qtools

qmetro.qtools.choi_from_krauses(krauses)[source]

Get Choi matrix of a channel defined using its Kraus operators.

For channel Phi: H_in -> H_out returned matrix acts on tensor product H_out (x) H_in.

Parameters:: krauses (list[np.ndarray]) – List of Kraus operators.
Returns:: choi – Choi matrix.
Return type:: np.ndarray

qmetro.qtools.choi_from_lindblad(lindblad, dlindblad, t, dim=None, lind_kwargs=None)[source]

Calculates Choi matrix and its derivative of a channel simulating evolution with a given Lindbladian for time t.

Lindbladian - L is a function, that for input density matrix returns its derivative over time e.g. lindblad(rho) = drho/dt. For example for rotation of Bloch ball around z-axis with angular velocity omega:

L(rho) = 0.5j * omega * (rho sigma_z - sigma_z rho),

where sigma_z is a Z Pauli matrix. If omega is an estimated parameter, then the derivative of Lindbladian over this parameter - dL is a function:

dL(rho) = 0.5j * (rho sigma_z - sigma_z rho).

choi_from_lindblad computes Choi matrix and its derivative over the estimated parameter for continuous time evolution with a Lindbladian constant in time. Computations are done algebraically, without numerical integration.

Parameters:

lindblad (Callable[..., np.ndarray] | tuple[np.ndarray, list[np.ndarray]]) – Argument representing Lindbladian. It can be: - A function L(rho, a0, a1, …) returning derivative drho/dt for input state rho and additional keyword arguments a0, a1,… In this case additional parameter dim representing dimension of rho has to be provided. - A tuple (H, Ls) where H is a Hamiltonian divided by hbar (np.ndarray) and Ls are jump operators (list[np.ndarray]).
dlindblad (Callable[[np.ndarray], np.ndarray] | tuple[np.ndarray, list[np.ndarray]] | np.ndarray) – Argument representing Lindbladian’s derivative. It can be: - A function dL(rho, b0, b1, …) returning derivative of drho/dt over paramter. - A tuple (dH, dLs) where dH and dLs are derivatives of H and Ls. - An array dH and dLs are assumed to be zero.
dim (int) – Dimension of Hilbert space on which Lindbladian acts
t (float) – Evolution time
lind_kwargs (Mapping[str, Any] | None = None) – Additional keyword arguments passed to lindblad and dlindblad

Returns:

choi – Choi matrix
dchoi – Derivative of Choi matrix over some parameter

Return type:

tuple[ndarray, ndarray]

qmetro.qtools.comb_variables(dims, hermitian=True, trace_constraint=1)[source]

Construct a sequence of CVXPY variables that represent quantum combs satisfying causality constraints.

Each element in the returned combs list represents a quantum comb matrix acting on progressively larger subsystems of a composite Hilbert space, where subsystems alternate as input (odd-indexed, starting from 1) and output (even-indexed) spaces.

Parameters:

dims (tuple of int) – Dimensions of each Hilbert space H_1, H_2, …, H_2N in the composite system. The number of spaces (length of dims) must be even, with N pairs of input-output spaces.
hermitian (bool, optional) – If True, each matrix in combs is constrained to be Hermitian, by default True.
trace_constraint (float or None, optional) – Specifies the trace constraint on the first comb operator. If None, a trace variable is used instead of a fixed trace, by default 1.

Returns:

A tuple containing:

combslist of cp.Variable
List of CVXPY variables representing the quantum comb operators.
constraintslist of cp.Constraint
List of CVXPY constraints on the comb operators ensuring causality.
trace_var or trace_constraintcp.Variable or float
If trace_constraint is None, returns a trace variable for the constraint. Otherwise, returns the specified trace_constraint value.

Return type:

tuple

Raises:

ValueError – If the length of dims is not even, as quantum comb spaces must have matching input-output pairs.

Notes

Element combs[i] is a Choi-Jamiolkowski operator acting on
H_2i+2 (x) … (x)H_2 (x) H_2i+1 (x) … H_1, so it belongs to set Comb[(H_1, H_2), …, (H_2i+1, H_2i+2)]
The full comb is the last element combs[N-1] Comb[(H_1, H_2), …,
(H_2N-1, H_2N)]
If trace_constraint is not None, it represents the trace of
the initial operator scaled by input space dimensions. This is not the overall comb trace, which includes all input dimensions.
The function does not assume positivity for the combs. A positivity
constraint can be added externally if needed.

Examples

>>> dims = (2, 2, 2, 2)
>>> combs, constraints, trace_constraint = comb_variables(dims)
>>> len(combs)
2
>>> len(constraints)
2

qmetro.qtools.dchoi_from_krauses(krauses, dkrauses)[source]

Get derivative of a Choi matrix of a channel defined using its Kraus operators and their derivatives.

For channel Phi: H_in -> H_out returnde matrix acts on tensor product H_out (x) H_in.

Parameters:

krauses (list[np.ndarray]) – List of Kraus operators.
dkrauses (list[np.ndarray]) – List of derivatives of Kraus operators.

Returns:

dchoi – Derivative of a Choi matrix.

Return type:

np.ndarray

qmetro.qtools.depolarization_krauses(p=None, noise_first=True, eta=None)[source]

Computes Kraus operators and their derivatives for a qubit channel where:

the signal is rotating Bloch sphere around the z-axis,
noise shrinks uniformly the whole Bloch ball.

See more details in the documentation.

Parameters:

p (float | None, optional) – Probability that the input state will remain unchanged.
noise_first (bool, optional) – Whether noise is before signal, by default True.
eta (float | None, optional) – Alternative method of determining the noise strength that when provided is used instead of p (either p or eta argument has to be provided). In this parametrisation eta is the factor by which Bloch sphere gets shrunken.

Returns:

krauses (list[np.ndarray]) – List of Kraus operators.
dkrauses (list[np.ndarray]) – List of derivatives of Kraus operators.

Raises:

ValueError – When poth p and eta are provided.

Return type:

tuple[list[ndarray], list[ndarray]]

qmetro.qtools.dkrauses_from_choi(choi, dchoi, dims, eps=1e-07)[source]

Calculates Kraus operators and derivaties of a channel with a given Choi and derivative.

Parameters:

choi (np.ndarray) – Choi-Jamiolkowski matrix of a channel : Lin(dout x din). It has to be a positive semi-definite matrix.
dchoi (np.ndarray) – Derivative of aChoi-Jamiolkowski matrix of a channel.
dims (tuple[int, int]) – Tuple [din, dout],`din`, dout are input/output dimensions of a channel
eps (float, optional) – Eigenvectors of choi with eigenvalues smaller than eps are ignored.

Returns:

krauses (list[np.ndarray]) – Kraus operators of a channel (arrays of dimensions dout x din).
dkrauses (list[np.ndarray]) – Derivatives of Kraus operators of a channel (arrays of dimensions dout x din).

Return type:

tuple[list[ndarray], list[ndarray]]

Notes

Normalization convention: choi trace is dimension of input space

qmetro.qtools.get_sld(rho, drho, return_qfi=False, **kwargs)[source]

Computes symmetric logarithmic derivative (SLD) of a parametrized state.

Parameters:

rho (np.ndarray) – Density matrix.
drho (np.ndarray) – Derivative of density matrix.
return_qfi (bool, optional) – Whether to return also a quantum Fisher information (QFI) of the state, by default False.
**kwargs – Additional keyword arguments passed to the CVXPY solve method (see docs).

Returns:

SLD matrix or a pair (SLD, QFI) if return_qfi is True.

Return type:

np.ndarray | tuple[float, np.ndarray]

qmetro.qtools.hc(x)[source]

Hermitian conjugation of a numpy matrix or a cvxpy expression.

Parameters:: x (np.ndarray | cp.Expression) – Matrix.
Returns:: x.conjugate().T
Return type:: np.ndarray | cp.Expression

qmetro.qtools.ket_bra(x, y)[source]

Parameters:

x (ndarray)
y (ndarray)

Return type:

ndarray

qmetro.qtools.krauses_from_choi(choi, dims, eps=1e-07, return_eigensystem=False)[source]

Calculates Kraus operators of a channel with a given Choi.

Parameters:

choi (np.ndarray) – Choi-Jamiolkowski matrix of a channel : Lin(dout (x) din)
dims (tuple[int, int]) – Tuple [din, dout],`din`, dout are input/output dimensions of a channel
eps (float, optional) – Eigenvectors of choi with eigenvalues smaller than eps are ignored.
return_eigensystem (bool, optional) – If true, returns additionally eigenvalues and eigenvectors.

Returns:

krauses (list[np.ndarray] | tuple[list[np.ndarray], np.ndarray,)
np.ndarray] – Kraus operators of a channel (arrays of dimensions dout x din) or if return_eigensystem=True a tuple (Kraus operators, eigenvalues, eigenvectors) where i-th eigenvector is eigenvectors[:, i].

Return type:

list[ndarray] | tuple[list[ndarray], ndarray, ndarray]

Notes

Normalization convention: Trace of the Choi matrix is equal to the dimension of the input space.

qmetro.qtools.krauses_kron(krauses1, dkrauses1, krauses2, dkrauses2)[source]

Computes the Kraus representation of the Kronecker product of two quantum channels and their derivatives.

This function creates a representation of the Kronecker product of two quantum channels, including the combined Kraus operators and their derivatives.

Parameters:

krauses1 (list of np.ndarray) – List of Kraus operators for the first channel.
dkrauses1 (list of np.ndarray) – List of derivatives of Kraus operators for the first channel.
krauses2 (list of np.ndarray) – List of Kraus operators for the second channel.
dkrauses2 (list of np.ndarray) – List of derivatives of Kraus operators for the second channel.

Returns:

krauses12 (list of np.ndarray) – The Kraus operators of the Kronecker product channel.
dkrauses12 (list of np.ndarray) – The derivatives of the Kraus operators for the Kronecker product channel.

Return type:

tuple[list[ndarray], list[ndarray]]

qmetro.qtools.krauses_sequential(krauses1, dkrauses1, krauses2, dkrauses2)[source]

Computes the Kraus representation of the sequential composition of two quantum channels and their derivatives.

This function creates a representation of the sequential composition of two quantum channels (krauses and their derivatives of a resulting channel). krauses1 go before krauses2

Parameters:

krauses1 (list of np.ndarray) – List of Kraus operators for the first channel.
dkrauses1 (list of np.ndarray) – List of derivatives of Kraus operators for the first channel.
krauses2 (list of np.ndarray) – List of Kraus operators for the second channel.
dkrauses2 (list of np.ndarray) – List of derivatives of Kraus operators for the second channel.

Returns:

krauses12 (list of np.ndarray) – The Kraus operators of the Kronecker product channel.
dkrauses12 (list of np.ndarray) – The derivatives of the Kraus operators for the Kronecker product channel.

Return type:

tuple[list[ndarray], list[ndarray]]

qmetro.qtools.minimize_alpha(krauses, dkrauses, **kwargs)[source]

Minimize the norm of alpha over all Kraus representations for a given channel [12, 27].

Given a list of Kraus operators and their derivatives, this function calculates the minimum norm of alpha over all possible Kraus representations for the input channel.

Parameters:

krauses (list[np.ndarray]) – List of Kraus operators, each represented as a 2D NumPy array.
dkrauses (list[np.ndarray]) – List of derivatives of Kraus operators, each represented as a 2D NumPy array.
**kwargs –
Additional keyword arguments passed to the CVXPY solve method (see docs).

Returns:

The minimum value of norm of alpha over all Kraus representations.

Return type:

float

qmetro.qtools.par_amp_damping_krauses(p, noise_first=True)[source]

Computes Kraus operators and their derivatives for a qubit channel where:

the signal is rotating Bloch sphere around the z-axis,
noise models decay from state |1⟩ to |0⟩.

See more details in the documentation.

Parameters:

p (float) – Noise parametrization. For p = 1 there is no noise for p = 0 the noise is maximal.
noise_first (bool, optional) – Whether the noise is before the signal, by default True.

Returns:

krauses (list[np.ndarray]) – List of Kraus operators.
dkrauses (list[np.ndarray]) – List of derivatives of Kraus operators.

Return type:

tuple[list[ndarray], list[ndarray]]

qmetro.qtools.par_dephasing_krauses(p=None, noise_first=True, eps=None, rot_like=False)[source]

Computes Kraus operators and their derivatives for a qubit channel where:

the signal is rotating Bloch sphere around the z-axis,
noise shrinks the xy-plane preserving the z-axis.

See more details in the documentation.

Parameters:

p (float | None, optional) – Probability that the input state will remain unchanged.
noise_first (bool, optional) – Whether noise is before signal, by default True.
eps (float | None, optional) –
Alternative way of determining the noise strength:

p = cos(eps/2)**2,

that when provided is used instead of p (p argument is ignored).
rot_like (bool, optional) –
If True then Kraus operators of noise are:

exp(-1j/2 * eps * sigma_z) / sqrt(2), exp(+1j/2 * eps * sigma_z) / sqrt(2),

where p = cos(eps/2)**2. By default False.

Returns:

krauses (list[np.ndarray]) – List of Kraus operators.
dkrauses (list[np.ndarray]) – List of derivatives of Kraus operators.

Raises:

ValueError – When both p and eps are provided.

Return type:

tuple[list[ndarray], list[ndarray]]

qmetro.qtools.parallel_krauses(krauses, dkrauses, n)[source]

Returns Kraus operators and their derivatives for n parallel channels.

Parameters:

krauses (list[np.ndarray]) – Kraus operators of single channel.
dkrauses (list[np.ndarray]) – Derivatives of Kraus operators of single channel.
n (int) – Number of parallel channels.

Returns:

krauses (list[np.ndarray]) – Kraus operators of n channels.
dkrauses (list[np.ndarray]) – Derivatives of Kraus operators of n channels.

Return type:

tuple[list[ndarray], list[ndarray]]

qmetro.qtools.per_amp_damping_krauses(p, noise_first=True)[source]

Computes Kraus operators and their derivatives for a qubit channel where:

the signal is rotating Bloch sphere around the z-axis,
noise models decay from state |+⟩ to |-⟩.

See more details in the documentation.

Parameters:

p (float) – Noise parametrization. For p = 1 there is no noise for p = 0 the noise is maximal.
noise_first (bool, optional) – Whether noise is before signal, by default True.

Returns:

krauses (list[np.ndarray]) – List of Kraus operators.
dkrauses (list[np.ndarray]) – List of derivatives of Kraus operators.

Return type:

tuple[list[ndarray], list[ndarray]]

qmetro.qtools.per_dephasing_krauses(p, noise_first=True)[source]

Computes Kraus operators and their derivatives for a qubit channel where:

the signal is rotating Bloch sphere around the z-axis,
noise shrinks the yz-plane preserving the x-axis.

See more details in the documentation.

Parameters:

p (float) – Probability that the input state will remain unchanged.
noise_first (bool, optional) – Whether noise is before signal, by default True.

Returns:

krauses (list[np.ndarray]) – List of Kraus operators.
dkrauses (list[np.ndarray]) – List of derivatives of Kraus operators.

Return type:

tuple[list[ndarray], list[ndarray]]

qmetro.qtools.povm_from_sld(sld)[source]

Calculates optimal measurement POVM from SLD matrix.

Parameters:: sld (np.ndarray) – Symmetric logarithmic derivative (SLD) matrix
Returns:: List of optimal measurement operators: projections on eigenvectors of SLD matrix.
Return type:: list[np.ndarray]

qmetro.qtools.swap_operator(dims, i1, i2)[source]

Construct a swap operator matrix for a tensor product space, swapping two specified subsystems.

This function returns a matrix that, when applied to a vector or matrix in the tensor product space H_1 x H_2 x … x H_N, swaps the specified subsystems with indices i1 and i2. The resulting space ordering will be:

Original structure: H_1 x … x H_i1 x … x H_i2 x … H_N
New structure: H_1 x … x H_i2 x … x H_i1 x … H_N

If v is a vector in the original structure, swap_operator @ v gives the vector in the swapped structure. If M is a matrix in the original structure, swap_operator @ M @ swap_operator.T gives the matrix in the swapped structure.

Parameters:

dims (tuple of int) – Dimensions of each subsystem H_1, …, H_N in the tensor product space.
i1 (int) – Index of the first subsystem to swap.
i2 (int) – Index of the second subsystem to swap.

Returns:

The swap operator matrix that swaps subsystems i1 and i2 in the tensor product space.

Return type:

np.ndarray

Notes

The swap operator constructs the necessary index mappings by decomposing the tensor product space indices and swapping the basis states of the specified subsystems.

Examples

>>> dims = (2, 2, 2)
>>> i1, i2 = 0, 1
>>> swap_operator(dims, i1, i2)
array([...])  # Swap operator matrix for a 3-qubit system