Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states

Abstract

Vanishing gradients can pose substantial obstacles for high-dimensional optimization problems. Here we consider energy minimization problems for quantum many-body systems with extensive Hamiltonians and finite-range interactions, which can be studied on classical computers or in the form of variational quantum eigensolvers on quantum computers. Barren plateaus correspond to scenarios where the average amplitude of the energy gradient decreases exponentially with increasing system size. This occurs, for example, for quantum neural networks and for brickwall quantum circuits when the depth increases polynomially in the system size. Here we prove that the variational optimization problems for matrix product states, tree tensor networks, and the multiscale entanglement renormalization ansatz are free of barren plateaus. The derived scaling properties for the gradient variance provide an analytical guarantee for the trainability of randomly initialized tensor network states (TNS) and motivate certain initialization schemes. In a suitable representation, unitary tensors that parametrize the TNS are sampled according to the uniform Haar measure. We employ a Riemannian formulation of the gradient based optimizations which simplifies the analytical evaluation.

30 pages main text, 14 pages appendix, 14 figures; added 6 figures concerning MERA and TTNS, an analysis for nonary 2D MERA and TTNS, and references; added appendices D.1 and D.2 explaining apparent discrepancies concerning ZX-MPS discussed in [Cervero Martín et al., Quantum 7, 974 (2023)] and QMPS simulated in [Liu et al., PRR 1, 023025 (2019)], further minor improvements; published version

References in corpus (31)

• The density-matrix renormalization group in the age of matrix product states
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• A class of quantum many-body states that can be efficiently simulated
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• Entanglement entropy of fermions in any dimension and the Widom conjecture
• Identifying and attacking the saddle point problem in high-dimensional non-convex optimization
• Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks
• The power of quantum systems on a line
• Entanglement scaling in critical two-dimensional fermionic and bosonic systems
• Entropy and Entanglement in Quantum Ground States
• Trainability of Dissipative Perceptron-Based Quantum Neural Networks
• Equivalence of quantum barren plateaus to cost concentration and narrow gorges
• Entanglement Devised Barren Plateau Mitigation
• Parallel Quantum Simulation of Large Systems on Small Quantum Computers
• On barren plateaus and cost function locality in variational quantum algorithms
• Holographic quantum algorithms for simulating correlated spin systems
• The Variational Power of Quantum Circuit Tensor Networks
• Barren plateaus in quantum tensor network optimization
• Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus
• Riemannian optimization of isometric tensor networks
• The Presence and Absence of Barren Plateaus in Tensor-network Based Machine Learning
• Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies
• Optimizing quantum circuits with Riemannian gradient flow
• Holographic simulation of correlated electrons on a trapped ion quantum processor
• Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus
• Quantum-classical eigensolver using multiscale entanglement renormalization
• Barren plateaus from learning scramblers with local cost functions
• On the closedness and geometry of tensor network state sets
• Probing critical states of matter on a digital quantum computer
• Equivalence of cost concentration and gradient vanishing for quantum circuits: An elementary proof in the Riemannian formulation
• Convergence and Quantum Advantage of Trotterized MERA for Strongly-Correlated Systems

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