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arXiv:1804.03613 · Physical review. B./Physical review. B

Efficient approximation for global functions of matrix product operators

Moritz August, Mari Carmen Bañuls

Published Aug 15, 2018• 1 citation
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Abstract

Building on a previously introduced block Lanczos method, we demonstrate how to approximate any operator function of the form $\mathsf{Tr}f(A)$ when the argument $A$ is given as a Hermitian matrix product operator (MPO). This gives access to quantities that, depending on the full spectrum, are difficult to access for standard tensor network techniques, such as the von Neumann entropy and the trace norm of an MPO. We present a modified, more efficient strategy for computing thermal properties of short- or long-range Hamiltonians, and we illustrate the performance of the method with numerical results for the thermal equilibrium states of the Lipkin-Meshkov-Glick and Ising Hamiltonians.

DOI
10.1103/physrevb.98.075128

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Building on a previously introduced block Lanczos method, we demonstrate how to approximate any operator function of the form Trf(A)\mathsf{Tr}f(A)Trf(A) when the argument AAA is given as a Hermitian matrix product operator (MPO). This gives access to quantities that, depending on the full spectrum, are difficult to access for standard tensor network techniques, such as the von Neumann entropy and the trace norm of an MPO. We present a modified, more efficient strategy for computing thermal properties of short- or long-range Hamiltonians, and we illustrate the performance of the method with numerical results for the thermal equilibrium states of the Lipkin-Meshkov-Glick and Ising Hamiltonians.
Trace norms are fundamental to quantum information theory, yet in many-body systems their evaluation remains a major computational bottleneck, as it generally requires diagonalizing exponentially large operators. Here, we overcome this bottleneck by introducing a controlled tensor-network algorithm for estimating the trace norm of matrix product operators without full diagonalization. The key idea is to combine Zolotarev's rational approximation to the sign function with a variational formulation solved using a density-matrix-renormalization-group-like algorithm. The resulting approximation is systematically improvable, with its accuracy controlled by the rational approximation parameters and the spectral weight near zero. Beyond the reach of exact diagonalization, we demonstrate controlled trace-norm calculations for entanglement negativity, quantum fidelity and quantum Fisher information, achieving substantially improved accuracy over polynomial-based Lanczos approaches. Our results establish trace-norm-based quantities as practical tensor-network observables, opening a route toward tensor-network studies of quantum information in mixed states.
We present a method to approximate functionals Tr f(A)\text{Tr} \, f(A)Trf(A) of very high-dimensional hermitian matrices AAA represented as Matrix Product Operators (MPOs). Our method is based on a reformulation of a block Lanczos algorithm in tensor network format. We state main properties of the method and show how to adapt the basic Lanczos algorithm to the tensor network formalism to allow for high-dimensional computations. Additionally, we give an analysis of the complexity of our method and provide numerical evidence that it yields good approximations of the entropy of density matrices represented by MPOs while being robust against truncations.
We recently introduced a method to approximate functions of Hermitian Matrix Product Operators or Tensor Trains that are of the form Trf(A)\mathsf{Tr} f(A)Trf(A). Functions of this type occur in several applications, most notably in quantum physics. In this work we aim at extending the theoretical understanding of our method by showing several properties of our algorithm that can be used to detect and correct errors in its results. Most importantly, we show that there exists a more computationally efficient version of our algorithm for certain inputs. To illustrate the usefulness of our finding, we prove that several classes of spin Hamiltonians in quantum physics fall into this input category. We finally support our findings with numerical results obtained for an example from quantum physics.
We propose a tensor-network (TN) approach for solving classical optimization problems that is inspired by spectral filtering and sampling on quantum states. We first shift and scale an Ising Hamiltonian of the cost function so that all eigenvalues become non-negative and the ground states correspond to the the largest eigenvalues, which are then amplified by power iteration. We represent the transformed Hamiltonian as a matrix product operator (MPO) and form an immense power of this object via truncated MPO-MPO contractions, embedding the resulting operator into a matrix product state for sampling in the computational basis. In contrast to the density-matrix renormalization group, our approach provides a straightforward route to systematic improvement by increasing the bond dimension and is better at avoiding local minima. We also study the performance of this power method in the context of a higher-order Ising Hamiltonian on a heavy-hexagonal lattice, making a comparison with simulated annealing. These results highlight the potential of quantum-inspired algorithms for solving optimization problems and provide a baseline for assessing and developing quantum algorithms.
We present a new method for compressing matrix product operators (MPOs) which represent sums of local terms, such as Hamiltonians. Just as with area law states, such local operators may be fully specified with a small amount of information per site. Standard matrix product state (MPS) tools are ill-suited to this case, due to extensive Schmidt values that coexist with intensive ones, and Jordan blocks in the transfer matrix. We ameliorate these issues by introducing an ``almost Schmidt decomposition'' that respects locality. Our method is ``$\ensuremath{\varepsilon}$-close'' to the accuracy of MPS-based methods for finite MPOs, and extends seamlessly to the thermodynamic limit, where MPS techniques are inapplicable. In the framework of control theory, our method generalizes Kung's algorithm for model order reduction. Several examples are provided, including an all-MPO version of the operator recursion method (Lanczos algorithm) directly in the thermodynamic limit. All results are accompanied by practical algorithms, well-suited for the large MPOs that arise in DMRG for long-range or quasi-2D models.
Matrix product states and matrix product operators (MPOs) are one-dimensional tensor networks that underlie the modern density matrix renormalization group (DMRG) algorithm. The use of MPOs accounts for the high level of generality and wide range of applicability of DMRG. However, current algorithms for two-dimensional (2D) tensor network states, known as projected entangled-pair states, rarely employ the associated 2D tensor network operators, projected entangled-pair operators (PEPOs), due to their computational cost and conceptual complexity. To lower these two barriers, we describe how to reformulate a PEPO into a set of tensor network operators that resemble MPOs by considering the different sets of local operators that are generated from sequential bipartitions of the 2D system. The expectation value of a PEPO can then be evaluated on the fly using only the action of MPOs and generalized MPOs at each step of the approximate contraction of the 2D tensor network. This technique allows for the simpler construction and more efficient energy evaluation of 2D Hamiltonians that contain finite-range interactions, and provides an improved strategy to encode long-range interactions that is orders of magnitude more accurate and efficient than existing schemes.