# RandomUnitaryEvolution¶

Inheritance Diagram

Methods

 RandomUnitaryEvolution.__init__(psi, ...) Draw new random two-site unitaries replacing the usual U of TEBD. RandomUnitaryEvolution.evolve(N_steps, dt) Apply N_steps random two-site unitaries to each bond (in even-odd pattern). RandomUnitaryEvolution.evolve_step(U_idx_dt, odd) Updates either even or odd bonds in unit cell. Return necessary data to resume a run() interrupted at a checkpoint. Do nothing, as we call calc_U() directly in update(). Resume a run that was interrupted. Time evolution with TEBD and random two-site unitaries (possibly conserving charges). TEBD algorithm in imaginary time to find the ground state. RandomUnitaryEvolution.run_evolution(N_steps, dt) Perform a (real-)time evolution of psi by N_steps * dt. Returns list of necessary steps for the suzuki trotter decomposition. Return time steps of U for the Suzuki Trotter decomposition of desired order. RandomUnitaryEvolution.switch_engine(...[, ...]) Initialize algorithm from another algorithm instance of a different class. RandomUnitaryEvolution.update_bond(i, U_bond) Updates the B matrices on a given bond. Update a bond with a (possibly non-unitary) U_bond. Perform an update suitable for imaginary time evolution.

Class Attributes and Properties

 RandomUnitaryEvolution.TEBD_params RandomUnitaryEvolution.time_dependent_H whether the algorithm supports time-dependent H RandomUnitaryEvolution.trunc_err_bonds truncation error introduced on each non-trivial bond. RandomUnitaryEvolution.verbose
class tenpy.algorithms.tebd.RandomUnitaryEvolution(psi, options, **kwargs)[source]

Bases: TEBDEngine

Evolution of an MPS with random two-site unitaries in a TEBD-like fashion.

Instead of using a model Hamiltonian, this TEBD engine evolves with random two-site unitaries. These unitaries are drawn according to the Haar measure on unitaries obeying the conservation laws dictated by the conserved charges. If no charge is preserved, this distribution is called circular unitary ensemble (CUE), see CUE(). The distribution can be changed through the RandomUnitaryEvolution.distribution_function.

On one hand, such an evolution is of interest in recent research (see eg. arXiv:1710.09827). On the other hand, it also comes in handy to “randomize” an initial state, e.g. for DMRG. Note that the entanglement grows very quickly, choose the truncation paramters accordingly!

Options

config RandomUnitaryEvolution
option summary

delta_tau_list (from TEBDEngine) in PurificationTEBD.run_GS

A list of floats: the timesteps to be used. [...]

distribution_func in RandomUnitaryEvolution.calc_U

Function or name for one of the matrix ensembles in [...]

distribution_func_kwargs in RandomUnitaryEvolution.calc_U

Extra keyword arguments for distribution_func.

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

Number of two-site unitaries to be applied on each bond.

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 step.

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

Truncation parameters as described in :cfg:config:truncate

option N_steps: int

Number of two-site unitaries to be applied on each bond.

option trunc_params: dict

Truncation parameters as described in truncate

Examples

One can initialize a “random” state with total Sz = L//2 as follows:

>>> from tenpy.algorithms.tebd import RandomUnitaryEvolution
>>> from tenpy.networks.mps import MPS
>>> L = 8
>>> spin_half = tenpy.networks.site.SpinHalfSite(conserve='Sz')
>>> psi = MPS.from_product_state([spin_half]*L, ["up", "down"]*(L//2), bc='finite')  # Neel
>>> print(psi.chi)
[1, 1, 1, 1, 1, 1, 1]
>>> options = dict(N_steps=2, trunc_params={'chi_max':10})
>>> eng = RandomUnitaryEvolution(psi, options)
>>> eng.run()
>>> print(psi.chi)
[2, 4, 8, 10, 8, 4, 2]
>>> psi.canonical_form()  # a good idea if there was a truncation necessary.


The “random” unitaries preserve the specified charges, e.g. here we have Sz-conservation. If you start in a sector of all up spins, the random unitaries can only apply a phase:

>>> psi2 = MPS.from_product_state([spin_half]*L, ["up"]*L, bc='finite')  # all spins up
>>> print(psi2.chi)
[1, 1, 1, 1, 1, 1, 1]
>>> eng2 = RandomUnitaryEvolution(psi2, options)
>>> eng2.run()  # random unitaries respect Sz conservation -> we stay in all-up sector
>>> print(psi2.chi)  # still a product state, not really random!!!
[1, 1, 1, 1, 1, 1, 1]

run()[source]

Time evolution with TEBD and random two-site unitaries (possibly conserving charges).

prepare_evolve(dt)[source]

Do nothing, as we call calc_U() directly in update().

calc_U()[source]

Draw new random two-site unitaries replacing the usual U of TEBD.

The parameter dt is only there for compatibility with parent classes and is ignored.

option RandomUnitaryEvolution.distribution_func: str | function

Function or name for one of the matrix ensembles in random_matrix which generates unitaries (or a subset of them). To be used as func for generating unitaries with from_func_square(), i.e. the U still preserves the charge block structure!

option RandomUnitaryEvolution.distribution_func_kwargs: dict

Extra keyword arguments for distribution_func.

evolve(N_steps, dt)[source]

Apply N_steps random two-site unitaries to each bond (in even-odd pattern).

Parameters
• dt (float) – Mostly ignored, but used as unit to update evolved_time.

• 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

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

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_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 (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
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 transfered. 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. The correponding 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

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