PurificationTEBD2¶

full name: tenpy.algorithms.purification.PurificationTEBD2
parent module: tenpy.algorithms.purification
type: class

Inheritance Diagram

Methods

`PurificationTEBD2.__init__`(psi, model, ...)
`PurificationTEBD2.calc_U`(order, delta_t[, ...])	see `calc_U()`
`PurificationTEBD2.disentangle`(theta)	Disentangle theta before splitting with svd.
`PurificationTEBD2.disentangle_global`([pair])	Try global disentangling by determining the maximally entangled pairs of sites.
`PurificationTEBD2.disentangle_global_nsite`([n])	Perform a sweep through the system and disentangle with `disentangle_n_site()`.
`PurificationTEBD2.disentangle_n_site`(i, n, theta)	Generalization of `disentangle()` to n sites.
`PurificationTEBD2.evolve`(N_steps, dt)	Evolve by `dt * N_steps`.
`PurificationTEBD2.evolve_step`(U_idx_dt, odd)	Updates either even or odd bonds in unit cell.
`PurificationTEBD2.get_resume_data`([...])	Return necessary data to resume a `run()` interrupted at a checkpoint.
`PurificationTEBD2.prepare_evolve`(dt)	Prepare an evolution step.
`PurificationTEBD2.resume_run`()	Resume a run that was interrupted.
`PurificationTEBD2.run`()	Perform a (real-)time evolution of `psi` by N_steps * dt.
`PurificationTEBD2.run_GS`()	TEBD algorithm in imaginary time to find the ground state.
`PurificationTEBD2.run_evolution`(N_steps, dt)	Perform a (real-)time evolution of `psi` by N_steps * dt.
`PurificationTEBD2.run_imaginary`(beta)	Run imaginary time evolution to cool down to the given beta.
`PurificationTEBD2.suzuki_trotter_decomposition`(...)	Returns list of necessary steps for the suzuki trotter decomposition.
`PurificationTEBD2.suzuki_trotter_time_steps`(order)	Return time steps of U for the Suzuki Trotter decomposition of desired order.
`PurificationTEBD2.switch_engine`(other_engine, *)	Initialize algorithm from another algorithm instance of a different class.
`PurificationTEBD2.update`(N_steps)	Evolve by `N_steps * U_param['dt']`.
`PurificationTEBD2.update_bond`(i, U_bond)	Updates the B matrices on a given bond.
`PurificationTEBD2.update_bond_imag`(i, U_bond)	Update a bond with a (possibly non-unitary) U_bond.
`PurificationTEBD2.update_imag`(N_steps)	Perform an update suitable for imaginary time evolution.
`PurificationTEBD2.update_step`(U_idx_dt, odd)	Updates bonds in unit cell.

Class Attributes and Properties

`PurificationTEBD2.TEBD_params`
`PurificationTEBD2.disent_iterations`	For each bond the total number of iterations performed in any Disentangler.
`PurificationTEBD2.time_dependent_H`	whether the algorithm supports time-dependent H
`PurificationTEBD2.trunc_err_bonds`	truncation error introduced on each non-trivial bond.
`PurificationTEBD2.verbose`

class tenpy.algorithms.purification.PurificationTEBD2(psi, model, options, **kwargs)[source]¶

Bases: PurificationTEBD

Similar as PurificationTEBD, but perform sweeps instead of brickwall.

Instead of the A-B pattern of even/odd bonds used in TEBD, perform sweeps similar as in DMRG for real-time evolution (similar as update_imag() does for imaginary time evolution).

update(N_steps)[source]¶

Evolve by N_steps * U_param['dt'].

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

update_step(U_idx_dt, odd)[source]¶

Updates bonds in unit cell.

Depending on the choice of odd, perform a sweep to the left or right, updating once per site with a time step given by U_idx_dt.

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

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

property disent_iterations¶: For each bond the total number of iterations performed in any Disentangler.

disentangle(theta)[source]¶

Disentangle theta before splitting with svd.

For the purification we write \(\rho_P = Tr_Q{|\psi_{P,Q}><\psi_{P,Q}|}\). Thus, we can actually apply any unitary to the auxiliar Q space of \(|\psi>\) without changing the result.

Note

We have to apply the same unitary to the ‘bra’ and ‘ket’ used for expectation values / correlation functions!

The behaviour of this function is set by used_disentangler, which in turn is obtained from get_disentangler(options['disentangle']), see get_disentangler() for details on the syntax.

Parameters

theta (Array) – Wave function to disentangle, with legs 'vL', 'vR', 'p0', 'p1', 'q0', 'q1'.

Returns

theta_disentangled (Array) – Disentangled theta; npc.tensordot(U, theta, axes=[['q0*', 'q1*'], ['q0', 'q1']]).
U (Array) – The unitary used to disentangle theta, with labels 'q0', 'q1', 'q0*', 'q1*'. If no unitary was found/applied, it might also be None.

disentangle_global(pair=None)[source]¶

Try global disentangling by determining the maximally entangled pairs of sites.

Calculate the mutual information (in the auxiliar space) between two sites and determine where it is maximal. Disentangle these two sites with disentangle()

disentangle_global_nsite(n=2)[source]¶

Perform a sweep through the system and disentangle with disentangle_n_site().

Parameters: n (int) – maximal number of sites to disentangle at once.

disentangle_n_site(i, n, theta)[source]¶

Generalization of disentangle() to n sites.

Simply group left and right n/2 physical legs, adjust labels, and apply disentangle() to disentangle the central bond. Recursively proceed to disentangle left and right parts afterwards. Scales (for even n) as \(O(\chi^3 d^n d^{n/2})\).

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.

run_imaginary(beta)[source]¶

Run imaginary time evolution to cool down to the given beta.

Note that we don’t change the norm attribute of the MPS, i.e. normalization is preserved.

Parameters: beta (float) – The inverse temperature beta = 1/T, by which we should cool down. We evolve to the closest multiple of options['dt'], see also evolved_time.

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_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. This would look something like:

|           |             |
|     ... - B1  -  s  -  B2 - ...
|           |             |
|           |-------------|
|           |      U      |
|           |-------------|
|           |             |

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*' for U_bond.

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

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