Variational optimization of continuous matrix product states

Abstract

Just as matrix product states represent ground states of one-dimensional quantum spin systems faithfully, continuous matrix product states (cMPS) provide faithful representations of the vacuum of interacting field theories in one spatial dimension. Unlike the quantum spin case however, for which the density matrix renormalization group and related matrix product state algorithms provide robust algorithms for optimizing the variational states, the optimization of cMPS for systems with inhomogeneous external potentials has been problematic. We resolve this problem by constructing a piecewise linear parameterization of the underlying matrix-valued functions, which enables the calculation of the exact reduced density matrices everywhere in the system by high-order Taylor expansions. This turns the variational cMPS problem into a variational algorithm from which both the energy and its backwards derivative can be calculated exactly and at a cost that scales as the cube of the bond dimension. We illustrate this by finding ground states of interacting bosons in external potentials, and by calculating boundary or Casimir energy corrections of continuous many-body systems with open boundary conditions.

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• Many-Body Physics with Ultracold Gases
• Continuous Matrix Product States for Quantum Fields
• Holographic quantum states
• Multigrid Algorithms for Tensor Network States
• The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions
• Continuous matrix product states with periodic boundary conditions and an application to atomtronics
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Cited by in corpus (9)

• Riemannian optimization of isometric tensor networks
• Variational Neural-Network Ansatz for Continuum Quantum Field Theory
• Density-Matrix Renormalization Group for Continuous Quantum Systems
• Continuous matrix-product states in inhomogeneous systems with long-range interactions
• Entanglement Renormalization of the class of Continuous Matrix Product States
• Symmetries and field tensor network states
• Kac-Moody symmetries in one-dimensional bosonic systems
• Continuous matrix product operators for quantum fields
• Holographic Tensor Networks as Tessellations of Geometry