Auxillary functions¶

This module contains auxiliary functions for extracting the parameters of quadratic and Bose-Hubbard Hamiltonians.

Quadratic Hamiltonian functions¶

quadratic_coefficients(operator) Return the quadratic coefficient matrix representing a Gaussian Hamiltonian.

Bose Hubbard functions¶

`BoseHubbardError`	Custom error function for invalid Bose-Hubbard Hamiltonians.
`extract_tunneling`(H)	Extracts the tunneling terms from Bose-Hubbard Hamiltonians.
`extract_onsite_chemical`(H)	Extracts the onsite interactions and chemical potential terms from Bose-Hubbard Hamiltonians.
`extract_dipole`(H)	Extracts the dipole terms from Bose-Hubbard Hamiltonians.
`trotter_layer`(H, t, k)	Returns a single Trotter layer for a Bose-Hubbard Hamiltonian.

Code details¶

sfopenboson.auxillary.quadratic_coefficients(operator)[source]¶

Return the quadratic coefficient matrix representing a Gaussian Hamiltonian.

A Gaussian Hamiltonian is any combination of quadratic operators that can be written in quadratic form:

\[H = \frac{1}{2}\mathbf{r}^T A\mathbf{r} + \mathbf{r}^T \mathbf{d}\]

where \(A\in\mathbb{R}^{2N\times 2N}\) is a symmetric matrix, \(\mathbf{d}\in\mathbb{R}^{2N}\) is a real vector, and \(\mathbf{r} = (\x_1,\dots,\x_N,\p_1,\dots,\p_N)\) is the vector of quadrature operators in \(xp\)-ordering.

This function accepts a bosonic Gaussian Hamiltonian, and returns the matrix \(A\) and vector \(\mathbf{d}\) representing the quadratic and linear coefficients.

Parameters:	operator (QuadOperator) – a bosonic Gaussian Hamiltonian
Returns:	a tuple contains a 2Nx2N real symmetric numpy array, and a length-2N real numpy array, where N is the number of modes the operator acts on.
Return type:	tuple (A, d)

exception sfopenboson.auxillary.BoseHubbardError[source]¶

Custom error function for invalid Bose-Hubbard Hamiltonians.

with_traceback(tb)[source]¶: This method sets argument tb as the new traceback for the exception and returns the exception object. See the Python documentation for more details.

sfopenboson.auxillary.extract_tunneling(H)[source]¶

Extracts the tunneling terms from Bose-Hubbard Hamiltonians.

Parameters:	H (BosonOperator) – A Bose-Hubbard Hamiltonian.
Returns:	Returns a length-2 list of the form `[[(i, j),...], t]` where `[(i, j),...]` is a list containing pairs of modes that are entangled by beamsplitters due to tunneling, and `t` (float) is the tunneling coefficient.
Return type:	list

sfopenboson.auxillary.extract_onsite_chemical(H)[source]¶

Extracts the onsite interactions and chemical potential terms from Bose-Hubbard Hamiltonians.

Parameters:	H (BosonOperator) – A Bose-Hubbard Hamiltonian.
Returns:	Returns a tuple containing two lists; the first list is the onsite interaction, and the second list is the chemical potential. Each list is a length-2 list of the form `[[i,j,...], t]` where `[i,j,...]` is a list containing modes operated on, and t is the onsite coefficient U or chemical potential mu.
Return type:	tuple(list, list)

sfopenboson.auxillary.extract_dipole(H)[source]¶

Extracts the dipole terms from Bose-Hubbard Hamiltonians.

Parameters:	H (BosonOperator) – A Bose-Hubbard Hamiltonian.
Returns:	Returns a length-2 list of the form `[[(i, j),...], V]` where `[(i, j),...]` is a list containing pairs of modes that are entangled due to nearest-neighbour interactions, and `V` (float) is the dipole coefficient.
Return type:	list

sfopenboson.auxillary.trotter_layer(H, t, k)[source]¶

Returns a single Trotter layer for a Bose-Hubbard Hamiltonian.

Parameters:

H (BosonOperator) – A Bose-Hubbard Hamiltonian.
t (float) – the time propagation duration.
k (int) – the number of products in the truncated Lie product formula.

Returns:

A dictionary containing the items:

'BS': (theta, phi, modes) corresponding to beamsplitters with parameters (theta, phi) acting on the list of modes provided.
'K': (kappa, modes) corresponding to Kerr gates with parameter kappa acting on the list of modes provided.
'R': (r, modes) corresponding to rotation gates with parameter r acting on the list of modes provided.
'CK': (kappa, modes) corresponding to a cross-Kerr interactions with parameter kappa acting on the list of modes provided.

Return type:

dict