MultiQubitDual

class MultiQubitDual(list_operators: list[Operator])

Bases: MultiQubitFrame, BaseDual

Class that collects all information that any Dual over multiple qubits should specify.

This is a representation of a dual frame. Its elements are specified as a list of Operator.

Initialize from explicit operators.

Parameters:

list_operators (list[Operator]) – list that contains the explicit frame operators. The length of the list is the number of operators of the frame.

Raises:

ValueError – if the frame operators do not have a correct shape. They should all be hermitian and of the same dimension.

Inherited Attributes

dimension

The dimension of the Hilbert space on which the effects act.

informationally_complete

If the frame spans the entire Hilbert space.

num_operators

The number of effects of the frame.

num_outcomes

The number of outcomes of the Dual.

num_subsystems

The number of subsystems which the frame operators act on.

For qubits, this is always ${\log }_2($dimension$)$.

operators

Return the list of frame operators.

pauli_operators

Convert the internal frame operators to Pauli form.

Warning

The conversion to Pauli form can be computationally intensive.

Returns:

The frame operators in Pauli form. Each frame operator is returned as a dictionary mapping Pauli labels to coefficients.

Raises:

QiskitError – when the frame operators could not be converted to Pauli form (e.g. when they are not N-qubit operators).

Methods

classmethod build_dual_from_frame(frame: BaseFrame, alphas: tuple[float, ...] | None = None) → MultiQubitDual

Construct a dual frame to another (primal) frame.

Parameters:

• frame (BaseFrame) – The primal frame from which we will build the dual frame.

• alphas (tuple[float, ...] | None) – parameters of the frame super-operator used to build the dual frame. If None, the parameters are set as the traces of each operator in the primal frame.

Returns:

A dual frame to the supplied frame.

Return type:

MultiQubitDual

is_dual_to(frame: BaseFrame) → bool

Check if self is a dual to another frame.

Parameters:

frame (BaseFrame) – the other frame to check duality against.

Returns:

Whether self is dual to frame.

Return type:

bool

Inherited Methods

analysis(hermitian_op: SparsePauliOp | Operator, frame_op_idx: int | set[int] | None = None) → float | dict[int, float] | ndarray

Return the frame coefficients of hermitian_op.

This method implements the analysis operator $A$ of the frame $\{F_k{\}}_k$:

$$A:\mathcal{O}\mapsto \{\mathrm{Tr}\left[F_k\mathcal{O}\right]{\}}_k,$$

where $c_k=\mathrm{Tr}\left[F_k\mathcal{O}\right]$ are called the frame coefficients of the Hermitian operator $\mathcal{O}$.

Parameters:

• hermitian_op (SparsePauliOp | Operator) – a hermitian operator whose frame coefficients to compute.

• frame_op_idx (int | set[int] | None) – label or set of labels indicating which coefficients are queried. If None, all coefficients are queried.

Returns:

Frame coefficients, specified by frame_op_idx, of the Hermitian operator hermitian_op. If a specific coefficient was queried, a float is returned. If a specific set of coefficients was queried, a dictionary mapping labels to coefficients is returned. If all coefficients were queried, an array with all coefficients is returned.

Raises:

• TypeError – when the provided single or sequence of labels frame_op_idx does not have a valid type.

• ValueError – when the dimension of the provided hermitian_op does not match the dimension of the frame operators.

Return type:

float | dict[int, float] | ndarray

classmethod from_vectors(frame_vectors: ndarray) → Self

Initialize a frame from non-normalized bloch vectors.

The non-normalized Bloch vectors are given by $|{\tilde{\psi }}_k⟩=\sqrt{{\gamma }_k}|{\psi }_k⟩$. The resulting frame operators are $F_k={\gamma }_k|{\psi }_k⟩⟨{\psi }_k|$ where ${\gamma }_k$ is the trace of the $k$’th frame operator.

Parameters:

frame_vectors (ndarray) – list of vectors $|\widetilde{{\psi }_k}⟩$. The length of the list corresponds to the number of operators of the frame. Each vector is of shape $(\dim ,)$ where $\dim$ is the dimension of the Hilbert space on which the frame acts.

Returns:

The frame corresponding to the vectors.

Return type:

Self

get_omegas(observable: SparsePauliOp | Operator, outcome_idx: LabelT | set[LabelT] | None = None) → float | dict[LabelT, float] | ndarray

Return the decomposition weights of the provided observable.

Computes the ${\omega }_k$ in

$$\mathcal{O}=\sum _{k=1}^n{\omega }_k{M}_k$$

where $\mathcal{O}$ is the observable and $M_k$ are the effects of the POVM of which self is the dual. The closed form for computing ${\omega }_k$ is

$${\omega }_k=\text{Tr}\left[\mathcal{O}{D}_k\right]$$

where $D_k$ make of this dual frame (i.e. self).

Note

In the frame theory formalism, the mapping $A:\mathcal{O}\mapsto \{\text{Tr}\left[\mathcal{O}{D}_k\right]{\}}_k$ is referred to as the analysis operator, which is implemented by the analysis() method.

Parameters:

• observable (SparsePauliOp | Operator) – the observable for which to compute the decomposition weights.

• outcome_idx (LabelT | set[LabelT] | None) – label or set of labels indicating which decomposition weights are queried. If None, all weights are queried.

Returns:

Decomposition weight(s) associated to the effect(s) specified by outcome_idx. If a specific outcome was queried, a float is returned. If a specific set of outcomes was queried, a dictionary mapping outcome labels to weights is returned. If all outcomes were queried, an array with all weights is returned.

Return type:

float | dict[LabelT, float] | ndarray