Barren plateaus in quantum tensor network optimization

Abstract

We analyze the barren plateau phenomenon in the variational optimization of quantum circuits inspired by matrix product states (qMPS), tree tensor networks (qTTN), and the multiscale entanglement renormalization ansatz (qMERA). We consider as the cost function the expectation value of a Hamiltonian that is a sum of local terms. For randomly chosen variational parameters we show that the variance of the cost function gradient decreases exponentially with the distance of a Hamiltonian term from the canonical centre in the quantum tensor network. Therefore, as a function of qubit count, for qMPS most gradient variances decrease exponentially and for qTTN as well as qMERA they decrease polynomially. We also show that the calculation of these gradients is exponentially more efficient on a classical computer than on a quantum computer.

arXiv comment26 pages, 7 figures

References in corpus (66)

• Quantum Computing in the NISQ era and beyond
• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• A variational eigenvalue solver on a quantum processor
• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Variational Quantum Algorithms
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Barren plateaus in quantum neural network training landscapes
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• A Quantum Approximate Optimization Algorithm
• Noisy intermediate-scale quantum (NISQ) algorithms
• Quantum computational chemistry
• Quantum Circuit Learning
• Quantum Chemistry in the Age of Quantum Computing
• Quantum Convolutional Neural Networks
• Strong quantum computational advantage using a superconducting quantum processor
• Parameterized quantum circuits as machine learning models
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Evaluating analytic gradients on quantum hardware
• The Variational Quantum Eigensolver: a review of methods and best practices
• Circuit-centric quantum classifiers
• A class of quantum many-body states that can be efficiently simulated
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Noise-Induced Barren Plateaus in Variational Quantum Algorithms
• Quantum algorithms for quantum chemistry and quantum materials science
• Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices
• Classical simulation of quantum many-body systems with a tree tensor network
• Tensor networks for complex quantum systems
• Theory of variational quantum simulation
• Training variational quantum algorithms is NP-hard
• An initialization strategy for addressing barren plateaus in parametrized quantum circuits
• Absence of Barren Plateaus in Quantum Convolutional Neural Networks
• Sequential generation of entangled multi-qubit states
• Layerwise learning for quantum neural networks
• Algorithms for entanglement renormalization
• Variational quantum algorithms for nonlinear problems
• Towards Quantum Machine Learning with Tensor Networks
• Interacting Quantum Observables: Categorical Algebra and Diagrammatics
• Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law
• Solving nonlinear differential equations with differentiable quantum circuits
• Hardware-efficient variational quantum algorithms for time evolution
• Variational Quantum Eigensolver with Fewer Qubits
• Tensor Network Algorithms: a Route Map
• Avoiding barren plateaus using classical shadows
• Filtering variational quantum algorithms for combinatorial optimization
• Crossing a topological phase transition with a quantum computer
• A Survey of Quantum Computing for Finance
• Holographic quantum algorithms for simulating correlated spin systems
• A case study of variational quantum algorithms for a job shop scheduling problem
• Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus
• Avoiding barren plateaus via transferability of smooth solutions in Hamiltonian Variational Ansatz
• Holographic dynamics simulations with a trapped ion quantum computer
• The Presence and Absence of Barren Plateaus in Tensor-network Based Machine Learning
• Variational inference with a quantum computer
• Variational quantum amplitude estimation
• Variational Hamiltonian simulation for translational invariant systems via classical pre-processing
• Mitigating Barren Plateaus with Transfer-learning-inspired Parameter Initializations
• Classical versus Quantum: comparing Tensor Network-based Quantum Circuits on LHC data
• Entanglement from tensor networks on a trapped-ion QCCD quantum computer
• Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits
• Holographic simulation of correlated electrons on a trapped ion quantum processor
• Entanglement Diagnostics for Efficient Quantum Computation
• A tensor network discriminator architecture for classification of quantum data on quantum computers
• Qubit-efficient simulation of thermal states with quantum tensor networks
• Matrix Product State Pre-Training for Quantum Machine Learning

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• Does provable absence of barren plateaus imply classical simulability?
• Towards adiabatic quantum computing using compressed quantum circuits
• The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ansätze
• Realization of quantum signal processing on a noisy quantum computer
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• Quantum computing for chemistry and physics applications from a Monte Carlo perspective
• Absence of barren plateaus in finite local-depth circuits with long-range entanglement
• Nonlinear dynamics as a ground-state solution on quantum computers
• On the Sample Complexity of Quantum Boltzmann Machine Learning
• Variational quantum simulation: a case study for understanding warm starts
• Training variational quantum algorithms with random gate activation
• Tensor networks for quantum computing
• Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus
• Problem specific classical optimization of Hamiltonian simulation
• Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and limit sets
• Efficient MPS representations and quantum circuits from the Fourier modes of classical image data
• Emergence of noise-induced barren plateaus in arbitrary layered noise models
• Hybrid Tree Tensor Networks for quantum simulation
• Optimizing ZX-Diagrams with Deep Reinforcement Learning
• Variational-quantum-eigensolver-inspired optimization for spin-chain work extraction
• Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states
• Speedy Contraction of ZX Diagrams with Triangles via Stabiliser Decompositions
• Application of Quantum Tensor Networks for Protein Classification
• Matrix product state ansatz for the variational quantum solution of the Heisenberg model on Kagome geometries
• Efficient quantum-enhanced classical simulation for patches of quantum landscapes
• Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learning
• Channel Attention for Quantum Convolutional Neural Networks
• Variational Quantum Imaginary Time Evolution for Matrix Product State Ansatz with Tests on Transcorrelated Hamiltonians
• Perturbative gadgets for gate-based quantum computing: Non-recursive constructions without subspace restrictions
• Pitfalls when tackling the exponential concentration of parameterized quantum models
• Survey on Computational Applications of Tensor Network Simulations
• Quantum circuits for partial differential equations in Fourier space
• Quantum tensor network algorithms for evaluation of spectral functions on quantum computers
• Quantitative convergence of trained quantum neural networks to a Gaussian process
• Performance analysis of a filtering variational quantum algorithm
• Estimates of loss function concentration in noisy parametrized quantum circuits
• Efficient Online Quantum Circuit Learning with No Upfront Training
• Exploiting many-body localization for scalable variational quantum simulation
• Batched Line Search Strategy for Navigating through Barren Plateaus in Quantum Circuit Training
• A quantum eigenvalue solver based on tensor networks
• Qubit frugal entanglement determination with the deep multi-scale entanglement renormalization ansatz