance of the parity-encoded quantum approximate optimization algorithm applied to spin-glass models
• A general approach to quantum integration of cross sections in high-energy physics
• Dynamical transition in controllable quantum neural networks with large depth
• Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms
• Notes on Quantum Computing for Thermal Science
• Consensus-based qubit configuration optimization for variational algorithms on neutral atom quantum systems
• Variational-toolbox-based separability detection of multiqubit states
• A variational quantum algorithm for entanglement quantification
• Lie groups for quantum complexity and barren plateau theory
• The Dual Role of Low-Weight Pauli Propagation: A Flawed Simulator but a Powerful Initializer for Variational Quantum Algorithms
• Orbital Expansion Variational Quantum Eigensolver: Enabling Efficient Simulation of Molecules with Shallow Quantum Circuit
• Quantum neural network autoencoder and classifier applied to an industrial case study
• Global optimization in variational quantum algorithms via dynamic tunneling method
• Exploiting many-body localization for scalable variational quantum simulation
• Quantum Encoding of Structured Data with Matrix Product States
• Exploring Entanglement and Parameter Sensitivity in QAOA through Quantum Fisher Information
• Equivalence between exponential concentration in quantum machine learning kernels and barren plateaus in variational algorithms
• Gradient Scalability and Taylor Surrogation of Quantum Cost Landscapes
• Schmidt quantum compressor
• Quantum circuit evolutionary framework applied on set partitioning problem
• Pulse engineering via projection of response functions
• Ancillary entangling Floquet kicks for accelerating quantum algorithms
• Role of overparametrization in quantum approximate optimization
• Adaptive Interpolating Quantum Transform: A Quantum-Native Framework for Efficient Transform Learning
• Data-driven Quantum Dynamical Embedding Method for Long-term Prediction on Near-term Quantum Computers
• Learning Feedback Mechanisms for Measurement-Based Variational Quantum State Preparation
• Hamiltonian-reconstruction distance as a success metric for the Variational Quantum Eigensolver
• Performance Guarantees for Quantum Neural Estimation of Entropies
• Information Processing in Quantum Thermodynamic Systems: an Autonomous Hamiltonian Approach
• Moments of Quantum Channel Ensembles
• Noisy Monitored Quantum Circuits
• Velocity Verlet-based optimization for variational quantum eigensolvers
• Quantum-Enhanced Neural Exchange-Correlation Functionals
• Trainability issues in quantum policy gradients
• Compression of quantum shallow-circuit states
• Hybrid Quantum-Classical Clustering for Preparing a Prior Distribution of Eigenspectrum
• Improving the trainability of VQE on NISQ computers for solving portfolio optimization using convex interpolation
• Mode connectivity in the QCBM loss landscape
• Trainable Quantum Neural Network for Multiclass Image Classification with the Power of Pre-trained Tree Tensor Networks
• From barren plateaus through fertile valleys: Conic extensions of parameterised quantum circuits
• Direct Gradient Computation for Barren Plateaus in Parameterized Quantum Circuits
• Fast suppression of classification error in variational quantum circuits
• Quantum Compressive Sensing: Mathematical Machinery, Quantum Algorithms, and Quantum Circuitry
• Multivariate unbounded quantum regression via log-ratio probabilities mitigating barren plateaus
• Clustering and enhanced classification using a hybrid quantum autoencoder