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Cambridge University Press eBooks

Determination of MPS

Tao Xiang

Published Aug 31, 2023• 0 citations
Publisher

Abstract

This chapter introduces two commonly used methods of determining the local tensors of an MPS. The first is the variational optimization method, which determines an MPS by minimizing the energy expectation value. This method is equivalent to solving a generalized eigenequation around the extreme point of the ground-state energy. The second is an update method based on an imaginary time evolution, which cools down a quantum state from finite to zero temperature. We discuss three update approaches: update via canonicalization, full update, and simple update. For an MPS, the canonicalization approach is accurate and easy to implement. However, the full and simple update can be generalized to higher dimensions and applied to, for example, PEPS. The full update is a global minimization approach. It is accurate but has a higher computational cost than the simple update. The simple update is a local optimization approach based on an entanglement mean-field approximation and is easy to implement. Finally, we discuss the purification technique and apply it to evaluate the thermal density matrix or solve a quenched disorder problem in the framework of MPS.

DOI
10.1017/9781009398671.013

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Cambridge University Press eBooks•Aug 31, 2023•0 citations
This chapter introduces two commonly used methods of determining the local tensors of an MPS. The first is the variational optimization method, which determines an MPS by minimizing the energy expectation value. This method is equivalent to solving a generalized eigenequation around the extreme point of the ground-state energy. The second is an update method based on an imaginary time evolution, which cools down a quantum state from finite to zero temperature. We discuss three update approaches: update via canonicalization, full update, and simple update. For an MPS, the canonicalization approach is accurate and easy to implement. However, the full and simple update can be generalized to higher dimensions and applied to, for example, PEPS. The full update is a global minimization approach. It is accurate but has a higher computational cost than the simple update. The simple update is a local optimization approach based on an entanglement mean-field approximation and is easy to implement. Finally, we discuss the purification technique and apply it to evaluate the thermal density matrix or solve a quenched disorder problem in the framework of MPS.
This chapter discusses the methods of solving PEPS or other two-dimensional tensor network states, including variational optimization and the annealing simulation. The variation optimization determines the local terms by minimizing the ground-state energy. The annealing simulation takes the full or simple update strategy to filter out the ground state through the imaginary time evolution. The nonlinear effect arises in evaluating the derivative of uniform PEPS and is avoided by utilizing automatic differentiation. Both variational optimization and the annealing simulation involve a contraction of double-layer tensor network states. This contraction is the primary technical barrier in the study of PEPS. A nested tensor network approach is introduced to combat this difficulty.
The tangent space of a tensor network state is a manifold that parameterizes the variational trajectory of local tensors. It leads to a framework to accommodate the time-dependent variational principle (TDVP), which unifies stationary and time-dependent methods for dealing with tensor networks. It also offers an ideal platform for investigating elementary excitations, including nontrivial topological excitations, in a quantum many-body system under the single-mode approximation. This chapter introduces the tangent-space approaches in the variational determination of MPS and PEPS, starting with a general discussion on the properties of the tangent vectors of uniform MPS. It then exemplifies TDVP by applying it to optimize the ground state MPS. Finally, the methods for calculating the excitation spectra in both one and two dimensions are explored and applied to the antiferromagnetic Heisenberg model on the square lattice.
These notes provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS). They develop the basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation. The main MPS algorithms are then introduced, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods. The notes also briefly discuss projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction. Finally, tensor-network representations of mixed states, quantum channels, and Lindblad dynamics are presented, with applications to thermal states and open quantum systems. The presentation is accompanied by short Julia code examples based on ITensor, ITensorMPS, and TensorMixedStates. These notes were written for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems, held in June 2026.
Within the Projected Entangled Pair State (PEPS) tensor network formalism, a simple update (SU) method has been used to investigate the time evolution of a two-dimensional U(1) critical spin-1/2 spin liquid under Hamiltonian quench [Phys. Rev. B 106, 195132 (2022)]. Here we introduce two different variational frameworks to describe the time dynamics of SU(2)-symmetric translationally-invariant PEPS, aiming to improve the accuracy. In one approach, after using a Trotter-Suzuki decomposition of the time evolution operator in term of two-site elementary gates, one considers a single bond embedded in an environment approximated by a Corner Transfer Matrix Renormalization Group (CTMRG). A variational update of the two tensors on the bond is performed under the application of the elementary gate and then, after symmetrization of the site tensors, the environment is updated. In the second approach, a cluster optimization is performed on a finite (periodic) cluster, maximizing the overlap of the exact time-evolved state with a symmetric finite-size PEPS ansatz. Observables are then computed on the infinite lattice contracting the infinite-PEPS (iPEPS) by CTMRG. We show that the variational schemes outperform the SU method and remain accurate over a significant time interval before hitting the entanglement barrier. Studying the spectrum of the transfer matrix, we find that the asymptotic correlations are very well preserved under time evolution, including the critical nature of the singlet correlations, as expected from the Lieb-Robinson (LR) bound theorem. Consistently, the system (asymptotic) boundary is found to be described by the same Conformal Field Theory of central charge c = 1 c = 1 during time evolution. We also compute the time-evolution of the short distance spin-spin correlations and estimate the LR velocity.
This chapter discusses the properties of tree tensor network states and the methods for evaluating the ground state and thermodynamic properties of quantum lattice models on a Bethe lattice or, more generally, a Husimi lattice. It starts with a brief discussion of the canonical form of a tree tensor network state. Then, a canonicalization scheme is proposed. To calculate the ground state through the imaginary time evolution, the full and simple update methods are introduced to renormalize the local tensors. Finally, as the correlation length of a quantum system is finite even at a critical point, an accurate and efficient method is described to compute the thermodynamic quantities of quantum lattice models on the Bethe lattice.