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CRC Press eBooks

Simulations Based on Matrix Product States and Projected Entangled Pair States

V. Murg, Ignacio Cirac, Frank Verstraete

Published Nov 2, 2010• 0 citations
Publisher

Abstract

Valentin Murg Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany Ignacio Cirac Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany Frank Verstraete Fakulta¨t fu¨r Physik, Universita¨t Wien, Boltzmanngasse 5, A-1090 Wien, Austria The formulation of renormalization group methods in terms of matrix product states (MPS) has the big advantage that it allows one to describe time evolution. This opens up the possibility of simulating the non-equilibrium properties of spin chains, a topic that is currently very relevant given the recent breakthroughs in creating strongly-correlated quantum spin systems, for example in optical lattices [1].

DOI
10.1201/b10273-32

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Simulations Based on Matrix Product States and Projected Entangled Pair States

V. Murg, Ignacio Cirac, Frank Verstraete

Series in condensed matter physics•Nov 2, 2010•0 citations
Valentin Murg Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany Ignacio Cirac Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany Frank Verstraete Fakulta¨t fu¨r Physik, Universita¨t Wien, Boltzmanngasse 5, A-1090 Wien, Austria The formulation of renormalization group methods in terms of matrix product states (MPS) has the big advantage that it allows one to describe time evolution. This opens up the possibility of simulating the non-equilibrium properties of spin chains, a topic that is currently very relevant given the recent breakthroughs in creating strongly-correlated quantum spin systems, for example in optical lattices [1].
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.
Time-dependent density matrix renormalization group method with a matrix product ansatz is employed for explicit computation of non-equilibrium steady state density operators of several integrable and non-integrable quantum spin chains, which are driven far from equilibrium by means of Markovian couplings to external baths at the two ends. It is argued that even though the time-evolution can not be simulated efficiently due to fast entanglement growth, the steady states in and out of equilibrium can be typically accurately approximated, so that chains of length of the order n ~ 100 are accessible. Our results are demonstrated by performing explicit simulations of steady states and calculations of energy/spin densities/currents in several problems of heat and spin transport in quantum spin chains. Previously conjectured relation between quantum chaos and normal transport is re-confirmed with high acurracy on much larger systems.
The Thesis explores the application of Matrix Product States (MPS) in the domain of quantum many-body systems and quantum computing, show casing the versatility and effectiveness of MPS for addressing complex com putational challenges. The opening chapter introduces essential tools for Tensor Networks, addressing the curse of dimensionality and elucidating the main characteristics of Matrix Product States and Matrix Product Op erators. The discussion encompasses also important numerical techniques, including Density Matrix Renormalization Group and Time Evolving Block Decimation. Furthermore, the relation between MPS and quantum circuits is discussed. The subsequent Chapters delve into specific applications of MPS. In the second Chapter, we introduce a Tensor Network ansatz inspired by the backflow transformation for correlated systems. This extension of the MPS representation ensures an area law for entanglement in dimensions one or greater. We employ an optimization scheme combining DMRG and variational Monte Carlo algorithms for efficient ground-state search. Bench marking against spin models demonstrates high accuracy. The third Chapter explores quantum annealing for optimizing complex classical spin Hamil tonians, as the Hopfield model and the binary perceptron. We introduce an efficient Tensor Network representation for the adiabatic time evolution of quantum annealing, enabling scalable classical simulations. The use of MPS in mitigating Trotter errors and mapping to quantum circuits is also explored. In the fourth Chapter we present a novel method for evaluating the amount of nonstabilizerness (also known as quantum magic) contained in an MPS. We overcome the exponentially hard evaluation of this quan tity by employing a simple perfect sampling technique of Pauli string. Our innovative MPS approach enables efficient computation. Benchmarked on magic states and the quantum Ising chain ground state, this method offers easy access to the non-equilibrium dynamics of nonstabilizerness following a quantum quench. The Appendices provide additional insights, including: an introduction to variational Monte Carlo, technical details on MPS simula tions for the quantum annealing and a comprehensive overview on stabilizer formalism and nonstabilizerness.