Time evolution#

Evolution step#

Time evolution in YASTN operates via the Trotterization of the evolution operator to a set of gates. Each gate acts on a path of sites, single site for a local gate, nearest-neighbor sites for a two-site gate, etc. A single evolution step consists of the application of trotterized gates, together with truncation of the bond dimension after each application of a non-local gate. For a detailed explanation, please refer to the Basic concepts/Time evolution.

The main method for performing a time evolution step is yastn.tn.fpeps.evolution_step_(). This function requires an environment containing the PEPS state (updated in place) and provides control over bond truncation via different ways to calculate bond metric. Four classes which support this are:

  • yastn.tn.fpeps.EnvNTU: Employ small local clusters, which can be contracted numerically exactly, resulting in a stable and positively defined bond metric.

  • yastn.tn.fpeps.EnvBP: Employ belief propagation, either to define a bipartite bond metric, or to gauge NTU-like clusters.

  • yastn.tn.fpeps.EnvCTM: Employ CTMRG environment to calculate the bond metric, with information from the entire network, allowing for a fast full update approach.

  • yastn.tn.fpeps.EnvApproximate: Employ local clusters of sizes beyond exact contraction, utilizing approximate boundary MPS to calculate the bond metric.

yastn.tn.fpeps.evolution_step_(env, gates, opts_svd, method='mpo', fix_metric=0, pinv_cutoffs=(1e-12, 1e-11, 1e-10, 1e-09, 1e-08, 1e-07, 1e-06, 1e-05, 0.0001), max_iter=100, tol_iter=1e-13, initialization='EAT_SVD')[source]#

Perform a single step of PEPS evolution by applying a list of gates. Truncate bond dimension after each application of a two-site gate.

Parameters:

  • env (EnvNTU | EnvCTM | EnvApproximate) – Environment class containing PEPS state (updated in place), and a method to calculate bond metric tensors employed during truncation.

  • gates (yastn.tn.fpeps.Gates) – The gates to be applied to PEPS.

  • opts_svd (dict | Sequence[dict]) – Options passed to yastn.linalg.svd_with_truncation() which are used to initially truncate the bond dimension before it is further optimized iteratively. In particular, it fixes bond dimensions (potentially, sectorial). It is possible to provide a list of dicts (with decreasing bond dimensions), in which case the truncation is done gradually in a few steps.

  • method (str) – For 'NN', split multi-site gates into a series of nearest-neighbor gates. Otherwise, apply mpo-gate first, and then sequentially truncate enlarged bonds (the default).

  • fix_metric (int | None) – Error measure of the metric tensor is a sum of: the norm of its non-hermitian part and absolute value of the most negative eigenvalue (if present). If fix_metric is a number, replace eigenvalues smaller than the error_measure with fix_metric * error_measure. Sensible values of fix_metric are \(0\) and \(1\). The default is \(0\). If None, do not perform eigh to test for negative eigenvalues and do not fix metric.

  • pinv_cutoffs (Sequence[float] | float) – List of pseudo-inverse cutoffs. The one that gives the smallest truncation error is used during iterative optimizations and EAT initialization.

  • max_iter (int) – The maximal number of iterative steps for each truncation optimization.

  • tol_iter (int) – Tolerance of truncation_error to stop iterative optimization.

  • initialization (str) – Tested initializations of iterative optimization. The one resulting in the smallest error is selected. Possible options are ‘SVD’ (svd initialization only), ‘EAT’ (EAT optimization only), ‘SVD_EAT’ (tries both).

Returns:

List of Namedtuple containing fields:

  • bond bond where the gate is applied.

  • truncation_error relative norm of the difference between untruncated and truncated bonds, calculated in metric specified by env.

  • best_method initialization/optimization method giving the best truncation_error. Possible values are ‘eat’, ‘eat_opt’, ‘svd’, ‘svd_opt’.

  • nonhermitian_part norm of the non-hermitian part of the bond metric, normalized by the bond metric norm. Estimator of metric error.

  • min_eigenvalue the smallest bond metric eigenvalue, normalized by the bond metric norm. Can be negative which gives an estimate of metric error.

  • wrong_eigenvalues a fraction of bond metrics eigenvalues that were below the error threshold; and were modified according to fix_metric argument.

  • eat_metric_error error of approximating metric tensor by a product via SVD-1 in EAT initialization.

  • truncation_errors dict with truncation_error-s for all tested initializations/optimizations.

  • iterations dict with number of iterations to converge iterative optimization.

  • pinv_cutoffs dict with optimal pinv_cutoffs for methods where pinv appears.

Return type:

list[Evolution_out(NamedTuple)]

Auxiliary functions, such as yastn.tn.fpeps.accumulated_truncation_error(), assist with tracking cumulative truncation error across multiple evolution steps, facilitating error analysis in time evolution simulations.

yastn.tn.fpeps.accumulated_truncation_error(infoss, statistics='mean')[source]#

Return accumulated truncation error \(\Delta\) calcuated from evolution output statistics.

\(\Delta = \sum_{steps} statistics_{bond} [\sum_{gate \in bond} truncation\_error(gate, step)]\), where statistics is mean or max.

Gives an estimate of errors accumulated during time evolution.

Parameters:

  • infoss (Sequence[Sequence[Evolution_out]]) – list of outputs of evolution_step_().

  • statistics (str) – ‘max’ or ‘mean’, whether to take the maximal value or a mean over the bonds in the lattice.

Example

infoss = []
for step in range(number_of_steps):
    infos = fpeps.evolution_step_(env, gates, opt_svd)
    infoss.append(infos)

# Accumulated runcation error
Delta = fpeps.accumulated_truncation_error(infoss)

To apply a gate to a PEPS, without truncating the bond dimension, one can use yastn.tn.fpeps.Peps.apply_gate_(). Truncation of the bond dimension can be done later, e.g., via yastn.tn.fpeps.truncate_(). Both operations modify the PEPS in place.

Peps.apply_gate_(gate)[source]#

Apply gate to PEPS state psi, changing respective tensors in place.

fpeps.truncate_(opts_svd, bond=None, fix_metric=0, pinv_cutoffs=(1e-12, 1e-11, 1e-10, 1e-09, 1e-08, 1e-07, 1e-06, 1e-05, 0.0001), max_iter=100, tol_iter=1e-13, initialization='EAT_SVD')#

Truncate virtual bond dimensions of PEPS.

Parameters:

  • env (EnvNTU | EnvCTM | EnvApproximate) – Environment class containing PEPS state (updated in place), and a method to calculate bond metric tensors employed during truncation.

  • opts_svd (dict | Sequence[dict]) – Options passed to yastn.linalg.svd_with_truncation() which are used to initially truncate the bond dimension before it is further optimized iteratively. In particular, it fixes bond dimensions (potentially, sectorial). It is possible to provide a list of dicts (with decreasing bond dimensions), in which case the truncation is done gradually in a few steps.

  • bond (tuple[Site, Site] | None) – Specify single bond to be truncated. If None, truncate all bonds. The default is None.

  • fix_metric (int | None) – Error measure of the metric tensor is a sum of: the norm of its non-hermitian part and absolute value of the most negative eigenvalue (if present). If fix_metric is a number, replace eigenvalues smaller than the error_measure with fix_metric * error_measure. Sensible values of fix_metric are \(0\) and \(1\). The default is \(0\). If None, do not perform eigh to test for negative eigenvalues and do not fix metric.

  • pinv_cutoffs (Sequence[float] | float) – List of pseudo-inverse cutoffs. The one that gives the smallest truncation error is used during iterative optimizations and EAT initialization.

  • max_iter (int) – The maximal number of iterative steps for each truncation optimization.

  • tol_iter (int) – Tolerance of truncation_error to stop iterative optimization.

  • initialization (str) – Tested initializations of iterative optimization. The one resulting in the smallest error is selected. Possible options are ‘SVD’ (svd initialization only), ‘EAT’ (EAT optimization only), ‘SVD_EAT’ (tries both).

Returns:

Namedtuple containing fields:

  • bond bond where the gate is applied.

  • truncation_error relative norm of the difference between untruncated and truncated bonds, calculated in metric specified by env.

  • best_method initialization/optimization method giving the best truncation_error. Possible values are ‘eat’, ‘eat_opt’, ‘svd’, ‘svd_opt’.

  • nonhermitian_part norm of the non-hermitian part of the bond metric, normalized by the bond metric norm. Estimator of metric error.

  • min_eigenvalue the smallest bond metric eigenvalue, normalized by the bond metric norm. Can be negative which gives an estimate of metric error.

  • wrong_eigenvalues a fraction of bond metrics eigenvalues that were below the error threshold; and were modified according to fix_metric argument.

  • eat_metric_error error of approximating metric tensor by a product via SVD-1 in EAT initialization.

  • truncation_errors dict with truncation_error-s for all tested initializations/optimizations.

  • iterations dict with number of iterations to converge iterative optimization.

  • pinv_cutoffs dict with optimal pinv_cutoffs for methods where pinv appears.

Return type:

Evolution_out(NamedTuple)

Gates#

yastn.tn.fpeps.evolution_step_() takes a list of gates to be applied on PEPS

class yastn.tn.fpeps.Gate(G: tuple = None, sites: tuple = None)[source]#

Gate to be applied on Peps state.

G contains operators for respective sites.

Operator can be given in the form of an MPO (yastn.tn.mps.MpsMpoOBC) of the same length as the number of provided sites. Sites should form a continuous path in the two-dimensional PEPS lattice. The fermionic order of MPO should be linear, with the first MPO site being first in the fermionic order, irrespective of the provided sites.

For a two-site operator acting on sites beyond nearest neighbor, it can be provided as G with two elements, and sites forming a path between the end sites where the provided elements of G will act.

It is also possible to provide G as a list of tensots. In this case, the convention of legs is (ket, bra, virtual_0, virtual_1) – i.e., the first two legs are always physical (operator) legs. For one site, there are no virtual legs. For two or more sites, the first and last elements of G have one virtual leg (3 in total). For three sites or more, the middle elements of G have two virtual legs connecting, respectively, to the preceding and following elements of G.

Create new instance of Gate(G, sites)

An auxiliary function yastn.tn.fpeps.gates.distribute() distribute a set of gates homogeneously over the entire lattice. By default, it complements gates with their adjoint (gates repeated in reverse order), forming a 2nd order approximation for a small time step. Individual gates should then correspond to half of the timestep.

yastn.tn.fpeps.gates.distribute(geometry, gates_nn=None, gates_local=None, symmetrize=True) → Gates[source]#

Distributes gates homogeneous over the lattice.

Parameters:

  • geometry (yastn.tn.fpeps.SquareLattice | yastn.tn.fpeps.CheckerboardLattice | yast.tn.fpeps.Peps) – Geometry of PEPS lattice. Can be any structure that includes geometric information about the lattice, like the Peps class.

  • gates_nn (Gate_nn | Sequence[Gate_nn]) – Nearest-neighbor gate, or a list of gates, to be distributed over all unique lattice bonds.

  • gates_local (Gate_local | Sequence[Gate_local]) – Local gate, or a list of local gates, to be distributed over all unique lattice sites.

  • symmetrize (bool) – Whether to iterate through provided gates forward and then backward, resulting in a 2nd order method. In that case, each gate should correspond to half of the desired timestep. The default is True.

Predefined gates#

Some predefined gates can be found in yastn.tn.fpeps.gates, including

yastn.tn.fpeps.gates.gate_nn_hopping(t, step, I, c, cdag, bond=None) → Gate_nn[source]#

Nearest-neighbor gate \(G = \exp(-step \cdot H)\) for \(H = -t \cdot (cdag_1 c_2 + cdag_2 c_1)\)

\(G = I + (\cosh(x) - 1) (n_1 h_2 + h_1 n_2) + \sinh(x) (cdag_1 c_2 + cdag_2 c_1)\), where \(x = t \cdot step\)

yastn.tn.fpeps.gates.gate_nn_Ising(J, step, I, X, bond=None) → Gate_nn[source]#

Nearest-neighbor gate \(G = \exp(-step \cdot H)\) for \(H = J X_1 X_2\), where \(X\) is a Pauli matrix.

\(G = \cosh(x) I - \sinh(x) X_1 X_2\), where \(x = step \cdot J\).

yastn.tn.fpeps.gates.gate_local_Coulomb(mu_up, mu_dn, U, step, I, n_up, n_dn, site=None) → Gate_local[source]#

Local gate \(\exp(-step \cdot H)\) for \(H = U \cdot (n_{up} - I / 2) \cdot (n_{dn} - I / 2) - mu_{up} \cdot n_{up} - mu_{dn} \cdot n_{dn}\)

We ignore a constant \(U / 4\) in the above Hamiltonian.

yastn.tn.fpeps.gates.gate_local_occupation(mu, step, I, n, site=None) → Gate_local[source]#

Local gate \(G = \exp(-step \cdot H)\) for \(H = -mu \cdot n\)

\(G = I + n \cdot (\exp(mu \cdot step) - 1)\)

yastn.tn.fpeps.gates.gate_local_field(h, step, I, X, site=None) → Gate_local[source]#

Local gate \(G = \exp(-step \cdot H)\) for \(H = -h \cdot X\) where \(X\) is a Pauli matrix.

\(G = \cosh(h \cdot step) \cdot I + \sinh(h \cdot step) \cdot X\)