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arXiv:1411.7950 · Physical Review A

Connecting entanglement in time and space: Improving the folding algorithm

M. B. Hastings, Raghu Mahajan

Published Mar 12, 2015• 61 citations
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Abstract

The ``folding algorithm'' [M. C. Ba\~nuls, M. B. Hastings, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 102, 240603 (2009)] is a matrix product state algorithm for simulating quantum systems that involves a spatial evolution of a matrix product state. Hence, the computational effort of this algorithm is controlled by the temporal entanglement. We show that this temporal entanglement is, in many cases, equal to the spatial entanglement of a modified Hamiltonian. This inspires a modification to the folding algorithm, which we call the ``hybrid algorithm.'' We find that this leads to improved accuracy for the same numerical effort. We then use these algorithms to study relaxation in a transverse plus parallel field Ising model, finding persistent quasiperiodic oscillations for certain choices of initial conditions.

DOI
10.1103/physreva.91.032306

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The folding algorithm'' [M. C. Ba\~nuls, M. B. Hastings, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 102, 240603 (2009)] is a matrix product state algorithm for simulating quantum systems that involves a spatial evolution of a matrix product state. Hence, the computational effort of this algorithm is controlled by the temporal entanglement. We show that this temporal entanglement is, in many cases, equal to the spatial entanglement of a modified Hamiltonian. This inspires a modification to the folding algorithm, which we call the hybrid algorithm.'' We find that this leads to improved accuracy for the same numerical effort. We then use these algorithms to study relaxation in a transverse plus parallel field Ising model, finding persistent quasiperiodic oscillations for certain choices of initial conditions.
We analyze the recently developed folding algorithm (Bañuls et al 2009 Phys. Rev. Lett. 102 240603) for simulating the dynamics of infinite quantum spin chains and we relate its performance to the kind of entanglement produced under the evolution of product states. We benchmark the accomplishments of this technique with respect to alternative strategies using Ising Hamiltonians with transverse and parallel fields, as well as XY models. Also, we evaluate its capability of finding ground and thermal equilibrium states.
The transverse folding algorithm [M. C. Ba\~nuls et al., Phys. Rev. Lett. 102, 240603 (2009)] is a tensor network method to compute time-dependent local observables in out-of-equilibrium quantum spin chains that can overcome the limitations of matrix product states when entanglement grows slower in the time than in the space direction. We present a contraction strategy that makes use of the exact light cone structure of the tensor network representing the observables. The strategy can be combined with the hybrid truncation proposed for global quenches by Hastings and Mahajan Phys. Rev. A 91, 032306 (2015), which significantly improves the efficiency of the method. We demonstrate the performance of this transverse light cone contraction also for transport coefficients, and discuss how it can be extended to other dynamical quantities.
Quantum entanglement offers a fundamentally different approach to matrix computation, potentially surpassing classical methods in certain scenarios. This paper proposes a novel optimization algorithm specifically tailored for quantum entanglement, aiming to enhance the efficiency of matrix operations. We introduce a quantum state manipulation scheme leveraging entanglement to achieve a significant speedup in calculations, addressing limitations inherent in current quantum computing architectures. The core mechanism centers around strategically manipulating entangled qubit states to represent and manipulate matrix elements, offering a pathway to dramatically accelerate matrix computations. The algorithm's novelty lies in its incorporation of quantum entanglement as a key computational resource, providing a unique advantage over existing optimization strategies. We detail the algorithm's architecture, the quantum circuit implementation, and preliminary results demonstrating improved performance compared to classical methods. This work explores the potential of entanglement to fundamentally alter matrix computation, presenting a promising avenue for advancing quantum computing capabilities.
The generalization of matrix product states (MPS) to continuous systems, as proposed in the breakthrough Letter of Verstraete and Cirac [Phys. Rev. Lett. 104, 190405 (2010).PRLTAO0031-900710.1103/PhysRevLett.104.190405], provides a powerful variational ansatz for the ground state of strongly interacting quantum field theories in one spatial dimension. A continuous MPS (cMPS) approximation to the ground state can be obtained by simulating a Euclidean time evolution. In this Letter we propose a cMPS optimization algorithm based instead on energy minimization by gradient methods and demonstrate its performance by applying it to the Lieb-Liniger model (an integrable model of an interacting bosonic field) directly in the thermodynamic limit. We observe a very significant computational speed-up, of more than 2 orders of magnitude, with respect to simulating a Euclidean time evolution. As a result, a much larger cMPS bond dimension D can be reached (e.g., D=256 with moderate computational resources), thus helping unlock the full potential of the cMPS representation for ground state studies.
This paper investigates the potential of topological quantum entanglement, a paradigm shift from traditional qubit-based systems, by exploring a novel 'folding' process to generate and manipulate persistent entanglement patterns. We propose a method that dynamically shapes the topology of a quantum system, creating persistent entanglement without the need for explicit qubit manipulation. This approach offers a fundamentally new pathway to robust and scalable entanglement, potentially unlocking applications in quantum computing and communication. The core claim is to develop a robust method for generating and manipulating topological quantum states, extending entanglement beyond traditional qubit-based systems. This research utilizes a dynamically shaped 'folding' process to establish persistent entanglement without requiring traditional qubit manipulation. The resulting system demonstrates enhanced stability and adaptability, offering a significant advancement in the field of topological quantum entanglement.
The simulation of strongly correlated quantum systems on classical computers is an essential tool for investigating and understanding quantum materials.Due to the inherent "curse of dimensionality" of quantum systems, such simulations pose a challenging task, and have spurred the development of several computational approaches and approximations.For quasi one-dimensional lattice systems, the density matrix renormalization group (DMRG) framework (Schollwöck, 2005;White, 1992White, , 1993) has emerged as one of the most successful methods.Modern formulations are based on matrix product states (MPS) and operators (MPO) (Schollwöck, 2011): briefly speaking, a matrix product state is an Ansatz for the quantum many-body wavefunction amendable to efficient numerical manipulation.The favorable approximation properties of matrix product states have been theoretically proven for ground states of gapped Hamiltonians with local interactions (Hastings, 2007;Verstraete & Cirac, 2006), and recent work has extended this to quantum systems at finite temperature (Barthel, 2017).Over the years many variations and generalizations of the DMRG framework have been devised (Barthel, 2013;Stoudenmire & White, 2010;Verstraete, Murg, & Cirac, 2008;Vidal, 2003Vidal, , 2008)), and further developments and improvements of associated algorithms remain an active field of research.