RandomUnitaryEvolution¶
full name: tenpy.algorithms.tebd.RandomUnitaryEvolution
parent module:
tenpy.algorithms.tebd
type: class
Inheritance Diagram
Methods
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Initialize self. |
Draw new random two-site unitaries replacing the usual U of TEBD. |
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Return necessary data to resume a |
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Resume a run that was interrupted. |
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Time evolution with TEBD and random two-site unitaries. |
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TEBD algorithm in imaginary time to find the ground state. |
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Returns list of necessary steps for the suzuki trotter decomposition. |
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Return time steps of U for the Suzuki Trotter decomposition of desired order. |
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Apply |
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Updates the B matrices on a given bond. |
Update a bond with a (possibly non-unitary) U_bond. |
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Perform an update suitable for imaginary time evolution. |
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Updates either even or odd bonds in unit cell. |
Class Attributes and Properties
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truncation error introduced on each non-trivial bond. |
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-
class
tenpy.algorithms.tebd.
RandomUnitaryEvolution
(psi, options)[source]¶ Bases:
tenpy.algorithms.tebd.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()
.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
psi (
MPS
) – Initial state to be time evolved. Modified in place.options (dict) – See below for details.
Options
-
config
RandomUnitaryEvolution
¶ option summary 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.
Number of two-site unitaries to be applied on each bond.
order (from TEBDEngine) in PurificationTEBD.run_GS
Order of the Suzuki-Trotter decomposition.
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`.
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
-
option
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, [0]*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]
-
update
(N_steps)[source]¶ Apply
N_steps
random two-site unitaries to each bond (in even-odd pattern).- 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
-
get_resume_data
()[source]¶ Return necessary data to resume a
run()
interrupted at a checkpoint.At a
checkpoint
, you can savepsi
,model
andoptions
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(resume_data)
An algorithm which doesn’t support this should override resume_run to raise an Error.
- 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.
- Return type
-
resume_run
()[source]¶ Resume a run that was interrupted.
In case we saved an intermediate result at a
checkpoint
, this function allows to resume therun()
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 callrun()
.
-
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
-
option
-
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. Order1
approximation is simply \(e^A a^B\). Order2
is the “leapfrog” e^{A/2} e^B e^{A/2}. Order4
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-stepsd = suzuki_trotter_time_step(order)
and the decomposition of H. They are chosen such that a subsequent application ofexp(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
-
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. The correponding tensor networks look like this:
| --S--B1--B2-- --B1--B2-- | | | | | | theta: U_bond C: U_bond | | | | |
- Parameters
- Returns
trunc_err – The error of the represented state which is introduced by the truncation during this update step.
- Return type
-
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
- Returns
trunc_err – The error of the represented state which is introduced by the truncation during this update step.
- Return type
-
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
-
update_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 withUs[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