Modules

MPI

Warning

This module will only operate correctly if called within an MPI instance.

class qsw_mpi.MPI.GKSL(H, Ls, MPI_communicator, taus=None)

Bases: qsw_mpi.MPI.walk

GKSL provides for the simulation of systems obeying the GKSL master equation in its diagonalised form (see Equation (10)).

Parameters:
  • H (\(N \times N\) complex SciPy CSR) – Hamiltonian, \(H\).
  • Ls (\(N \times N\) complex SciPy CSR matrix) – Generalised Lindblad operators \(L_k\).
  • taus (float array, optional) – Coefficients \(\tau_k\) of the Lindblad operators \(L_k\) (see Equation (11)). If None, \(\tau_k = 1\).
  • MPI_communicator – MPI communicator object provided by mpi4py.
Returns:

A walk object with \(\tilde{\mathcal{L}}\) distributed over an MPI communicator.
initial_state(state)

Sets the initial state \(\rho(t_0)\) and generates \(\tilde{\rho}(t_0)\).

Parameters:state –

One of the following:

  • \(\rho(t_0)\) as a complex NumPy array.
  • \(p(t_0)\). An array describing the diagonal of \(\rho(t_0)\) which is then assumed to have no initial inter-vertex coherences.
  • ’mixed’ - Equally distribute \(\rho(t_0)\) over all sites, including specified sources and sinks.
set_omega(omega)

Re-defines \(\omega\) in the QSW master equation. This requires that \(\tilde{\mathcal{L}}\) be re-built, its communication pattern re-optimised and its one-norm series re-calculated.

Parameters:omega (float) – Interpolation parameter, \(\omega\)
class qsw_mpi.MPI.GQSW(omega, H, Ls, MPI_communicator, H_loc=None, vsets=None)

Bases: qsw_mpi.MPI.walk

GQSW provides for the simulation of G-QSWs and NM-G-QSWs (see Equations (18) or (22)).

Parameters:
  • omega (float) – Interpolation parameter, \(\omega\).
  • H (\(\tilde{N} \times \tilde{N}\) SciPy CSR) – Hamiltonian, \(H\).
  • Ls (\(\tilde{N} \times \tilde{N}\) SciPy CSR matrix) – Generalised Lindblad operators \(L_k\).
  • MPI_communicator – MPI communicator object provided by mpi4py.
  • H_loc (\(\tilde{N} \times \tilde{N}\) SciPy CSR, optional) – NM-G-QSW only. Hamiltonian acting in the subspaces of \(V^D\), produced by nm_H_loc() (see Equation (22)).
  • vsets (integer lists, optional) – NM-G-QSW only. Set of vertex subspaces \(V^D\) created by nm_vsets(), see (Equation (22)).
Returns:

A walk object with a \(\tilde{\mathcal{L}}\) distributed over an MPI communicator.
gather_populations(root=0)

Return \(p(t)\) or \((p(t_0),...,p(t_q))\) following a call to step() or series(). In the case of a G-QSW, \(p(t)\) corresponds to the originating graph \(G\) (see Equation (22)).

Parameters:root (integer, optional.) – MPI process rank at which to gather \(p(t)\) or \((p(t_0),...,p(t_q))\).
Returns:\(p(t)\) or \((p(t_0),...,p(t_q))\).
Return type:\(N\), or \((q + 1) \times N\), complex NumPy array.
initial_state(state)

Sets the initial state \(\rho(t_0)\) and generates \(\tilde{\rho}(t_0)\).

Parameters:state –

One of the following:

  • \(\rho(t_0)\) as a complex NumPy array.
  • \(p(t_0)\). An array describing the diagonal of \(\rho(t_0)\) which is then assumed to have no initial inter-vertex coherences.
  • ’mixed’ - Equally distribute \(\rho(t_0)\) over all sites, including specified sources and sinks.
save_vsets(filename, vsetsname, action='a')

Save \(V^D\) (see Equation (22)) created by nm_vsets() to a HDF5 file.

Parameters:
  • filename (string) – File basename.
  • vsetname (string) – Group/dataset name under which to save \(V^D\).
  • action (string) – File access type, append (‘a’), overwrite (‘w’).
class qsw_mpi.MPI.LQSW(omega, H, L, MPI_communicator, sources=(None, None), sinks=(None, None))

Bases: qsw_mpi.MPI.walk

Defines an L-QSW, a QSW with locally interacting Lindblad operators, with or without appended sources and sinks (see Equation (13) or (17)).

Parameters:
  • omega (float) – Interpolation parameter, \(\omega\)
  • H (\(N \times N\) SciPy CSR) – Hamiltonian, \(H\)
  • L (\(N \times N\) SciPy CSR) – \(M_L = \sum_k L_k\) where the Lindblad operators are defined as in Equation (12).
  • MPI_communicator – MPI communicator object provided by mpi4py.
  • sources (tuple ([integers],[floats]), optional) – Contains a list of absorption sites and the corresponding absorption rates.
  • sinks (tuple ([integers],[floats]), optional) – Contains a list of the emission sites and the corresponding absorption rates.
Returns:

A walk object with a \(\tilde{\mathcal{L}}\) distributed over an MPI communicator.
initial_state(state)

Sets the initial state, \(\rho(t_0)\). For an LQSW, this method accepts an additional keyword argument.

Parameters:state –

One of the following:

  • \(\rho(t_0)\) as a complex NumPy array.
  • \(p(t_0)\). An array describing the diagonal of \(\rho(t_0)\) which is then assumed to have no initial inter-vertex coherences.
  • ’sources’ - Equally distribute \(\rho(t_0)\) over all specified sources.
  • ’mixed’ - Equally distribute \(\rho(t_0)\) over all sites, including specified sources and sinks.
class qsw_mpi.MPI.walk

Bases: object

The walk class provides a framework for Lindbladian quantum walk simulation. This proceeds by creating a walk sub-class containing a private method with the signature,

def _construct_superoperator(self):

which is responsible for defining:

  • self.SO_nnz (integer): The number of non-zeros in \(\tilde{\mathcal{L}}\)

  • self.SO_rows (integer): The global number of rows in \(\tilde{\mathcal{L}}\)

  • self.partition_table (integer array): A 1-based array describing the number of \(\tilde{\mathcal{L}}\) rows per MPI rank.

    Example:
    An evenly distributed \(\tilde{\mathcal{L}}\) with an MPI communicator of size of 4 and self.SO_nnz = 16 would have,

    self.partition_table = [1,5,9,13,17]

    where the number of rows at MPI rank i (self.SO_local_rows) is equal to self.partition_table[i + 1] - self.partition_table[i]

  • self.SO_local_rows (integer) the number of \(\tilde{\mathcal{L}}\) stored locally, \(N_\text{local}\).

  • self.SO_n_row_starts (\(N_\text{local} + 1\) integer NumPy array): Local partition of the 1-based row index pointer array for \(\tilde{\mathcal{L}}\), part of the CSR matrix format.

  • self.SO_col_indexes (\(N_\text{local}\) integer NumPy array): Local partition of the 1-based column index array for \(\tilde{\mathcal{L}}\), part of the CSR matrix format.

  • self.SO_values (\(N_\text{local}\) complex NumPy array): Local partition of non-zero values corresponding to self.SO_col_indexes (in 1-based indexing) for \(\tilde{\mathcal{L}}\), part of the CSR matrix format.

  • self.size (integer tuple): Dimensions of \(\rho(t), H\), Lindblad and other non-vectorised operators.

  • self.MPI_communicator: MPI communicator object as defined by mpi4py.

  • self.rank (integer): MPI process rank

_construct_superoperator() must be called in the ‘__init__’ function of the walk subclass, followed by a call to the __init__ method of the parent class:

super().__init__()

The following walk sub-classes are defined in QSW_MPI:

  • LQSW: QSWs with locally interacting Lindbad operators.
  • GQSW: QSWs with globally interacting Lindblad operators (including non-moralising QSWs).
  • GKSL: Quantum walks defined using the diagonal form of the GKSL equation.
gather_populations(root=0)

Gathers \(p(t)\) or \(((p(t_0),...,p(t_q))\) following execution of qsw_mpi.MPI.walk.step() or qsw_mpi.MPI.walk.series() at a specified MPI rank. None` is returned elsewhere.

Parameters:root (integer) – MPI process rank at which to gather \(p(t)\) or \((p(t_1),...,p(t_q))\).
Returns:\(p(t)\) or \((p(t_0),...,p(t_q))\).
Return type:\(N\), or \((q + 1) \times N\), complex NumPy array.
gather_result(root=0)

Gathers \(\rho(t)\) or \((\rho(t_1),...,\rho(t_q))\) following execution of qsw_mpi.MPI.walk.step() or qsw_mpi.MPI.walk.series() at a specified MPI rank. None is returned elsewhere.

Parameters:root (integer) – MPI process rank at which to gather \(\rho(t)\) or \((\rho(t_1),...,\rho(t_q))\).
Returns:\(\tilde{N} \times \tilde{N}\) complex NumPy array at root, otherwise None.
gather_superoperator(root=0)

Gathers \(\tilde{\mathcal{L}}\) to the root MPI process as a SciPy CSR matrix.

Parameters:root (integer) – MPI process rank at which to gather \(\tilde{\mathcal{L}}\).
Returns:\(\tilde{\mathcal{L}}\)
Return type:\(\tilde{N}^2 \times \tilde{N}^2\) SciPY CSR matrix
initial_state(state)

Sets the initial state \(\rho(t_0)\) and generates \(\tilde{\rho}(t_0)\).

Parameters:state –

One of the following:

  • \(\rho(t_0)\) as a complex NumPy array.
  • \(p(t_0)\). An array describing the diagonal of \(\rho(t_0)\) which is then assumed to have no initial inter-vertex coherences.
  • ’mixed’ - Equally distribute \(\rho(t_0)\) over all sites, including specified sources and sinks.
save_populations(filename, popname, action='a')

Saves \(p(t)\) or \((p(t_1),...,p(t_q))\) to a HDF5 file.

Parameters:
  • filename (string) – File basename.
  • walkname (string) – Group/dataset name for \(p(t)\) or \((p(t),...,p(t_q))\).
  • action – File access type, append (‘a’), overwrite (‘w’).
  • action – string
save_result(filename, walkname, action='a')

Saves \(\rho(t)\) or \(\vec{\rho}(t)\) to a HDF5 file.

Parameters:
  • filename (string) – File basename.
  • walkname (string) – Group/dataset name for \(\rho(t)\) or \(\vec{\rho}(t)\)
  • action – File access type, append (‘a’), overwrite (‘w’).
  • action – string
save_superoperator(filename, operatorname, action='a')

Save \(\tilde{\mathcal{L}}\) to a HDF5 file as a CSR matrix.

Parameters:
  • filename (string) – File basename
  • operatorname (string) – Group/dataset name for \(\tilde{\mathcal{L}}\).
  • action (string) – File access type, append (‘a’), overwrite (‘w’).
series(t1, tq, steps, precision='dp')

Calculate \(\rho(t)\) over \([t_1, t_q]\) (where \(t_q > t_1\)) at \(q\) evenly spaced steps with \(\Delta t = \frac{t_q - t_1}{q}\). This is done via an implementation of Algorithm 5.2 (without optional balancing or minimisation of the Frobenius norm) as detailed in:

  1. Al-Mohy and N. Higham, SIAM Journal on Scientific Computing 33, 488 (2011).
Parameters:
  • t1 (float) – Starting time, \(t_1\).
  • tq (float) – Ending time, \(t_q\).
  • steps (integer) – \(q\).
  • precision (string, optional) – Target precision, accepts “sp” for single precision and “dp” for double precision. Defaults to “dp”.

Note

To gather or save \((\rho(t_0),...,\rho(t_q))\) or \((p(t_0),...,p(t_0))\):

set_omega(omega)

Re-defines \(\omega\) in the QSW master equation. This requires that \(\tilde{\mathcal{L}}\) be re-built, its communication pattern re-optimised and its one-norm series re-calculated.

Parameters:omega (float) – Interpolation parameter, \(\omega\)
step(t, precision='dp')

Calculate \(\rho(t)\). This is done via an implementation of Algorithm 3.2 (without optional balancing or minimisation of the Frobenius norm) as detailed in:

  1. Al-Mohy and N. Higham, SIAM Journal on Scientific Computing 33, 488 (2011).
Parameters:
  • t (float) – time, \(t\).
  • precision (string, optional) – Target precision, accepts “sp” for single precision and “dp” for double precision.

Note

To gather or save \(\rho(t)\) or \(p(t)\):

operators

qsw_mpi.operators.check_indices(G)

Ensure that as SciPY CSR matrix has column ordered indices and data arrays.

Parameters:G – SciPy CSR matrix
qsw_mpi.operators.import_superoperator(File, operatorname)

Import a \(\tilde{\mathcal{L}}\) from a HDF5 file.

Note

This method does not distrubted \(\tilde{\mathcal{L}}\).

Parameters:
  • File – H5Py File object.
  • operatorname (string) – Group containing the \(\tilde{\mathcal{L}}\) CSR arrays.
Returns:

Superoperator, \(\tilde{\mathcal{L}}\).
Return type:

\(\tilde{N}^2 \times \tilde{N}^2\) SciPy CSR matrix
qsw_mpi.operators.import_vsets(File, vsetsname)

Import \(V^D\) (see Step 1 of the demoralisation procedure) from a HDF5 file.

Parameters:File – H5Py File object.
Returns:Vertex subspaces, \(V^D\).
Return type:integer arrays
qsw_mpi.operators.local_lindblads(G)

Generate a Lindblad operator matrix from a graph transition matrix or adjacency matrix for use with the LQSW class (see Equation (15)).

Parameters:G (\(N \times N\) SciPy CSR matrix) – Adjacency matrix \(G\).
Return L:\(M_L\).
Return type:\(N \times N\) SciPy CSR matrix
qsw_mpi.operators.markov_chain(G)

Generate the cannonical Markov chain transition matrix (see Equation (14)) of a CSR adjacency matrix for use with the LQSW class (see Equation (15)).

Parameters:G (\(N \times N\) SciPy CSR matrix.) – Adjacency matrix, \(G\).
Returns:The cannonical Markov chain transition matrix of \(G\), \(C\).
Return type:\(N \times N\) SciPy CSR matrix
qsw_mpi.operators.nm_G(G, vsets)

Returns the demoralised graph \(G^D\), as per step 2 of the graph demoralisation procedure.

Parameters:
  • G (\(N \times N\) SciPy CSR matrix) – Adjacency matrix, \(G\).
  • vsets (integer arrays) – Vertex subspaces, \(V^D\), generated via nm_vsets().
Returns:

The demoralised graph, \(G^D\).
Return type:

\(\tilde{N} \times \tilde{N}\) SciPy CSR matrix
qsw_mpi.operators.nm_H_loc(vsets)

Returns the locally rotating Hamiltonian, \(H_\text{loc}\), as per step 4 of the graph demoralised procedure.

Parameters:vsets (integer arrays) – Vertex subspaces, \(V^D\), generated via nm_vsets().
Returns:The locally rotating Hamiltonian, \(H_\text{loc}\).
Return type:\(\tilde{N} \times \tilde{N}\) SciPy CSR matrix
qsw_mpi.operators.nm_L(G, vsets, perm_i=0)

Returns a global Lindblad operator, \(L^D\), capable of generating a non-moralising QSW, as per step 3 of the demoralisation procedure.

Parameters:
  • G (\(N \times N\) SciPy CSR matrix) – Adjacency matrix, \(G\).
  • vsets (integer arrays) – Vertex subspaces, \(V^D\), generated via nm_vsets().
  • perm_i (integer, optional) – Use the \(i^\text{th}\) permutation of the Fourier matrices to orthogonalise the vertex subspaces of \(L^D\).
Returns:

A demoralised global Lindblad operator, \(L^D\).
Return type:

\(\tilde{N} \times \tilde{N}\) SciPy CSR matrix
qsw_mpi.operators.nm_measure(rho, vsets)

Maps the vertex populations of \(\rho^D(t)\) or \((\rho^D(t_0),...,\rho^D(t_q))\) to \(p(t)\) or \((p(t_0),...,p(t_q))\) (see Equation (23)).

Parameters:
  • rho – \(\rho^D(t)\) or \((\rho^D(t_0),...,\rho^D(t_q))\).
  • vsets (integer arrays) – Vertex subspaces, \(V^D\), generated via nm_vsets().
Returns:

\(p(t)\) or \((p(t_0),...,p(t_q))\).
Return type:

\(N \times N\) or \(q \times N \times N\) NumPy complex array.
qsw_mpi.operators.nm_pop_inv_map(populations, vsets)

Maps the vertex populations of \(\rho^D(t)\) to \(\rho(t)\) (see Equation (23)) given \(p^D(t)\) or \((p^D(t_0),...,p^D(t_q))\).

Parameters:
  • rho – Vertex populations, \(p^D(t)\) or \((p^D(t_0),...,p^D(t_q))\).
  • vsets (integer arrays) – Vertex subspaces, \(V^D\), generated via nm_vsets().
Returns:

\(p(t)\) or \((p(t_0),...,p(t_q))\).
Return type:

\(N \times N\) or \(q \times N \times N\) NumPy complex array.
qsw_mpi.operators.nm_rho_map(probs, vsets)

Maps the vertex populations, \(p(t_0)\), of a QSW on graph \(G\), to the corresponding vertex subspaces of the demoralised graph, \(G^D\) (see Equation (23)).

Parameters:
  • probs (\(1 \times N\) float array) – Vertex populations of \(\rho(t_0)\).
  • vsets (integer arrays) – Vertex subspaces, \(V^D\), generated via nm_vsets().
Returns:

Density operator, \(\rho^D(t_0)\)
Return type:

\(\tilde{N} \times \tilde{N}\) SciPy CSR matrix
qsw_mpi.operators.nm_vsets(G)

Creates vertex subspaces, \(V^D\), as per Step 1 of the graph demoralisation procedure.

Parameters:G (\(N \times N\) SciPy CSR matrix) – Adjacency matrix, \(G\).
Returns:Vertex subspaces, \(V^D\).
Return type:integer arrays
qsw_mpi.operators.symmetrise(G)

Symmetrise a CSR matrix with a symmetric structure.

G is altered so as to satisfy \(G_{ij} = max(G_{ij}, G_{ji})\).

Parameters:G (SciPy real CSR matrix.) – CSR matrix to symmetrise.

Warning

The matrix must be structurally symmetric in terms of its non-zero entires.

qsw_mpi.operators.transition(gamma, G)

Generate the graph transition matrix (see Equation (5)) from a CSR adjacency matrix.

Parameters:
  • gamma (float) – Transition rate, \(\gamma\).
  • G (\(N \times N\) SciPy CSR matrix) – Adjacency matrix, \(G\).
Returns:

The transition matrix of \(G\), \(M\)
Return type:

\(N \times N\) SciPy CSR matrix