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arXiv:2001.11882 · SciPost Physics Core

Tangent-space methods for truncating uniform MPS

Bram Vanhecke, Maarten Van Damme, Jutho Haegeman, Laurens Vanderstraeten, Frank Verstraete

Published Feb 19, 2021• 2 citations
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Abstract

An essential primitive in quantum tensor network simulations is the problem of approximating a matrix product state with one of a smaller bond dimension. This problem forms the central bottleneck in algorithms for time evolution and for contracting projected entangled pair states. We formulate a tangent-space based variational algorithm to achieve this goal for uniform (infinite) matrix product states. The algorithm exhibits a favourable scaling of the computational cost, and we demonstrate its usefulness by several examples involving the multiplication of a matrix product state with a matrix product operator.

DOI
10.21468/scipostphyscore.4.1.004

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An essential primitive in quantum tensor network simulations is the problem of approximating a matrix product state with one of a smaller bond dimension. This problem forms the central bottleneck in algorithms for time evolution and for contracting projected entangled pair states. We formulate a tangent-space based variational algorithm to achieve this goal for uniform (infinite) matrix product states. The algorithm exhibits a favourable scaling of the computational cost, and we demonstrate its usefulness by several examples involving the multiplication of a matrix product state with a matrix product operator.
A central primitive in quantum tensor network simulations is the problem of approximating a matrix product state with one of a lower bond dimension. This problem forms the central bottleneck in algorithms for time evolution and for contracting projected entangled pair states. We formulate a tangent-space based variational algorithm to achieve this for uniform (infinite) matrix product states. The algorithm exhibits a favourable scaling of the computational cost, and we demonstrate its usefulness by several examples involving the multiplication of a matrix product state with a matrix product operator.
Time evolution of an infinite two-dimensional (2D) many body quantum lattice system can be described by the Suzuki-Trotter decomposition applied to the infinite projected entangled pair state (iPEPS). Each Trotter gate increases the bond dimension of the tensor network, DDD, that has to be truncated back in a way that minimizes a suitable error measure. This paper goes beyond simplified error measures--like the one used in the full update, the simple update, and their intermediate neighborhood tensor update--and directly maximizes an overlap between the exact iPEPS with the increased bond dimension and the new iPEPS with the truncated one. The optimization is performed in a tangent space of the iPEPS variational manifold. This gradient tensor update is benchmarked by a simulation of a sudden quench of a transverse field in the 2D quantum Ising model and the quantum Kibble-Zurek mechanism in the same 2D system.
Simulation of quantum systems is challenging due to the exponential size of the state space. Tensor networks provide a systematically improvable approximation for quantum states. 2D tensor networks such as Projected Entangled Pair States (PEPS) are well-suited for key classes of physical systems and quantum circuits. However, direct contraction of PEPS networks has exponential cost, while approximate algorithms require computations with large tensors. We propose new scalable algorithms and software abstractions for PEPS-based methods, accelerating the bottleneck operation of contraction and refactorization of a tensor subnetwork. We employ randomized SVD with an implicit matrix to reduce cost and memory footprint asymptotically. Further, we develop a distributed-memory PEPS library and study accuracy and efficiency of alternative algorithms for PEPS contraction and evolution on the Stampede2 supercomputer. We also simulate a popular near-term quantum algorithm, the Variational Quantum Eigensolver (VQE), and benchmark Imaginary Time Evolution (ITE), which compute ground states of Hamiltonians.
Classical simulation of quantum circuits is an essential tool in quantum information science, but its applicability is constrained by the exponential growth of the Hilbert space and the entanglement structure of quantum states. In this work, we introduce Matrix Product Evolution (MPE), a tensor-train representation of quantum circuits constructed along the circuit depth rather than along the qubit index. Within this formulation, the simulation of a quantum circuit is modeled as the contraction of multiple MPE tensors. We develop an efficient contraction strategy based on a zip-up procedure to carry out this contraction in practice. We investigate the numerical behavior of this MPE-based contraction framework through simulations of random quantum circuits and the time evolution of a quantum many-body state. Our results characterize the growth of temporal bond dimensions, clarify how post-selection modifies the contraction cost and approximation accuracy, and identify regimes in which depth-oriented tensor-network contractions provide a useful complement to standard MPS-based simulation approaches.
An infinite projected entangled-pair state (iPEPS) is a powerful variational tensor network ansatz for two-dimensional ground states in the thermodynamic limit, and can be seen as a natural generalization of a matrix-product state to two dimensions. One of the main challenges in iPEPS simulations is the optimization of the tensors, i.e., finding the optimal variational parameters, in order to have the best representation of the ground state of a given Hamiltonian. The author presents a variational optimization scheme, in which the energy is minimized in an iterative way by sweeping over all the tensors in the ansatz, in a similar spirit as done in the density-matrix renormalization group method. Benchmark results for challenging problems are presented that show that the variational scheme yields considerably more accurate results than the previously best imaginary-time evolution algorithm, with a similar computational cost and with a faster convergence towards the ground state.
In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.