QRBasedTEBDEngine¶

full name: tenpy.algorithms.tebd.QRBasedTEBDEngine
parent module: tenpy.algorithms.tebd
type: class

Inheritance Diagram

Methods

`QRBasedTEBDEngine.__init__`(psi, model, ...)
`QRBasedTEBDEngine.calc_U`(order, delta_t[, ...])	Calculate `self.U_bond` from `self.model.H_bond`.
`QRBasedTEBDEngine.evolve`(N_steps, dt)	Evolve by `dt * N_steps`.
`QRBasedTEBDEngine.evolve_step`(U_idx_dt, odd)	Updates either even or odd bonds in unit cell.
`QRBasedTEBDEngine.get_resume_data`([...])	Return necessary data to resume a `run()` interrupted at a checkpoint.
`QRBasedTEBDEngine.prepare_evolve`(dt)	Prepare an evolution step.
`QRBasedTEBDEngine.resume_run`()	Resume a run that was interrupted.
`QRBasedTEBDEngine.run`()	Perform a (real-)time evolution of `psi` by N_steps * dt.
`QRBasedTEBDEngine.run_GS`()	TEBD algorithm in imaginary time to find the ground state.
`QRBasedTEBDEngine.run_evolution`(N_steps, dt)	Perform a (real-)time evolution of `psi` by N_steps * dt.
`QRBasedTEBDEngine.suzuki_trotter_decomposition`(...)	Returns list of necessary steps for the suzuki trotter decomposition.
`QRBasedTEBDEngine.suzuki_trotter_time_steps`(order)	Return time steps of U for the Suzuki Trotter decomposition of desired order.
`QRBasedTEBDEngine.switch_engine`(other_engine, *)	Initialize algorithm from another algorithm instance of a different class.
`QRBasedTEBDEngine.update_bond`(i, U_bond)	Updates the B matrices on a given bond.
`QRBasedTEBDEngine.update_bond_imag`(i, U_bond)	Update a bond with a (possibly non-unitary) U_bond.
`QRBasedTEBDEngine.update_imag`(N_steps)	Perform an update suitable for imaginary time evolution.

Class Attributes and Properties

`QRBasedTEBDEngine.TEBD_params`
`QRBasedTEBDEngine.time_dependent_H`	whether the algorithm supports time-dependent H
`QRBasedTEBDEngine.trunc_err_bonds`	truncation error introduced on each non-trivial bond.
`QRBasedTEBDEngine.verbose`

class tenpy.algorithms.tebd.QRBasedTEBDEngine(psi, model, options, **kwargs)[source]¶

Bases: TEBDEngine

Version of TEBD that relies on QR decompositions rather than SVD.

As introduced in arXiv:2212.09782.

Todo

To use use_eig_based_svd == True, which makes sense on GPU only, we need to implement the _eig_based_svd for “non-square” matrices. This means that \(M^{\dagger} M\) and \(M M^{\dagger}\) dont have the same size, and we need to disregard those eigenvectors of the larger one, that have eigenvalue zero, since we dont have corresponding eigenvalues of the smaller one.

Options

config QRBasedTEBDEngine¶

option	summary
cbe_expand	Expansion rate. The QR-based decomposition is carried out at an expanded bo [...]
cbe_expand_0	Expansion rate at low ``chi``. [...]
cbe_min_block_increase	Minimum bond dimension increase for each block. Default is `1`.
compute_err	Whether the truncation error should be computed exactly. [...]
delta_tau_list (from TEBDEngine) in PurificationTEBD.run_GS	A list of floats: the timesteps to be used. [...]
dt (from TimeEvolutionAlgorithm) in TimeEvolutionAlgorithm	Minimal time step by which to evolve.
E_offset (from TEBDEngine) in TEBDEngine	Energy offset to be applied in :meth:`calc_U`, see doc there. [...]
N_steps (from TEBDEngine) in PurificationTEBD.run_GS	Number of steps before measurement can be performed
order (from TEBDEngine) in PurificationTEBD.run_GS	Order of the Suzuki-Trotter decomposition.
preserve_norm (from TimeEvolutionAlgorithm) in TimeEvolutionAlgorithm	Whether the state will be normalized to its initial norm after each time st [...]
start_time (from TimeEvolutionAlgorithm) in TimeEvolutionAlgorithm	Initial value for :attr:`evolved_time`.
start_trunc_err (from TEBDEngine) in TEBDEngine	Initial truncation error for :attr:`trunc_err`.
trunc_params (from Algorithm) in Algorithm	Truncation parameters as described in :cfg:config:`truncation`.
use_eig_based_svd	Whether the SVD of the bond matrix :math:`\Xi` should be carried out numeri [...]

option cbe_expand: float¶: Expansion rate. The QR-based decomposition is carried out at an expanded bond dimension eta = (1 + cbe_expand) * chi, where chi is the bond dimension before the time step. Default is 0.1.

option cbe_expand_0: float¶: Expansion rate at low chi. If given, the expansion rate decreases linearly from cbe_expand_0 at chi == 1 to cbe_expand at chi == trunc_params['chi_max'], then remains constant. If not given, the expansion rate is cbe_expand at all chi.

option cbe_min_block_increase: int¶: Minimum bond dimension increase for each block. Default is 1.

option use_eig_based_svd: bool¶: Whether the SVD of the bond matrix \(\Xi\) should be carried out numerically via the eigensystem. This is faster on GPUs, but less accurate. It makes no sense to do this on CPU. It is currently not supported for update_imag. Default is False.

option compute_err: bool¶: Whether the truncation error should be computed exactly. Compared to SVD-based TEBD, computing the truncation error is significantly more expensive. If True (default), the full error is computed. Otherwise, the truncation error is set to NaN.

update_bond(i, U_bond)[source]¶

Updates the B matrices on a given bond.

Function that updates the B matrices, the bond matrix s between and the bond dimension chi for bond i. The corresponding tensor networks look like this:

|           --S--B1--B2--           --B1--B2--
|                |   |                |   |
|     theta:     U_bond        C:     U_bond
|                |   |                |   |

Parameters

i (int) – Bond index; we update the matrices at sites i-1, i.
U_bond (Array) – The bond operator which we apply to the wave function. We expect labels 'p0', 'p1', 'p0*', 'p1*'.

Returns

trunc_err – The error of the represented state which is introduced by the truncation during this update step.

Return type

TruncationError

update_bond_imag(i, U_bond)[source]¶

Update a bond with a (possibly non-unitary) U_bond.

Similar as update_bond(); but after the SVD just keep the A, S, B canonical form. In that way, one can sweep left or right without using old singular values, thus preserving the canonical form during imaginary time evolution.

Parameters

i (int) – Bond index; we update the matrices at sites i-1, i.
U_bond (Array) – The bond operator which we apply to the wave function. We expect labels 'p0', 'p1', 'p0*', 'p1*'.

Returns

trunc_err – The error of the represented state which is introduced by the truncation during this update step.

Return type

TruncationError

calc_U(order, delta_t, type_evo='real', E_offset=None)[source]¶

Calculate self.U_bond from self.model.H_bond.

This function calculates

U_bond = exp(-i dt (H_bond-E_offset_bond)) for type_evo='real', or
U_bond = exp(- dt H_bond) for type_evo='imag'.

For first order (in delta_t), we need just one dt=delta_t. Higher order requires smaller dt steps, as given by suzuki_trotter_time_steps().

Parameters

order (int) – Trotter order calculated U_bond. See update for more information.
delta_t (float) – Size of the time-step used in calculating U_bond
type_evo ('imag' | 'real') – Determines whether we perform real or imaginary time-evolution.
E_offset (None | list of float) – Possible offset added to H_bond for real-time evolution.

evolve(N_steps, dt)[source]¶

Evolve by dt * N_steps.

Parameters

N_steps (int) – The number of steps for which the whole lattice should be updated.
dt (float) – The time step; but really this was already used in prepare_evolve().

Returns

trunc_err – The error of the represented state which is introduced due to the truncation during this sequence of evolvution steps.

Return type

TruncationError

evolve_step(U_idx_dt, odd)[source]¶

Updates either even or odd bonds in unit cell.

Depending on the choice of p, this function updates all even (E, odd=False,0) or odd (O) (odd=True,1) bonds:

|     - B0 - B1 - B2 - B3 - B4 - B5 - B6 -
|       |    |    |    |    |    |    |
|       |    |----|    |----|    |----|
|       |    |  E |    |  E |    |  E |
|       |    |----|    |----|    |----|
|       |----|    |----|    |----|    |
|       |  O |    |  O |    |  O |    |
|       |----|    |----|    |----|    |

Note that finite boundary conditions are taken care of by having Us[0] = None.

Parameters

U_idx_dt (int) – Time step index in self._U, evolve with Us[i] = self.U[U_idx_dt][i] at bond (i-1,i).
odd (bool/int) – Indication of whether to update even (odd=False,0) or even (odd=True,1) sites

Returns

trunc_err – The error of the represented state which is introduced due to the truncation during this sequence of update steps.

Return type

TruncationError

get_resume_data(sequential_simulations=False)[source]¶

Return necessary data to resume a run() interrupted at a checkpoint.

At a checkpoint, you can save psi, model and options along with the data returned by this function. When the simulation aborts, you can resume it using this saved data with:

eng = AlgorithmClass(psi, model, options, resume_data=resume_data)
eng.resume_run()

An algorithm which doesn’t support this should override resume_run to raise an Error.

Parameters: sequential_simulations (bool) – If True, return only the data for re-initializing a sequential simulation run, where we “adiabatically” follow the evolution of a ground state (for variational algorithms), or do series of quenches (for time evolution algorithms); see run_seq_simulations().
Returns: resume_data – Dictionary with necessary data (apart from copies of psi, model, options) that allows to continue the simulation from where we are now. It might contain an explicit copy of psi.
Return type: dict

prepare_evolve(dt)[source]¶

Prepare an evolution step.

This method is used to prepare repeated calls of evolve() given the model. For example, it may generate approximations of U=exp(-i H dt). To avoid overhead, it may cache the result depending on parameters/options; but it should always regenerate it if force_prepare_evolve is set.

Parameters: dt (float) – The time step to be used.

resume_run()[source]¶

Resume a run that was interrupted.

In case we saved an intermediate result at a checkpoint, this function allows to resume the run() of the algorithm (after re-initialization with the resume_data). Since most algorithms just have a while loop with break conditions, the default behaviour implemented here is to just call run().

run()[source]¶

Perform a (real-)time evolution of psi by N_steps * dt.

You probably want to call this in a loop along with measurements. The recommended way to do this is via the RealTimeEvolution.

run_GS()[source]¶

TEBD algorithm in imaginary time to find the ground state.

Note

It is almost always more efficient (and hence advisable) to use DMRG. This algorithms can nonetheless be used quite well as a benchmark and for comparison.

option TEBDEngine.delta_tau_list: list¶: A list of floats: the timesteps to be used. Choosing a large timestep delta_tau introduces large (Trotter) errors, but a too small time step requires a lot of steps to reach exp(-tau H) --> |psi0><psi0|. Therefore, we start with fairly large time steps for a quick time evolution until convergence, and then gradually decrease the time step.

option TEBDEngine.order: int¶: Order of the Suzuki-Trotter decomposition.

option TEBDEngine.N_steps: int¶: Number of steps before measurement can be performed

run_evolution(N_steps, dt)[source]¶

Perform a (real-)time evolution of psi by N_steps * dt.

This is the inner part of run() without the logging. For parameters see TimeEvolutionAlgorithm.

static suzuki_trotter_decomposition(order, N_steps)[source]¶

Returns list of necessary steps for the suzuki trotter decomposition.

We split the Hamiltonian as \(H = H_{even} + H_{odd} = H[0] + H[1]\). The Suzuki-Trotter decomposition is an approximation \(\exp(t H) \approx prod_{(j, k) \in ST} \exp(d[j] t H[k]) + O(t^{order+1 })\).

Parameters: order (1, 2, 4, '4_opt') – The desired order of the Suzuki-Trotter decomposition. Order 1 approximation is simply \(e^A a^B\). Order 2 is the “leapfrog” e^{A/2} e^B e^{A/2}. Order 4 is the fourth-order from [suzuki1991] (also referenced in [schollwoeck2011]), and '4_opt' gives the optmized version of Equ. (30a) in [barthel2020].
Returns: ST_decomposition – Indices j, k of the time-steps d = suzuki_trotter_time_step(order) and the decomposition of H. They are chosen such that a subsequent application of exp(d[j] t H[k]) to a given state |psi> yields (exp(N_steps t H[k]) + O(N_steps t^{order+1}))|psi>.
Return type: list of (int, int)

static suzuki_trotter_time_steps(order)[source]¶

Return time steps of U for the Suzuki Trotter decomposition of desired order.

See suzuki_trotter_decomposition() for details.

Parameters: order (int) – The desired order of the Suzuki-Trotter decomposition.
Returns: time_steps – We need U = exp(-i H_{even/odd} delta_t * dt) for the dt returned in this list.
Return type: list of float

classmethod switch_engine(other_engine, *, options=None, **kwargs)[source]¶

Initialize algorithm from another algorithm instance of a different class.

You can initialize one engine from another, not too different subclasses. Internally, this function calls get_resume_data() to extract data from the other_engine and then initializes the new class.

Note that it transfers the data without making copies in most case; even the options! Thus, when you call run() on one of the two algorithm instances, it will modify the state, environment, etc. in the other. We recommend to make the switch as engine = OtherSubClass.switch_engine(engine) directly replacing the reference.

Parameters

cls (class) – Subclass of Algorithm to be initialized.
other_engine (Algorithm) – The engine from which data should be transferred. Another, but not too different algorithm subclass-class; e.g. you can switch from the TwoSiteDMRGEngine to the OneSiteDMRGEngine.
options (None | dict-like) – If not None, these options are used for the new initialization. If None, take the options from the other_engine.
**kwargs – Further keyword arguments for class initialization. If not defined, resume_data is collected with get_resume_data().

time_dependent_H = False¶: whether the algorithm supports time-dependent H

property trunc_err_bonds¶: truncation error introduced on each non-trivial bond.

update_imag(N_steps)[source]¶

Perform an update suitable for imaginary time evolution.

Instead of the even/odd brick structure used for ordinary TEBD, we ‘sweep’ from left to right and right to left, similar as DMRG. Thanks to that, we are actually able to preserve the canonical form.

Parameters: N_steps (int) – The number of steps for which the whole lattice should be updated.
Returns: trunc_err – The error of the represented state which is introduced due to the truncation during this sequence of update steps.
Return type: TruncationError