Unifying projected entangled pair state contractions

The approximate contraction of a tensor network of projected entangled pair states (PEPS) is a fundamental ingredient of any PEPS algorithm, required for the optimization of the tensors in ground state search or time evolution, as well as for the evaluation of expectation values. An exact contraction is in general impossible, and the choice of the approximating procedure determines the efficiency and accuracy of the algorithm. We analyze different previous proposals for this approximation, and show that they can be understood via the form of their environment, i.e. the operator that results from contracting part of the network. This provides physical insight into the limitation of various approaches, and allows us to introduce a new strategy, based on the idea of clusters, that unifies previous methods. The resulting contraction algorithm interpolates naturally between the cheapest and most imprecise and the most costly and most precise method. We benchmark the different algorithms with finite PEPS, and show how the cluster strategy can be used for both the tensor optimization and the calculation of expectation values. Additionally, we discuss its applicability to the parallelization of PEPS and to infinite systems.

[1]  Glen Evenbly,et al.  Improving the efficiency of variational tensor network algorithms , 2014 .

[2]  J. Ignacio Cirac,et al.  Purifications of multipartite states: limitations and constructive methods , 2013, 1308.1914.

[3]  G. Guo,et al.  Size consistency of tensor network methods for quantum many-body systems , 2013, 1304.6150.

[4]  F. Verstraete,et al.  Cluster update for tensor network states , 2011, 1110.4362.

[5]  M. Troyer,et al.  Stripes in the two-dimensional t-J model with infinite projected entangled-pair states , 2011, 1104.5463.

[6]  J. Ignacio Cirac,et al.  Entanglement spectrum and boundary theories with projected entangled-pair states , 2011, 1103.3427.

[7]  Ling Wang,et al.  Time evolution of projected entangled pair states in the single-layer picture , 2011 .

[8]  Ling Wang,et al.  Monte Carlo simulation with tensor network states , 2010, 1010.5450.

[9]  G. Vidal,et al.  Simulation of fermionic lattice models in two dimensions with projected entangled-pair states: Next-nearest neighbor Hamiltonians , 2010, 1008.3937.

[10]  F. Verstraete,et al.  Fermionic implementation of projected entangled pair states algorithm , 2010, 1003.2743.

[11]  G. Guo,et al.  Efficient numerical method to calculate the three-tangle of mixed states , 2009, 1001.0067.

[12]  Bela Bauer,et al.  Simulation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states , 2009, 0912.0646.

[13]  J. Cirac,et al.  Simulating two- and three-dimensional frustrated quantum systems with string-bond states , 2009, 0908.4036.

[14]  J. Eisert,et al.  Unitary circuits for strongly correlated fermions , 2009, 0905.0669.

[15]  J. Ignacio Cirac,et al.  Fermionic projected entangled pair states , 2009, 0904.4667.

[16]  Jens Eisert,et al.  Contraction of fermionic operator circuits and the simulation of strongly correlated fermions , 2009, 0907.3689.

[17]  Roman Orus,et al.  Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction , 2009, 0905.3225.

[18]  F. Verstraete,et al.  Exploring frustrated spin systems using projected entangled pair states , 2009, 0901.2019.

[19]  Z. Y. Xie,et al.  Second renormalization of tensor-network states. , 2008, Physical review letters.

[20]  T. Xiang,et al.  Accurate determination of tensor network state of quantum lattice models in two dimensions. , 2008, Physical review letters.

[21]  Hui Li,et al.  Entanglement spectrum as a generalization of entanglement entropy: identification of topological order in non-Abelian fractional quantum Hall effect states. , 2008, Physical review letters.

[22]  F. Verstraete,et al.  Matrix product operator representations , 2008, 0804.3976.

[23]  F. Verstraete,et al.  Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems , 2008, 0907.2796.

[24]  Michael M. Wolf,et al.  Sequentially generated states for the study of two-dimensional systems , 2008, 0802.2472.

[25]  Norbert Schuch,et al.  Simulation of quantum many-body systems with strings of operators and Monte Carlo tensor contractions. , 2008, Physical review letters.

[26]  F. Verstraete,et al.  Classical simulation of infinite-size quantum lattice systems in two spatial dimensions. , 2007, Physical review letters.

[27]  G. Vidal.  Class of quantum many-body states that can be efficiently simulated. , 2006, Physical review letters.

[28]  A W Sandvik,et al.  Variational quantum Monte Carlo simulations with tensor-network states. , 2007, Physical review letters.

[29]  F. Verstraete,et al.  Computational complexity of projected entangled pair states. , 2007, Physical review letters.

[30]  S. Todo,et al.  The ALPS project release 2.0: open source software for strongly correlated systems , 2011, 1101.2646.

[31]  Michael Levin,et al.  Tensor renormalization group approach to two-dimensional classical lattice models. , 2006, Physical review letters.

[32]  F. Verstraete,et al.  Variational study of hard-core bosons in a two-dimensional optical lattice using projected entangled pair states , 2006, cond-mat/0611522.

[33]  G. Vidal.  Classical simulation of infinite-size quantum lattice systems in one spatial dimension. , 2006, Physical review letters.

[34]  G. Vidal,et al.  Classical simulation of quantum many-body systems with a tree tensor network , 2005, quant-ph/0511070.

[35]  M. Hastings.  Solving gapped Hamiltonians locally , 2005, cond-mat/0508554.

[36]  F. Verstraete,et al.  Efficient evaluation of partition functions of inhomogeneous many-body spin systems. , 2005, Physical review letters.

[37]  D Porras,et al.  Density matrix renormalization group and periodic boundary conditions: a quantum information perspective. , 2004, Physical review letters.

[38]  F. Verstraete,et al.  Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions , 2004, cond-mat/0407066.

[39]  G. Vidal,et al.  Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm. , 2004, Physical review letters.

[40]  F. Verstraete,et al.  Matrix product density operators: simulation of finite-temperature and dissipative systems. , 2004, Physical review letters.

[41]  S. White,et al.  Real-time evolution using the density matrix renormalization group. , 2004, Physical review letters.

[42]  G. Vidal,et al.  A pr 2 00 4 Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces , 2022 .

[43]  G. Vidal.  Efficient simulation of one-dimensional quantum many-body systems. , 2003, Physical review letters.

[44]  G. Vidal.  Efficient classical simulation of slightly entangled quantum computations. , 2003, Physical review letters.

[45]  S. Todo,et al.  Cluster algorithms for general-S quantum spin systems. , 1999, Physical review letters.

[46]  H. Everts,et al.  First excitations of the spin 1/2 Heisenberg antiferromagnet on the kagomé lattice , 1998, cond-mat/9802168.

[47]  Östlund,et al.  Thermodynamic limit of density matrix renormalization. , 1995, Physical review letters.

[48]  T. Nishino.  Density Matrix Renormalization Group Method for 2D Classical Models , 1995, cond-mat/9508111.

[49]  A ug 1 99 5 Density Matrix Renormalization Group Method for 2 D Classical Models , 1995 .

[50]  J. M. Haile,et al.  Molecular dynamics simulation : elementary methods / J.M. Haile , 1992 .

[51]  M. Fannes,et al.  Finitely correlated states on quantum spin chains , 1992 .

[52]  Wang,et al.  Replica Monte Carlo simulation of spin glasses. , 1986, Physical review letters.

[53]  E. L. Battiste,et al.  Computer Science And Statistics , 1978, ACM Annual Conference.