sf.backends.BaseState

class BaseState(num_modes, mode_names=None)

Bases: abc.ABC

Abstract base class for the representation of quantum states.

Attributes

EQ_TOLERANCE
data	Returns the underlying numerical (or symbolic) representation of the state.
hbar	Returns the value of \(\hbar\) used in the generation of the state.
is_pure	Checks whether the state is a pure state.
mode_indices	Returns a dictionary mapping the mode names to mode indices.
mode_names	Returns a dictionary mapping the mode index to mode names.
num_modes	Gets the number of modes that the state represents.

EQ_TOLERANCE = 1e-10

data

Returns the underlying numerical (or symbolic) representation of the state. The form of this data differs for different backends.

hbar

Returns the value of \(\hbar\) used in the generation of the state.

The value of \(\hbar\) is a convention chosen in the definition of \(\x\) and \(\p\). See Operators for more details.

Returns

\(\hbar\) value.

Return type

float

is_pure

Checks whether the state is a pure state.

Returns

True if and only if the state is pure.

Return type

bool

mode_indices

Returns a dictionary mapping the mode names to mode indices.

The mode names are determined from the initialization argument mode_names. If these were not supplied, the names are generated automatically based on the mode indices.

Returns

dictionary of the form {"mode name":i,...}

Return type

dict

mode_names

Returns a dictionary mapping the mode index to mode names.

The mode names are determined from the initialization argument mode_names. If these were not supplied, the names are generated automatically based on the mode indices.

Returns

dictionary of the form {i:"mode name",...}

Return type

dict

num_modes

Gets the number of modes that the state represents.

Returns

the number of modes in the state

Return type

int

Methods

all_fock_probs(**kwargs)	Probabilities of all possible Fock basis states for the current circuit state.
dm(**kwargs)	The numerical density matrix for the quantum state in the Fock basis.
fidelity(other_state, mode, **kwargs)	Fidelity of the reduced state in the specified mode with a user supplied state.
fidelity_coherent(alpha_list, **kwargs)	The fidelity of the state with a product of coherent states.
fidelity_vacuum(**kwargs)	The fidelity of the state with the vacuum state.
fock_prob(n, **kwargs)	Probability of a particular Fock basis state.
ket(**kwargs)	The numerical state vector for the quantum state in the Fock basis.
mean_photon(mode, **kwargs)	Returns the mean photon number of a particular mode.
number_expectation(modes)	Calculates the expectation value of the product of the number operators of the modes.
p_quad_values(mode, xvec, pvec)	Calculates the discretized p-quadrature probability distribution of the specified mode.
parity_expectation(modes)	Calculates the expectation value of a product of parity operators acting on given modes
poly_quad_expectation(A[, d, k, phi])	The multi-mode expectation values and variance of arbitrary 2nd order polynomials of quadrature operators.
quad_expectation(mode[, phi])	The \(\x_{\phi}\) operator expectation values and variance for the specified mode.
reduced_dm(modes, **kwargs)	Returns a reduced density matrix in the Fock basis.
wigner(mode, xvec, pvec)	Calculates the discretized Wigner function of the specified mode.
x_quad_values(mode, xvec, pvec)	Calculates the discretized x-quadrature probability distribution of the specified mode.

abstract all_fock_probs(**kwargs)

Probabilities of all possible Fock basis states for the current circuit state.

For example, in the case of 3 modes, this method allows the Fock state probability \(|\braketD{0,2,3}{\psi}|^2\) to be returned via

probs = state.all_fock_probs()
probs[0,2,3]

Returns

array of dimension \(\underbrace{D\times D\times D\cdots\times D}_{\text{num modes}}\)

containing the Fock state probabilities, where \(D\) is the Fock basis cutoff truncation

Return type

array

Keyword Arguments

cutoff (int) – Specifies where to truncate the computation (default value is 10). Note that the cutoff argument only applies for Gaussian representation; states represented in the Fock basis will use their own internal cutoff dimension.

abstract dm(**kwargs)

The numerical density matrix for the quantum state in the Fock basis.

Keyword Arguments

cutoff (int) – Specifies where to truncate the returned density matrix (default value is 10). Note that the cutoff argument only applies for Gaussian representation; states represented in the Fock basis will use their own internal cutoff dimension.

Returns

the numerical density matrix in the Fock basis

Return type

array

abstract fidelity(other_state, mode, **kwargs)

Fidelity of the reduced state in the specified mode with a user supplied state. Note that this method only supports single-mode states.

Parameters

other_state – a pure state vector array represented in the Fock basis (for Fock backends) or a Sequence (mu, cov) containing the means and covariance matrix (for Gaussian backends)

Returns

The fidelity of the circuit state with other_state.

abstract fidelity_coherent(alpha_list, **kwargs)

The fidelity of the state with a product of coherent states.

The fidelity is defined by

\[\bra{\vec{\alpha}}\rho\ket{\vec{\alpha}}\]

Parameters

alpha_list (Sequence[complex]) – list of coherent state parameters, one for each mode

Returns

the fidelity value

Return type

float

abstract fidelity_vacuum(**kwargs)

The fidelity of the state with the vacuum state.

Returns

the fidelity of the state with the vacuum

Return type

float

abstract fock_prob(n, **kwargs)

Probability of a particular Fock basis state.

Computes the probability \(|\braket{\vec{n}|\psi}|^2\) of measuring the given multi-mode Fock state based on the state \(\ket{\psi}\).

Warning

Computing the Fock probabilities of states has exponential scaling in the Gaussian representation (for example states output by a Gaussian backend as a BaseGaussianState). This shouldn’t affect small-scale problems, where only a few Fock basis state probabilities need to be calculated, but will become evident in larger scale problems.

Parameters

n (Sequence[int]) – the Fock state \(\ket{\vec{n}}\) that we want to measure the probability of

Keyword Arguments

cutoff (int) – Specifies where to truncate the computation (default value is 10). Note that the cutoff argument only applies for Gaussian representation; states represented in the Fock basis will use their own internal cutoff dimension.

Returns

measurement probability

Return type

float

abstract ket(**kwargs)

The numerical state vector for the quantum state in the Fock basis. Note that if the state is mixed, this method returns None.

Keyword Arguments

cutoff (int) – Specifies where to truncate the returned density matrix (default value is 10). Note that the cutoff argument only applies for Gaussian representation; states represented in the Fock basis will use their own internal cutoff dimension.

Returns

the numerical state vector. Returns None if the state is mixed.

Return type

array/None

abstract mean_photon(mode, **kwargs)

Returns the mean photon number of a particular mode.

Parameters

• mode (int) – specifies the mode

• **kwargs –

  • cutoff (int): (default 10) Fock basis trunction for calculation of mean photon number. Note that the cutoff argument only applies for Gaussian representation; states represented in the Fock basis will use their own internal cutoff dimension.

Returns

the mean photon number and variance

Return type

tuple

abstract number_expectation(modes)

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

This method computes the analytic expectation value \(\langle \hat{n}_{i_0} \hat{n}_{i_1}\dots \hat{n}_{i_{N-1}}\rangle\) for a (sub)set of modes \([i_0, i_1, \dots, i_{N-1}]\) of the system.

Parameters

modes (list) – list of modes for which one wants the expectation of the product of their number operator.

Returns

the expectation value and variance

Return type

tuple[float, float]

Example

Consider the following program:

prog = sf.Program(3)

with prog.context as q:
    ops.Sgate(0.5) | q[0]
    ops.Sgate(0.5) | q[1]
    ops.Sgate(0.5) | q[2]
    ops.BSgate(np.pi/3, 0.1) |  (q[0], q[1])
    ops.BSgate(np.pi/3, 0.1) |  (q[1], q[2])

Executing this on the Fock backend,

>>> eng = sf.Engine("fock", backend_options={"cutoff_dim": 10})
>>> state = eng.run(prog).state

we can compute the expectation value \(\langle \hat{n}_0\hat{n}_2\rangle\):

>>> state.number_expectation([0, 2])[0]
0.07252895071309405

Executing the same program on the Gaussian backend,

>>> eng = sf.Engine("gaussian")
>>> state = eng.run(prog).state
>>> state.number_expectation([0, 2])[0]
0.07566984755267293

This slight difference in value compared to the result from the Fock backend above is due to the finite Fock basis truncation when using the Fock backend; this can be avoided by increasing the value of cutoff_dim.

Warning

This method only supports at most two modes in the Gaussian backend.

p_quad_values(mode, xvec, pvec)

Calculates the discretized p-quadrature probability distribution of the specified mode.

Parameters

• mode (int) – the mode to calculate the p-quadrature probability values of

• xvec (array) – array of discretized \(x\) quadrature values

• pvec (array) – array of discretized \(p\) quadrature values

Returns

1D array of size len(pvec), containing reduced p-quadrature probability values for a specified range of x and p.

Return type

array

abstract parity_expectation(modes)

Calculates the expectation value of a product of parity operators acting on given modes

abstract poly_quad_expectation(A, d=None, k=0, phi=0, **kwargs)

The multi-mode expectation values and variance of arbitrary 2nd order polynomials of quadrature operators.

An arbitrary 2nd order polynomial of quadrature operators over $N$ modes can always be written in the following form:

\[P(\mathbf{r}) = \mathbf{r}^T A\mathbf{r} + \mathbf{r}^T \mathbf{d} + k I\]

where:

• \(A\in\mathbb{R}^{2N\times 2N}\) is a symmetric matrix representing the quadratic coefficients,

• \(\mathbf{d}\in\mathbb{R}^{2N}\) is a real vector representing the linear coefficients,

• \(k\in\mathbb{R}\) represents the constant term, and

• \(\mathbf{r} = (\x_1,\dots,\x_N,\p_1,\dots,\p_N)\) is the vector of quadrature operators in \(xp\)-ordering.

This method returns the expectation value of this second-order polynomial,

\[\langle P(\mathbf{r})\rangle,\]

as well as the variance

\[\Delta P(\mathbf{r})^2 = \braket{P(\mathbf{r})^2} - \braket{P(\mathbf{r})}^2\]

Parameters

• A (array) – a real symmetric 2Nx2N NumPy array, representing the quadratic coefficients of the second order quadrature polynomial.

• d (array) – a real length-2N NumPy array, representing the linear coefficients of the second order quadrature polynomial. Defaults to the zero vector.

• k (float) – the constant term. Default 0.

• phi (float) – quadrature angle, clockwise from the positive \(x\) axis. If provided, the vectori of quadrature operators \(\mathbf{r}\) is first rotated by angle \(\phi\) in the phase space.

Returns

expectation value and variance

Return type

tuple (float, float)

abstract quad_expectation(mode, phi=0, **kwargs)

The \(\x_{\phi}\) operator expectation values and variance for the specified mode.

The \(\x_{\phi}\) operator is defined as follows,

\[\x_{\phi} = \cos\phi~\x + \sin\phi~\p\]

with corresponding expectation value

\[\bar{x_{\phi}}=\langle x_{\phi}\rangle = \text{Tr}(\x_{\phi}\rho_{mode})\]

and variance

\[\Delta x_{\phi}^2 = \langle x_{\phi}^2\rangle - \braket{x_{\phi}}^2\]

Parameters

• mode (int) – the requested mode

• phi (float) –

  quadrature angle, clockwise from the positive \(x\) axis.

  • \(\phi=0\) corresponds to the \(x\) expectation and variance (default)

  • \(\phi=\pi/2\) corresponds to the \(p\) expectation and variance

Returns

expectation value and variance

Return type

tuple (float, float)

abstract reduced_dm(modes, **kwargs)

Returns a reduced density matrix in the Fock basis.

Parameters

modes (int or Sequence[int]) – specifies the mode(s) to return the reduced density matrix for.

Keyword Arguments

cutoff (int) – Specifies where to truncate the returned density matrix (default value is 10). Note that the cutoff argument only applies for Gaussian representation; states represented in the Fock basis will use their own internal cutoff dimension.

Returns

the reduced density matrix for the specified modes

Return type

array

abstract wigner(mode, xvec, pvec)

Calculates the discretized Wigner function of the specified mode.

Parameters

• mode (int) – the mode to calculate the Wigner function for

• xvec (array) – array of discretized \(x\) quadrature values

• pvec (array) – array of discretized \(p\) quadrature values

Returns

2D array of size [len(xvec), len(pvec)], containing reduced Wigner function values for specified x and p values.

Return type

array

x_quad_values(mode, xvec, pvec)

Calculates the discretized x-quadrature probability distribution of the specified mode.

Parameters

• mode (int) – the mode to calculate the x-quadrature probability values of

• xvec (array) – array of discretized \(x\) quadrature values

• pvec (array) – array of discretized \(p\) quadrature values

Returns

1D array of size len(xvec), containing reduced x-quadrature probability values for a specified range of x and p.

Return type

array

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