Quantum algorithms¶
This submodule provides access to various utility functions that act on Gaussian quantum states.
For more details on how the hafnian relates to various properties of Gaussian quantum states, see:
Kruse, R., Hamilton, C. S., Sansoni, L., Barkhofen, S., Silberhorn, C., & Jex, I. “Detailed study of Gaussian boson sampling.” Physical Review A 100, 032326 (2019)
Hamilton, C. S., Kruse, R., Sansoni, L., Barkhofen, S., Silberhorn, C., & Jex, I. “Gaussian boson sampling.” Physical Review Letters, 119(17), 170501 (2017)
Quesada, N. “Franck-Condon factors by counting perfect matchings of graphs with loops.” Journal of Chemical Physics 150, 164113 (2019)
Quesada, N., Helt, L. G., Izaac, J., Arrazola, J. M., Shahrokhshahi, R., Myers, C. R., & Sabapathy, K. K. “Simulating realistic non-Gaussian state preparation.” Physical Review A 100, 022341 (2019)
Fock states and tensors¶
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Returns the \(\langle i | \psi\rangle\) element of the state ket of a Gaussian state defined by covariance matrix cov. |
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Returns the state vector of a (PNR post-selected) Gaussian state. |
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Returns the \(\langle i | \rho | j \rangle\) element of the density matrix of a Gaussian state defined by covariance matrix cov. |
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Returns the density matrix of a (PNR post-selected) Gaussian state. |
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Calculates the Fock representation of a Gaussian unitary parametrized by the symplectic matrix S and the displacements alpha up to cutoff in Fock space. |
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Generate the Fock space probabilities of a Gaussian state up to a Fock space cutoff. |
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Given a list of transmissivities a tensor of probabilitites, calculate an updated tensor of probabilities after loss is applied. |
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Given a list of noise probability distributions for each of the modes and a tensor of probabilitites, calculate an updated tensor of probabilities after noise is applied. |
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Calculates the fidelity between two Gaussian quantum states. |
Details¶
-
pure_state_amplitude
(mu, cov, i, include_prefactor=True, tol=1e-10, hbar=2, check_purity=True)[source]¶ Returns the \(\langle i | \psi\rangle\) element of the state ket of a Gaussian state defined by covariance matrix cov.
- Parameters
mu (array) – length-\(2N\) quadrature displacement vector
cov (array) – length-\(2N\) covariance matrix
i (list) – list of amplitude elements
include_prefactor (bool) – if
True
, the prefactor is automatically calculated used to scale the result.tol (float) – tolerance for determining if displacement is negligible
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\).
check_purity (bool) – if
True
, the purity of the Gaussian state is checked before calculating the state vector.
- Returns
the pure state amplitude
- Return type
complex
-
state_vector
(mu, cov, post_select=None, normalize=False, cutoff=5, hbar=2, check_purity=True, **kwargs)[source]¶ Returns the state vector of a (PNR post-selected) Gaussian state.
The resulting density matrix will have shape
\[\underbrace{D\times D \times \cdots \times D}_M\]where \(D\) is the Fock space cutoff, and \(M\) is the number of non post-selected modes, i.e.
M = len(mu)//2 - len(post_select)
.If post_select is None then the density matrix elements are calculated using the multidimensional Hermite polynomials which provide a significantly faster evaluation.
- Parameters
mu (array) – length-\(2N\) means vector in xp-ordering
cov (array) – \(2N\times 2N\) covariance matrix in xp-ordering
post_select (dict) – dictionary containing the post-selected modes, of the form
{mode: value}
.normalize (bool) – If
True
, a post-selected density matrix is re-normalized.cutoff (dim) – the final length (i.e., Hilbert space dimension) of each mode in the density matrix.
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\).
check_purity (bool) – if
True
, the purity of the Gaussian state is checked before calculating the state vector.
- Keyword Arguments
choi_r (float or None) – Value of the two-mode squeezing parameter used in Choi-Jamiolkoski trick in
fock_tensor()
. This keyword argument should only be used whenstate_vector
is called byfock_tensor()
.- Returns
the state vector of the Gaussian state
- Return type
np.array[complex]
-
density_matrix_element
(mu, cov, i, j, include_prefactor=True, tol=1e-10, hbar=2)[source]¶ Returns the \(\langle i | \rho | j \rangle\) element of the density matrix of a Gaussian state defined by covariance matrix cov.
- Parameters
mu (array) – length-\(2N\) quadrature displacement vector
cov (array) – length-\(2N\) covariance matrix
i (list) – list of density matrix rows
j (list) – list of density matrix columns
include_prefactor (bool) – if
True
, the prefactor is automatically calculated used to scale the result.tol (float) – tolerance for determining if displacement is negligible
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\).
- Returns
the density matrix element
- Return type
complex
-
density_matrix
(mu, cov, post_select=None, normalize=False, cutoff=5, hbar=2)[source]¶ Returns the density matrix of a (PNR post-selected) Gaussian state.
The resulting density matrix will have shape
\[\underbrace{D\times D \times \cdots \times D}_{2M}\]where \(D\) is the Fock space cutoff, and \(M\) is the number of non post-selected modes, i.e.
M = len(mu)//2 - len(post_select)
.Note that we use the Strawberry Fields convention for indexing the density matrix; the first two dimensions correspond to subsystem 1, the second two dimensions correspond to subsystem 2, etc. If post_select is None then the density matrix elements are calculated using the multidimensional Hermite polynomials which provide a significantly faster evaluation.
- Parameters
mu (array) – length-\(2N\) means vector in xp-ordering
cov (array) – \(2N\times 2N\) covariance matrix in xp-ordering
post_select (dict) – dictionary containing the post-selected modes, of the form
{mode: value}
. If post_select is None the whole non post-selected density matrix is calculated directly using (multidimensional) Hermite polynomials, which is significantly faster than calculating one hafnian at a time.normalize (bool) – If
True
, a post-selected density matrix is re-normalized.cutoff (dim) – the final length (i.e., Hilbert space dimension) of each mode in the density matrix.
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\).
- Returns
the density matrix of the Gaussian state
- Return type
np.array[complex]
-
fock_tensor
(S, alpha, cutoff, choi_r=0.881373587019543, check_symplectic=True, sf_order=False, rtol=1e-05, atol=1e-08)[source]¶ Calculates the Fock representation of a Gaussian unitary parametrized by the symplectic matrix S and the displacements alpha up to cutoff in Fock space.
- Parameters
S (array) – symplectic matrix
alpha (array) – complex vector of displacements
cutoff (int) – cutoff in Fock space
choi_r (float) – squeezing parameter used for the Choi expansion
check_symplectic (boolean) – checks whether the input matrix is symplectic
sf_order (boolean) – reshapes the tensor so that it follows the sf ordering of indices
rtol (float) – the relative tolerance parameter used in np.allclose
atol (float) – the absolute tolerance parameter used in np.allclose
- Returns
Tensor containing the Fock representation of the Gaussian unitary
- Return type
(array)
-
probabilities
(mu, cov, cutoff, hbar=2.0, rtol=1e-05, atol=1e-08)[source]¶ Generate the Fock space probabilities of a Gaussian state up to a Fock space cutoff.
- Parameters
mu (array) – vector of means of length
2*n_modes
cov (array) – covariance matrix of shape
[2*n_modes, 2*n_modes]
cutoff (int) – cutoff in Fock space
hbar (float) – value of \(\hbar\) in the commutation relation :math;`[hat{x}, hat{p}]=ihbar`
rtol (float) – the relative tolerance parameter used in np.allclose
atol (float) – the absolute tolerance parameter used in np.allclose
- Returns
Fock space probabilities up to cutoff. The shape of this tensor is
[cutoff]*num_modes
.- Return type
(array)
-
update_probabilities_with_loss
(etas, probs)[source]¶ Given a list of transmissivities a tensor of probabilitites, calculate an updated tensor of probabilities after loss is applied.
- Parameters
etas (list) – List of transmissitivities describing the loss in each of the modes
probs (array) – Array of probabilitites in the different modes
- Returns
List of loss-updated probabilities with the same shape as probs.
- Return type
array
-
update_probabilities_with_noise
(probs_noise, probs)[source]¶ Given a list of noise probability distributions for each of the modes and a tensor of probabilitites, calculate an updated tensor of probabilities after noise is applied.
- Parameters
probs_noise (list) – List of probability distributions describing the noise in each of the modes
probs (array) – Array of probabilitites in the different modes
- Returns
List of noise-updated probabilities with the same shape as probs.
- Return type
array
Utility functions¶
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Returns the vector of means and the covariance matrix of the specified modes. |
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Returns the matrix \(X_n = \begin{bmatrix}0 & I_n\\ I_n & 0\end{bmatrix}\) |
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Returns the \(Q\) Husimi matrix of the Gaussian state. |
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Returns the Wigner covariance matrix in the \(xp\)-ordering of the Gaussian state. |
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Returns the \(A\) matrix of the Gaussian state whose hafnian gives the photon number probabilities. |
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Returns the vector of complex displacements and conjugate displacements. |
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Returns the vector of real quadrature displacements. |
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Returns the prefactor. |
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Returns the scaling parameter by which the adjacency matrix A should be rescaled so that the Gaussian state that endodes it has a total mean photon number n_mean. |
Given an adjacency matrix this function calculates the mean number of clicks. |
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Returns the scaling parameter by which the adjacency matrix A should be rescaled so that the Gaussian state that encodes it has give a mean number of clicks equal to |
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Returns the Qmat xp-covariance matrix associated to a graph with adjacency matrix \(A\) and with mean photon number \(n_{mean}\). |
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Calculate the mean photon number of mode j of a Gaussian state. |
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Calculate the mean photon number of each of the modes in a Gaussian state |
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Calculate the variance/covariance of the photon number distribution of a Gaussian state. |
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Calculate the covariance matrix of the photon number distribution of a Gaussian state. |
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Checks if the covariance matrix is a valid quantum covariance matrix. |
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Checks if the covariance matrix is a valid quantum covariance matrix that corresponds to a quantum pure state |
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Checks if the covariance matrix can be efficiently sampled. |
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Calculates the total photon number distribution of a pure state with zero mean. |
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Generate the photon number distribution of \(N\) identical single mode squeezed states. |
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Generates the total photon number distribution of single mode squeezed states with different squeezing values. |
Details¶
-
Amat
(cov, hbar=2, cov_is_qmat=False)[source]¶ Returns the \(A\) matrix of the Gaussian state whose hafnian gives the photon number probabilities.
- Parameters
cov (array) – \(2N\times 2N\) covariance matrix
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\).
cov_is_qmat (bool) – if
True
, it is assumed thatcov
is in fact the Q matrix.
- Returns
the \(A\) matrix.
- Return type
array
-
Beta
(mu, hbar=2)[source]¶ Returns the vector of complex displacements and conjugate displacements.
- Parameters
mu (array) – length-\(2N\) means vector
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\).
- Returns
the expectation values \([\langle a_1\rangle, \langle a_2\rangle,\dots,\langle a_N\rangle, \langle a^\dagger_1\rangle, \dots, \langle a^\dagger_N\rangle]\)
- Return type
array
-
Covmat
(Q, hbar=2)[source]¶ Returns the Wigner covariance matrix in the \(xp\)-ordering of the Gaussian state. This is the inverse function of Qmat.
- Parameters
Q (array) – \(2N\times 2N\) Husimi Q matrix
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\).
- Returns
the \(xp\)-ordered covariance matrix in the xp-ordering.
- Return type
array
-
Means
(beta, hbar=2)[source]¶ Returns the vector of real quadrature displacements.
- Parameters
beta (array) – length-\(2N\) means bivector
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\).
- Returns
the quadrature expectation values \([\langle q_1\rangle, \langle q_2\rangle,\dots,\langle q_N\rangle, \langle p_1\rangle, \dots, \langle p_N\rangle]\)
- Return type
array
-
Qmat
(cov, hbar=2)[source]¶ Returns the \(Q\) Husimi matrix of the Gaussian state.
- Parameters
cov (array) – \(2N\times 2N xp-\) Wigner covariance matrix
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\).
- Returns
the \(Q\) matrix.
- Return type
array
-
Xmat
(N)[source]¶ Returns the matrix \(X_n = \begin{bmatrix}0 & I_n\\ I_n & 0\end{bmatrix}\)
- Parameters
N (int) – positive integer
- Returns
\(2N\times 2N\) array
- Return type
array
-
fidelity
(mu1, cov1, mu2, cov2, hbar=2, rtol=1e-05, atol=1e-08)[source]¶ Calculates the fidelity between two Gaussian quantum states.
Note that if the covariance matrices correspond to pure states this function reduces to the modulus square of the overlap of their state vectors. For the derivation see ‘Quantum Fidelity for Arbitrary Gaussian States’, Banchi et al..
- Parameters
mu1 (array) – vector of means of the first state
cov1 (array) – covariance matrix of the first state
mu2 (array) – vector of means of the second state
cov2 (array) – covariance matrix of the second state
hbar (float) – value of hbar in the uncertainty relation
rtol (float) – the relative tolerance parameter used in np.allclose
atol (float) – the absolute tolerance parameter used in np.allclose
- Returns
value of the fidelity between the two states
- Return type
(float)
-
find_scaling_adjacency_matrix
(A, n_mean)[source]¶ Returns the scaling parameter by which the adjacency matrix A should be rescaled so that the Gaussian state that endodes it has a total mean photon number n_mean.
- Parameters
A (array) – Adjacency matrix
n_mean (float) – Mean photon number of the Gaussian state
- Returns
Scaling parameter
- Return type
float
-
find_scaling_adjacency_matrix_torontonian
(A, c_mean)[source]¶ Returns the scaling parameter by which the adjacency matrix A should be rescaled so that the Gaussian state that encodes it has give a mean number of clicks equal to
c_mean
when measured with threshold detectors.- Parameters
A (array) – adjacency matrix
c_mean (float) – mean photon number of the Gaussian state
- Returns
scaling parameter
- Return type
float
-
fock_tensor
(S, alpha, cutoff, choi_r=0.881373587019543, check_symplectic=True, sf_order=False, rtol=1e-05, atol=1e-08)[source] Calculates the Fock representation of a Gaussian unitary parametrized by the symplectic matrix S and the displacements alpha up to cutoff in Fock space.
- Parameters
S (array) – symplectic matrix
alpha (array) – complex vector of displacements
cutoff (int) – cutoff in Fock space
choi_r (float) – squeezing parameter used for the Choi expansion
check_symplectic (boolean) – checks whether the input matrix is symplectic
sf_order (boolean) – reshapes the tensor so that it follows the sf ordering of indices
rtol (float) – the relative tolerance parameter used in np.allclose
atol (float) – the absolute tolerance parameter used in np.allclose
- Returns
Tensor containing the Fock representation of the Gaussian unitary
- Return type
(array)
-
gen_Qmat_from_graph
(A, n_mean)[source]¶ Returns the Qmat xp-covariance matrix associated to a graph with adjacency matrix \(A\) and with mean photon number \(n_{mean}\).
- Parameters
A (array) – a \(N\times N\)
np.float64
(symmetric) adjacency matrixn_mean (float) – mean photon number of the Gaussian state
- Returns
the \(2N\times 2N\) Q matrix.
- Return type
array
-
gen_multi_mode_dist
(s, cutoff=50, padding_factor=2)[source]¶ Generates the total photon number distribution of single mode squeezed states with different squeezing values.
- Parameters
s (array) – array of squeezing parameters
cutoff (int) – Fock cutoff
- Returns
total photon number distribution
- Return type
(array[int])
-
gen_single_mode_dist
(s, cutoff=50, N=1)[source]¶ Generate the photon number distribution of \(N\) identical single mode squeezed states.
- Parameters
s (float) – squeezing parameter
cutoff (int) – Fock cutoff
N (float) – number of squeezed states
- Returns
Photon number distribution
- Return type
(array)
-
is_classical_cov
(cov, hbar=2, atol=1e-08)[source]¶ Checks if the covariance matrix can be efficiently sampled.
- Parameters
cov (array) – a covariance matrix
hbar (float) – value of hbar in the uncertainty relation
- Returns
whether the given covariance matrix corresponds to a classical state
- Return type
(bool)
-
is_pure_cov
(cov, hbar=2, rtol=1e-05, atol=1e-08)[source]¶ Checks if the covariance matrix is a valid quantum covariance matrix that corresponds to a quantum pure state
- Parameters
cov (array) – a covariance matrix
hbar (float) – value of hbar in the uncertainty relation
rtol (float) – the relative tolerance parameter used in np.allclose
atol (float) – the absolute tolerance parameter used in np.allclose
- Returns
whether the given covariance matrix corresponds to a pure state
- Return type
(bool)
-
is_valid_cov
(cov, hbar=2, rtol=1e-05, atol=1e-08)[source]¶ Checks if the covariance matrix is a valid quantum covariance matrix.
- Parameters
cov (array) – a covariance matrix
hbar (float) – value of hbar in the uncertainty relation
rtol (float) – the relative tolerance parameter used in np.allclose
atol (float) – the absolute tolerance parameter used in np.allclose
- Returns
whether the given covariance matrix is a valid covariance matrix
- Return type
(bool)
-
loss_mat
(eta, cutoff)[source]¶ Constructs a binomial loss matrix with transmission eta up to n photons.
- Parameters
eta (float) – Transmission coefficient.
eta=0.0
corresponds to complete loss andeta=1.0
corresponds to no loss.cutoff (int) – cutoff in Fock space.
- Returns
\(n\times n\) matrix representing the loss.
- Return type
array
-
mean_number_of_clicks
(A)[source]¶ Given an adjacency matrix this function calculates the mean number of clicks. For this to make sense the user must provide a matrix with singular values less than or equal to one. See Appendix A.3 of <https://arxiv.org/abs/1902.00462>`_ by Banchi et al.
- Parameters
A (array) – rescaled adjacency matrix
- Returns
mean number of clicks
- Return type
float
-
photon_number_covar
(mu, cov, j, k, hbar=2)[source]¶ Calculate the variance/covariance of the photon number distribution of a Gaussian state.
Implements the covariance matrix of the photon number distribution of a Gaussian state according to the Last two eq. of Part II. in ‘Multidimensional Hermite polynomials and photon distribution for polymode mixed light’, Dodonov et al.
\[\begin{split}\sigma_{n_j n_j} &= \frac{1}{2}\left(T_j^2 - 2d_j - \frac{1}{2}\right) + \left<\mathbf{Q}_j\right>\mathcal{M}_j\left<\mathbf{Q}_j\right>, \\ \sigma_{n_j n_k} &= \frac{1}{2}\mathrm{Tr}\left(\Lambda_j \mathbf{M} \Lambda_k \mathbf{M}\right) + \frac{1}{2}\left<\mathbf{Q}\right>\Lambda_j \mathbf{M} \Lambda_k\left<\mathbf{Q}\right>,\end{split}\]where \(T_j\) and \(d_j\) are the trace and the determinant of \(2 \times 2\) matrix \(\mathcal{M}_j\) whose elements coincide with the nonzero elements of matrix \(\mathbf{M}_j = \Lambda_j \mathbf{M} \Lambda_k\) while the two-vector \(\mathbf{Q}_j\) has the components \((q_j, p_j)\). \(2N \times 2N\) projector matrix \(\Lambda_j\) has only two nonzero elements: \(\left(\Lambda_j\right)_{jj} = \left(\Lambda_j\right)_{j+N,j+N} = 1\). Note that the convention for
mu
used here differs from the one used in Dodonov et al., They both provide the same results in this particular case.- Parameters
mu (array) – vector of means of the Gaussian state using the ordering \([q_1, q_2, \dots, q_n, p_1, p_2, \dots, p_n]\)
cov (array) – the covariance matrix of the Gaussian state
j (int) – the j th mode
k (int) – the k th mode
hbar (float) – the
hbar
convention used in the commutation relation \([q, p]=i\hbar\)
- Returns
the covariance for the photon numbers at modes \(j\) and \(k\).
- Return type
float
-
photon_number_covmat
(mu, cov, hbar=2)[source]¶ Calculate the covariance matrix of the photon number distribution of a Gaussian state.
- Parameters
mu (array) – vector of means of the Gaussian state using the ordering \([q_1, q_2, \dots, q_n, p_1, p_2, \dots, p_n]\)
cov (array) – the covariance matrix of the Gaussian state
hbar (float) – the
hbar
convention used in the commutation relation \([q, p]=i\hbar\)
- Returns
the covariance matrix of the photon number distribution
- Return type
array
-
photon_number_mean
(mu, cov, j, hbar=2)[source]¶ Calculate the mean photon number of mode j of a Gaussian state.
- Parameters
mu (array) – vector of means of the Gaussian state using the ordering \([q_1, q_2, \dots, q_n, p_1, p_2, \dots, p_n]\)
cov (array) – the covariance matrix of the Gaussian state
j (int) – the j th mode
hbar (float) – the
hbar
convention used in the commutation relation \([q, p]=i\hbar\)
- Returns
the mean photon number in mode \(j\).
- Return type
float
-
photon_number_mean_vector
(mu, cov, hbar=2)[source]¶ Calculate the mean photon number of each of the modes in a Gaussian state
- Parameters
mu (array) – vector of means of the Gaussian state using the ordering \([q_1, q_2, \dots, q_n, p_1, p_2, \dots, p_n]\)
cov (array) – the covariance matrix of the Gaussian state
hbar (float) – the
hbar
convention used in the commutation relation \([q, p]=i\hbar\)
- Returns
the vector of means of the photon number distribution
- Return type
array
-
prefactor
(mu, cov, hbar=2)[source]¶ Returns the prefactor.
\[prefactor = \frac{e^{-\beta Q^{-1}\beta^*/2}}{n_1!\cdots n_m! \sqrt{|Q|}}\]- Parameters
mu (array) – length-\(2N\) vector of mean values \([\alpha,\alpha^*]\)
cov (array) – length-\(2N\) xp-covariance matrix
- Returns
the prefactor
- Return type
float
-
probabilities
(mu, cov, cutoff, hbar=2.0, rtol=1e-05, atol=1e-08)[source] Generate the Fock space probabilities of a Gaussian state up to a Fock space cutoff.
- Parameters
mu (array) – vector of means of length
2*n_modes
cov (array) – covariance matrix of shape
[2*n_modes, 2*n_modes]
cutoff (int) – cutoff in Fock space
hbar (float) – value of \(\hbar\) in the commutation relation :math;`[hat{x}, hat{p}]=ihbar`
rtol (float) – the relative tolerance parameter used in np.allclose
atol (float) – the absolute tolerance parameter used in np.allclose
- Returns
Fock space probabilities up to cutoff. The shape of this tensor is
[cutoff]*num_modes
.- Return type
(array)
-
reduced_gaussian
(mu, cov, modes)[source]¶ Returns the vector of means and the covariance matrix of the specified modes.
- Parameters
mu (array) – a length-\(2N\)
np.float64
vector of means.cov (array) – a \(2N\times 2N\)
np.float64
covariance matrix representing an \(N\) mode quantum state.modes (int of Sequence[int]) – indices of the requested modes
- Returns
where means is an array containing the vector of means, and cov is a square array containing the covariance matrix.
- Return type
tuple (means, cov)
-
total_photon_num_dist_pure_state
(cov, cutoff=50, hbar=2, padding_factor=2)[source]¶ Calculates the total photon number distribution of a pure state with zero mean.
- Parameters
cov (array) – \(2N\times 2N\) covariance matrix in xp-ordering
cutoff (int) – Fock cutoff
tol (float) – tolerance for determining if displacement is negligible
hbar (float) – the value of \(\hbar\) in the commutation
padding_factor (int) – expanded size of the photon distribution to avoid accumulation of errors
- Returns
Total photon number distribution
- Return type
(array)
-
update_probabilities_with_loss
(etas, probs)[source] Given a list of transmissivities a tensor of probabilitites, calculate an updated tensor of probabilities after loss is applied.
- Parameters
etas (list) – List of transmissitivities describing the loss in each of the modes
probs (array) – Array of probabilitites in the different modes
- Returns
List of loss-updated probabilities with the same shape as probs.
- Return type
array
-
update_probabilities_with_noise
(probs_noise, probs)[source] Given a list of noise probability distributions for each of the modes and a tensor of probabilitites, calculate an updated tensor of probabilities after noise is applied.
- Parameters
probs_noise (list) – List of probability distributions describing the noise in each of the modes
probs (array) – Array of probabilitites in the different modes
- Returns
List of noise-updated probabilities with the same shape as probs.
- Return type
array
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