RandomUnitaryEvolution¶

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

class tenpy.algorithms.tebd.RandomUnitaryEvolution(psi, TEBD_params)[source]¶

Bases: tenpy.algorithms.tebd.Engine

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().

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!

Parameters

psiMPS: Initial state to be time evolved. Modified in place.
TEBD_paramsdict: Use verbose=1 to print the used parameters during runtime. See run() and run_GS() for more details.

Examples

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

>>> L = 8
>>> spin_half = SpinHalfSite(conserve='Sz')
>>> psi = MPS.from_product_state([spin_half]*L, [0, 1]*(L//2), bc='finite')  # Neel state
>>> print(psi.chi)
[1, 1, 1, 1, 1, 1, 1]
>>> TEBD_params = dict(N_steps=2, trunc_params={'chi_max':10})
>>> eng = RandomUnitaryEvolution(psi, TEBD_params)
>>> eng.run()
>>> print(psi.chi)
[2, 4, 8, 10, 8, 4, 2]
>>> psi.canonical_form()  # necessary if you need to truncate (strongly) during the evolution

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, [0]*L, bc='finite')  # all spins up
>>> print(psi2.chi)
[1, 1, 1, 1, 1, 1, 1]
>>> eng2 = RandomUnitaryEvolution(psi2, TEBD_params)
>>> 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]

Attributes

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

Methods

`calc_U`(self)	Draw new random two-site unitaries replacing the usual U of TEBD.
`run`(self)	Time evolution with TEBD (time evolving block decimation) and random two-site unitaries.
`run_GS`(self)	TEBD algorithm in imaginary time to find the ground state.
`suzuki_trotter_decomposition`(order, N_steps)	Returns list of necessary steps for the suzuki trotter decomposition.
`suzuki_trotter_time_steps`(order)	Return time steps of U for the Suzuki Trotter decomposition of desired order.
`update`(self, N_steps)	Apply `N_steps` random two-site unitaries to each bond (in even-odd pattern).
`update_bond`(self, i, U_bond)	Updates the B matrices on a given bond.
`update_bond_imag`(self, i, U_bond)	Update a bond with a (possibly non-unitary) U_bond.
`update_imag`(self, N_steps)	Perform an update suitable for imaginary time evolution.
`update_step`(self, U_idx_dt, odd)	Updates either even or odd bonds in unit cell.

run(self)[source]¶

Time evolution with TEBD (time evolving block decimation) and random two-site unitaries.

The following (optional) parameters are read out from the TEBD_params.

key	type	description
N_steps	int	Number of two-site unitaries to be applied on each bond.
trunc_params	dict	Truncation parameters as described in `truncate()`

calc_U(self)[source]¶: Draw new random two-site unitaries replacing the usual U of TEBD.

update(self, N_steps)[source]¶

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

Parameters

N_stepsint: The number of steps for which the whole lattice should be updated.

Returns

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

run_GS(self)¶

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.

The following (optional) parameters are read out from the TEBD_params. Use verbose=1 to print the used parameters during runtime.

key	type	description
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.
order	int	Order of the Suzuki-Trotter decomposition.
N_steps	int	Number of steps before measurement can be performed
trunc_params	dict	Truncation parameters as described in `truncate()`

static suzuki_trotter_decomposition(order, N_steps)¶

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

orderint: The desired order of the Suzuki-Trotter decomposition.

Returns

ST_decompositionlist of (int, int): 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>.

static suzuki_trotter_time_steps(order)¶

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

See suzuki_trotter_decomposition() for details.

Parameters

orderint: The desired order of the Suzuki-Trotter decomposition.

Returns

time_stepslist of float: We need U = exp(-i H_{even/odd} delta_t * dt) for the dt returned in this list.

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

update_bond(self, i, U_bond)¶

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

iint: Bond index; we update the matrices at sites i-1, i.
U_bondArray: The bond operator which we apply to the wave function. We expect labels 'p0', 'p1', 'p0*', 'p1*'.

Returns

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

update_bond_imag(self, i, U_bond)¶

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

iint: Bond index; we update the matrices at sites i-1, i.
U_bondArray: The bond operator which we apply to the wave function. We expect labels 'p0', 'p1', 'p0*', 'p1*'.

Returns

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

update_imag(self, N_steps)¶

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_stepsint: The number of steps for which the whole lattice should be updated.

Returns

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

update_step(self, U_idx_dt, odd)¶

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

Returns

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