netket.operator.LocalLiouvillian¶
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class
netket.operator.LocalLiouvillian(ham, jump_ops=[], dtype=<class 'complex'>)[source]¶ Bases:
netket.operator._abstract_super_operator.AbstractSuperOperatorLocalLiouvillian super-operator, acting on the DoubledHilbert (tensor product) space ℋ⊗ℋ.
Internally it uses netket.operator.LocalOperator everywhere.
The Liouvillian is defined according to the definition:
\[\mathcal{L} = -i \left[ \hat{H}, \hat{\rho}\right] + \sum_i \left[ \hat{L}_i\hat{\rho}\hat{L}_i^\dagger - \left\{ \hat{L}_i^\dagger\hat{L}_i, \hat{\rho} \right\} \right]\]which generates the dynamics according to the equation
\[\frac{d\hat{\rho}}{dt} = \mathcal{L}\hat{\rho}\]Internally, it stores the non-hermitian hamiltonian
\[\hat{H}_{nh} = \hat{H} - \sum_i \frac{i}{2}\hat{L}_i^\dagger\hat{L}_i\]That is then composed with the jump operators in the inner kernel with the formula:
\[\mathcal{L} = -i \hat{H}_{nh}\hat{\rho} +i\hat{\rho}\hat{H}_{nh}^\dagger + \sum_i \hat{L}_i\hat{\rho}\hat{L}_i^\dagger\]-
__init__(ham, jump_ops=[], dtype=<class 'complex'>)[source]¶ Initialize a super-operator by passing it the physical hilbert space on which it acts.
This init method constructs the doubled-hilbert space and pass it down to the fundamental abstractoperator.
- Parameters
ham (netket.operator._abstract_operator.AbstractOperator) –
jump_ops (List[netket.operator._abstract_operator.AbstractOperator]) –
- Attributes
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H¶ Returns the Conjugate-Transposed operator
- Return type
AbstractOperator
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T¶ Returns the transposed operator
- Return type
AbstractOperator
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dtype¶
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hamiltonian¶ The hamiltonian of this Liouvillian
- Return type
LocalOperator
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hamiltonian_nh¶ The non hermitian Local Operator part of the Liouvillian
- Return type
LocalOperator
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hilbert¶ The hilbert space associated to this operator.
- Return type
AbstractHilbert
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hilbert_physical¶ The physical hilbert space on which this super-operator acts.
- Return type
AbstractHilbert
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is_hermitian¶
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- Methods
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__call__(v)¶ Call self as a function.
- Return type
- Parameters
v (numpy.ndarray) –
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apply(v)¶ - Return type
- Parameters
v (numpy.ndarray) –
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collect()¶ Returns a guranteed concrete instancce of an operator.
As some operations on operators return lazy wrapperes (such as transpose, hermitian conjugate…), this is used to obtain a guaranteed non-lazy operator.
- Return type
AbstractOperator
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conj(*, concrete=False)¶ - Return type
AbstractOperator
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conjugate(*, concrete=False)¶ Returns the complex-conjugate of this operator.
- Parameters
concrete – if True returns a concrete operator and not a lazy wrapper
- Return type
AbstractOperator- Returns
if concrete is not True, self or a lazy wrapper; the complex-conjugated operator otherwise
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get_conn(x)[source]¶ Finds the connected elements of the Operator. Starting from a given quantum number x, it finds all other quantum numbers x’ such that the matrix element \(O(x,x')\) is different from zero. In general there will be several different connected states x’ satisfying this condition, and they are denoted here \(x'(k)\), for \(k=0,1...N_{\mathrm{connected}}\).
- Parameters
x (array) – An array of shape (hilbert.size) containing the quantum numbers x.
- Returns
The connected states x’ of shape (N_connected,hilbert.size) array: An array containing the matrix elements \(O(x,x')\) associated to each x’.
- Return type
matrix
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get_conn_flattened(x, sections, pad=False)[source]¶ Finds the connected elements of the Operator. Starting from a given quantum number x, it finds all other quantum numbers x’ such that the matrix element \(O(x,x')\) is different from zero. In general there will be several different connected states x’ satisfying this condition, and they are denoted here \(x'(k)\), for \(k=0,1...N_{\mathrm{connected}}\).
This is a batched version, where x is a matrix of shape (batch_size,hilbert.size).
- Parameters
x (matrix) – A matrix of shape (batch_size,hilbert.size) containing the batch of quantum numbers x.
sections (array) – An array of sections for the flattened x’. See numpy.split for the meaning of sections.
- Returns
The connected states x’, flattened together in a single matrix. array: An array containing the matrix elements \(O(x,x')\) associated to each x’.
- Return type
matrix
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get_conn_padded(x)¶ Finds the connected elements of the Operator. Starting from a batch of quantum numbers x={x_1, … x_n} of size B x M where B size of the batch and M size of the hilbert space, finds all states y_i^1, …, y_i^K connected to every x_i. Returns a matrix of size B x Kmax x M where Kmax is the maximum number of connections for every y_i.
- Parameters
x (matrix) – A matrix of shape (batch_size,hilbert.size) containing the batch of quantum numbers x.
- Returns
The connected states x’, in a 3D tensor. array: A matrix containing the matrix elements \(O(x,x')\) associated to each x’ for every batch.
- Return type
matrix
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n_conn(x, out=None)¶ Return the number of states connected to x.
- Parameters
x (matrix) – A matrix of shape (batch_size,hilbert.size) containing the batch of quantum numbers x.
out (array) – If None an output array is allocated.
- Returns
The number of connected states x’ for each x[i].
- Return type
array
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to_dense()¶ Returns the dense matrix representation of the operator. Note that, in general, the size of the matrix is exponential in the number of quantum numbers, and this operation should thus only be performed for low-dimensional Hilbert spaces or sufficiently sparse operators.
This method requires an indexable Hilbert space.
- Return type
- Returns
The dense matrix representation of the operator as a Numpy array.
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to_linear_operator(*, sparse=True, append_trace=False)[source]¶ Returns a lazy scipy linear_operator representation of the Lindblad Super-Operator.
The returned operator behaves like the M**2 x M**2 matrix obtained with to_dense/sparse, and accepts vectorised density matrices as input.
- Parameters
sparse (
bool) – If True internally uses sparse matrices for the hamiltonian and jump operators, dense otherwise (default=True)append_trace (
bool) – If True (default=False) the resulting operator has size M**2 + 1, and the last element of the input vector is the trace of the input density matrix. This is useful when implementing iterative methods.
- Return type
LinearOperator- Returns
A linear operator taking as input vectorised density matrices and returning the product L*rho
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to_sparse()¶ Returns the sparse matrix representation of the operator. Note that, in general, the size of the matrix is exponential in the number of quantum numbers, and this operation should thus only be performed for low-dimensional Hilbert spaces or sufficiently sparse operators.
This method requires an indexable Hilbert space.
- Return type
csr_matrix- Returns
The sparse matrix representation of the operator.
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transpose(*, concrete=False)¶ Returns the transpose of this operator.
- Parameters
concrete – if True returns a concrete operator and not a lazy wrapper
- Return type
AbstractOperator- Returns
if concrete is not True, self or a lazy wrapper; the transposed operator otherwise
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