# PurificationTEBD2¶

Inheritance Diagram

Methods

 PurificationTEBD2.__init__(psi, model, ...) PurificationTEBD2.calc_U(order, delta_t[, ...]) see calc_U() Disentangle theta before splitting with svd. Try global disentangling by determining the maximally entangled pairs of sites. 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. Return necessary data to resume a run() interrupted at a checkpoint. Resume a run that was interrupted. Run TEBD real time evolution by N_steps*dt. TEBD algorithm in imaginary time to find the ground state. Run imaginary time evolution to cool down to the given beta. Returns list of necessary steps for the suzuki trotter decomposition. Return time steps of U for the Suzuki Trotter decomposition of desired order. 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. 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]

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]

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.

Caclulate 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 $$O(\chi^3 d^n d^{n/2})$$.

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

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]

Run TEBD real time evolution by N_steps*dt.

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 the 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_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 (also referenced in ), and '4_opt' gives the optmized version of Equ. (30a) in .

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

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