Quantum algorithms¶

Module name: thewalrus.quantum

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¶

`pure_state_amplitude`(mu, cov, i[, ...])	Returns the \(\langle i \| \psi\rangle\) element of the state ket of a Gaussian state defined by covariance matrix cov.
`state_vector`(mu, cov[, post_select, ...])	Returns the state vector of a (PNR post-selected) Gaussian state.
`density_matrix_element`(mu, cov, i, j[, ...])	Returns the \(\langle i \| \rho \| j \rangle\) element of the density matrix of a Gaussian state defined by covariance matrix cov.
`density_matrix`(mu, cov[, post_select, ...])	Returns the density matrix of a (PNR post-selected) Gaussian state.
`fock_tensor`(S, alpha, cutoff[, choi_r, ...])	Calculates the Fock representation of a Gaussian unitary parametrized by the symplectic matrix S and the displacements alpha up to cutoff in Fock space.
`probabilities`(mu, cov, cutoff[, parallel, ...])	Generate the Fock space probabilities of a Gaussian state up to a Fock space cutoff.
`loss_mat`(eta, cutoff)	Constructs a binomial loss matrix with transmission eta up to n photons.
`update_probabilities_with_loss`(etas, probs)	Given a list of transmissivities a tensor of probabilitites, calculate an updated tensor of probabilities after loss is applied.
`update_probabilities_with_noise`(probs_noise, ...)	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.
`find_classical_subsystem`(cov[, hbar, atol])	Find the largest integer `k` so that subsystem in modes `[0,1,...,k-1]` is a classical state.
`tvd_cutoff_bounds`(mu, cov, cutoff[, hbar, ...])	Gives bounds of the total variation distance between the exact Gaussian Boson Sampling distribution extending to infinity in Fock space and the ones truncated by any value between 0 and the user provided cutoff.
`n_body_marginals`(mean, cov, cutoff, n[, hbar])	Calculates the first n-body marginals of a Gaussian state.

Adjacency matrices¶

`adj_scaling`(A, n_mean)	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.
`adj_scaling_torontonian`(A, c_mean)	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.
`adj_to_qmat`(A, n_mean)	Returns the Qmat xp-covariance matrix associated to a graph with adjacency matrix \(A\) and with mean photon number \(n_{mean}\).

Gaussian checks¶

`is_valid_cov`(cov[, hbar, rtol, atol])	Checks if the covariance matrix is a valid quantum covariance matrix.
`is_pure_cov`(cov[, hbar, rtol, atol])	Checks if the covariance matrix is a valid quantum covariance matrix that corresponds to a quantum pure state
`is_classical_cov`(cov[, hbar, atol])	Checks if the covariance matrix can be efficiently sampled.
`fidelity`(mu1, cov1, mu2, cov2[, hbar, rtol, ...])	Calculates the fidelity between two Gaussian quantum states.
`is_symplectic`(S[, rtol, atol])	Checks if the matrix is symplectic.

Conversions¶

`reduced_gaussian`(mu, cov, modes)	Returns the vector of means and the covariance matrix of the specified modes.
`Xmat`(N)	Returns the matrix \(X_n = \begin{bmatrix}0 & I_n\\ I_n & 0\end{bmatrix}\)
`Qmat`(cov[, hbar])	Returns the \(Q\) Husimi matrix of the Gaussian state.
`Covmat`(Q[, hbar])	Returns the Wigner covariance matrix in the \(xp\)-ordering of the Gaussian state.
`Amat`(cov[, hbar, cov_is_qmat])	Returns the \(A\) matrix of the Gaussian state whose hafnian gives the photon number probabilities.
`complex_to_real_displacements`(mu[, hbar])	Returns the vector of complex displacements and conjugate displacements.
`real_to_complex_displacements`(beta[, hbar])	Returns the vector of real quadrature displacements.

Means and variances¶

`photon_number_mean`(mu, cov, j[, hbar])	Calculate the mean photon number of mode j of a Gaussian state.
`photon_number_mean_vector`(mu, cov[, hbar])	Calculate the mean photon number of each of the modes in a Gaussian state
`photon_number_covar`(mu, cov, j, k[, hbar])	Calculate the variance/covariance of the photon number distribution of a Gaussian state.
`photon_number_covmat`(mu, cov[, hbar])	Calculate the covariance matrix of the photon number distribution of a Gaussian state.
`photon_number_expectation`(mu, cov, modes[, hbar])	Calculates the expectation value of the product of the number operator of the modes in a Gaussian state.
`photon_number_squared_expectation`(mu, cov, modes)	Calculates the expectation value of the square of the product of the number operator of the modes in a Gaussian state.
`normal_ordered_expectation`(mu, cov, rpt[, hbar])	Calculates the expectation value of the normal ordered product \(\prod_{i=0}^{N-1} a_i^{\dagger n_i} \prod_{j=0}^{N-1} a_j^{m_j}\) with respect to an N-mode Gaussian state, where \(\text{rpt}=(n_0, n_1, \ldots, n_{N-1}, m_0, m_1, \ldots, m_{N-1})\).
`photon_number_moment`(mu, cov, indices[, hbar])	Calculates the expectation value of product of powers of photon number operators of a Gaussian state.
`s_ordered_expectation`(mu, cov, rpt[, hbar, s])	Calculates the expectation value of the s-ordered product obtained by taking deirvatives of the characteristic function of a Gaussian states, Here, \(\text{rpt}=(n_0, n_1, \ldots, n_{N-1}, m_0, m_1, \ldots, m_{N-1})\).
`mean_clicks`(cov[, hbar])	Calculates the total mean number of clicks when a zero-mean gaussian state is measured using threshold detectors.
`variance_clicks`(cov[, hbar])	Calculates the variance of the total number of clicks when a zero-mean gaussian state is measured using threshold detectors.
`photon_number_cumulant`(mu, cov, modes[, hbar])	Calculates the photon-number cumulant of the modes in the Gaussian state.
`click_cumulant`(mu, cov, modes[, hbar])	Calculates the click cumulant of the modes in the Gaussian state.

Photon number distributions¶

`pure_state_distribution`(cov[, cutoff, hbar, ...])	Calculates the total photon number distribution of a pure state with zero mean.
`total_photon_number_distribution`(n, k, s, eta)	Probability of observing a total of \(n\) photons when \(k\) identical single-mode squeezed vacua with squeezing parameter \(s\) undergo loss by transmission \(\eta\).
`characteristic_function`(k, s, eta, mu[, ...])	Calculates the expectation value of the characteristic function \(\langle n^m \exp(mu n) \rangle\) where \(n\) is the total photon number of \(k\) identical single-mode squeezed vacua with squeezing parameter \(s\) undergoing loss by transmission \(\eta\).

Entanglement¶

`vonneumann_entropy`(cov[, hbar])	Returns the vonNeumann entropy of a covariance matrix.
`entanglement_entropy`(cov[, modes_A, split, hbar])	Returns the entanglement entropy of a covariance matrix under a given bipartition.
`log_negativity`(cov[, modes_A, split, hbar])	Returns the logarithmic negativity of a covariance matrix under a given bipartition.

Code 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 that cov is in fact the Q matrix.

Returns:

the \(A\) matrix.

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

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

adj_scaling(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

adj_scaling_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

adj_to_qmat(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 matrix
n_mean (float) – mean photon number of the Gaussian state

Returns:

the \(2N\times 2N\) Q matrix.

Return type:

array

characteristic_function(k, s, eta, mu, max_iter=10000, delta=1e-14, poly_corr=None)[source]¶

Calculates the expectation value of the characteristic function \(\langle n^m \exp(mu n) \rangle\) where \(n\) is the total photon number of \(k\) identical single-mode squeezed vacua with squeezing parameter \(s\) undergoing loss by transmission \(\eta\).

Parameters:

k (int) – number of squeezed modes
s (float) – squeezing parameter
eta (float) – transmission parameter, between 0 and 1 inclusive
mu (float) – value at which to evaluate the characteristic function
max_iter (int) – maximum number of terms allowed in the sum
delta (float) – fractional change in the sum after which the sum is stopped
poly_corr (int) – give the value of the exponent \(m\) of the polynomial correction

Returns:

the expected value of the moment generation function

Return type:

(float)

complex_to_real_displacements(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

entanglement_entropy(cov, modes_A=None, split=None, hbar=2)[source]¶

Returns the entanglement entropy of a covariance matrix under a given bipartition.

Parameters:

cov (array) – a covariance matrix
modes_A (iterable or int) – the subset of modes used for the bipartition
split (int) – the index of the mode separating the two partitions (alternative to modes_A)
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\)

Returns:

logarithmic negativity

Return type:

(float)

fidelity(mu1, cov1, mu2, cov2, hbar=2, rtol=1e-05, atol=1e-08)[source]¶

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..

The actual implementation used here corresponds to the square of Eq. 96 of ‘Gaussian states and operations - a quick reference’, Brask.

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_classical_subsystem(cov, hbar=2, atol=1e-08)[source]¶

Find the largest integer k so that subsystem in modes [0,1,...,k-1] is a classical state.

Parameters:

cov (array) – a covariance matrix
hbar (float) – value of hbar in the uncertainty relation
atol (float) – the absolute tolerance parameter used when determining if the state is classical

Returns:

the largest k so that modes [0,1,...,k-1] are in a classical state.

Return type:

int

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)

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
atol (float) – the absolute tolerance parameter used in np.allclose

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_symplectic(S, rtol=1e-05, atol=1e-08)[source]¶

Checks if the matrix is symplectic.

Parameters:

S (array) – a matrix
rtol (float) – the relative tolerance parameter used in np.allclose
atol (float) – the absolute tolerance parameter used in np.allclose

Returns:

whether the given matrix is symplectic

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)

log_negativity(cov, modes_A=None, split=None, hbar=2)[source]¶

Returns the logarithmic negativity of a covariance matrix under a given bipartition.

Parameters:

cov (array) – a covariance matrix
modes_A (iterable or int) – the subset of modes used for the bipartition
split (int) – the index of the mode separating the two partitions (alternative to modes_A)

Returns:

entanglement entropy

Return type:

(float)

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 and eta=1.0 corresponds to no loss.
cutoff (int) – cutoff in Fock space.

Returns:

\(n\times n\) matrix representing the loss.

Return type:

array

mean_clicks(cov, hbar=2)[source]¶

Calculates the total mean number of clicks when a zero-mean gaussian state is measured using threshold detectors.

Args: cov (array): \(2N\times 2N\) covariance matrix in xp-ordering hbar (float): the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\)
Returns: float: mean number of clicks

n_body_marginals(mean, cov, cutoff, n, hbar=2)[source]¶

Calculates the first n-body marginals of a Gaussian state.

For an M-mode Gaussian state there exists a photon number distribution with probability mass function \(p[i_0,i_1,\ldots, i_{M-1}]\). The function n_body_marginals calculates the first n-body marginals of the (all-mode) probability distribution \(p\). The \(n=1\) marginals or single body marginals are simply the probability that mode \(k\) has \(i\) photons, i.e. \(p_k[i]\). For \(n=2\) one obtains the two-body probabilities. For two modes \(k\) and \(l\) this is a two dimensional probability distribution \(p_{k,l}[i,j]\). Note that these marginals have interesting permutation properties, for example \(p_{k,l}[i,j] = p_{l,k}[j,i]\).

The function provided here takes advantage of these symmetries to minimize the amount of calculations. The return of this function is a list of tensors where the first entry contains the one-body marginals of the \(M\) modes (giving a tensor of shape (M, cutoff)), the second entry contains the two-body marginals (giving a tensor of shape (M,M,cutoff, cutoff)) and so on and so forth.

To be clever about not calculating things that can be obtained by permutations it checks whether the index vector representing the modes is sorted. From the way itertools.product works we know that it will always produce a sorted index vector before generating any of its unordered permutations. Thus whenever the index vector is ordered we perform the numerical calculation.

If it is an unsorted index vector it realizes, in the if statement, that it can be obtained by permuting the marginal distribution of something that has already been calculated.

Parameters:

mean (array) – length-\(2N\) quadrature displacement vector
cov (array) – length-\(2N\) covariance matrix
cutoff (int) – cutoff in Fock space
n (int) – order of the correlations
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\).

Returns:

List with arrays containing the \(1,..,n\) body marginal: distributions of the modes

Return type:

list[array]

normal_ordered_expectation(mu, cov, rpt, hbar=2)[source]¶

Calculates the expectation value of the normal ordered product \(\prod_{i=0}^{N-1} a_i^{\dagger n_i} \prod_{j=0}^{N-1} a_j^{m_j}\) with respect to an N-mode Gaussian state, where \(\text{rpt}=(n_0, n_1, \ldots, n_{N-1}, m_0, m_1, \ldots, m_{N-1})\).

Parameters:

mu (array) – length-\(2N\) means vector in xp-ordering.
cov (array) – \(2N\times 2N\) covariance matrix in xp-ordering.
rpt (list) – integers specifying the terms to calculate.
hbar (float) – value of hbar in the uncertainty relation.

Returns:

expectation value of the normal ordered product of operators

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.

\[\begin{split}\sigma_{n_j n_j} &= \frac{1}{2}\left(T_j^2 - 2d_j - \frac{1}{2}\right) + \left<\mathbf{Q}_j\right>\mathcal{M}_j\left<\mathbf{Q}_j\right>, \\ \sigma_{n_j n_k} &= \frac{1}{2}\mathrm{Tr}\left(\Lambda_j \mathbf{M} \Lambda_k \mathbf{M}\right) + \left<\mathbf{Q}\right>\Lambda_j \mathbf{M} \Lambda_k\left<\mathbf{Q}\right>,\end{split}\]

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. Also note that the original reference of Dodonov et al. has an incorrect prefactor of 1/2 in the last terms of the last equation above.

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_cumulant(mu, cov, modes, hbar=2)[source]¶

Calculates the photon-number cumulant of the modes in the Gaussian state.

Parameters:

mu (array) – length-\(2N\) means vector in xp-ordering.
cov (array) – \(2N\times 2N\) covariance matrix in xp-ordering.
modes (list or array) – list of modes. Note that it can have repetitions.
hbar (float) – value of hbar in the uncertainty relation.

Returns:

the cumulant

Return type:

(float)

photon_number_expectation(mu, cov, modes, hbar=2)[source]¶

Calculates the expectation value of the product of the number operator of the modes in a Gaussian state.

Parameters:

mu (array) – length-\(2N\) means vector in xp-ordering.
cov (array) – \(2N\times 2N\) covariance matrix in xp-ordering.
modes (list) – list of modes
hbar (float) – value of hbar in the uncertainty relation.

Returns:

expectation value of the product of the number operators of the modes.

Return type:

(float)

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

photon_number_moment(mu, cov, indices, hbar=2)[source]¶

Calculates the expectation value of product of powers of photon number operators of a Gaussian state. The powers are specified by a dictionary with modes as keys and powers as values.

The calculation is performed by first writing any power of the photon number as

\((a^\dagger a)^m = \sum_{k=1}^m c_k a^{\dagger k} a^k\)

where the coefficients \(c_i\) are provided by the function _coeff_normal_ordered.

Parameters:

mu (array) – length-\(2N\) means vector in xp-ordering.
cov (array) – \(2N\times 2N\) covariance matrix in xp-ordering.
indices (dictionary) – specification of the different modes and their power of their photon number
hbar (float) – value of hbar in the uncertainty relation.

Returns:

the expectation value of the photon number powers.

Return type:

float

photon_number_squared_expectation(mu, cov, modes, hbar=2)[source]¶

Calculates the expectation value of the square of the product of the number operator of the modes in a Gaussian state.

Parameters:

mu (array) – length-\(2N\) means vector in xp-ordering.
cov (array) – \(2N\times 2N\) covariance matrix in xp-ordering.
modes (list) – list of modes
hbar (float) – value of hbar in the uncertainty relation.

Returns:

expectation value of the square of the product of the number operator of the modes.

Return type:

(float)

probabilities(mu, cov, cutoff, parallel=False, 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
parallel (bool) – if True, uses dask for parallelization instead of OpenMP
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)

pure_state_distribution(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)

real_to_complex_displacements(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

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)

s_ordered_expectation(mu, cov, rpt, hbar=2, s=0)[source]¶

Calculates the expectation value of the s-ordered product obtained by taking deirvatives of the characteristic function of a Gaussian states, Here, \(\text{rpt}=(n_0, n_1, \ldots, n_{N-1}, m_0, m_1, \ldots, m_{N-1})\). indicates how many derivatives are taken with respect to the complex argument and its conjugate. The values \(s=\{1,0,-1\}\) correspond respectively to normal, symmetric and antinormal order.

Parameters:

mu (array) – length-\(2N\) means vector in xp-ordering.
cov (array) – \(2N\times 2N\) covariance matrix in xp-ordering.
rpt (list) – integers specifying the terms to calculate.
hbar (float) – value of hbar in the uncertainty relation.
s (float) – value setting the ordering it must be between -1 and 1.

Returns:

expectation value of the normal ordered product of operators

Return type:

(float)

total_photon_number_distribution(n, k, s, eta, pref=1.0)[source]¶

Probability of observing a total of \(n\) photons when \(k\) identical single-mode squeezed vacua with squeezing parameter \(s\) undergo loss by transmission \(\eta\).

For the derivation see Appendix E of ‘Quantum Computational Supremacy via High-Dimensional Gaussian Boson Sampling’, Deshpande et al..

Parameters:

n (int) – number of photons
k (int) – number of squeezed modes
s (float) – squeezing parameter
eta (float) – transmission parameter, between 0 and 1 inclusive
pref (float) – use to return the probability times pref**n

Returns:

probability of observing a total of n photons or the probability times pref ** n.

Return type:

(float)

tvd_cutoff_bounds(mu, cov, cutoff, hbar=2, check_is_valid_cov=True, rtol=1e-05, atol=1e-08)[source]¶

Gives bounds of the total variation distance between the exact Gaussian Boson Sampling distribution extending to infinity in Fock space and the ones truncated by any value between 0 and the user provided cutoff.

For the derivation see Appendix B of ‘Exact simulation of Gaussian boson sampling in polynomial space and exponential time’, Quesada and Arrazola.

Parameters:

mu (array) – vector of means of the Gaussian state
cov (array) – covariance matrix of the Gaussian state
cutoff (int) – cutoff in Fock space
check_is_valid_cov (bool) – verify that the covariance matrix is physical
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:

values of the bound for different local Fock space dimensions up to cutoff

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

variance_clicks(cov, hbar=2)[source]¶

Calculates the variance of the total number of clicks when a zero-mean gaussian state is measured using threshold detectors.

Args: cov (array): \(2N\times 2N\) covariance matrix in xp-ordering hbar (float): the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\)
Returns: float: variance in the total number of clicks

vonneumann_entropy(cov, hbar=2)[source]¶

Returns the vonNeumann entropy of a covariance matrix.

Parameters:

cov (array) – a covariance matrix
hbar (float) – the value of \(\hbar\) in the commutation relation \([\x,\p]=i\hbar\)

Returns:

vonNeumann entropy

Return type:

(float)

code/quantum

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Quantum algorithms¶

Fock states and tensors¶

Adjacency matrices¶

Gaussian checks¶

Conversions¶

Means and variances¶

Photon number distributions¶

Entanglement¶

Code details¶

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