qiskit.quantum_info.SuperOp

class SuperOp(data, input_dims=None, output_dims=None)

Superoperator representation of a quantum channel.

The Superoperator representation of a quantum channel \(\mathcal{E}\) is a matrix \(S\) such that the evolution of a DensityMatrix \(\rho\) is given by

\[|\mathcal{E}(\rho)\rangle\!\rangle = S |\rho\rangle\!\rangle\]

where the double-ket notation \(|A\rangle\!\rangle\) denotes a vector formed by stacking the columns of the matrix \(A\) (column-vectorization).

See reference [1] for further details.

References

1. C.J. Wood, J.D. Biamonte, D.G. Cory, Tensor networks and graphical calculus for open quantum systems, Quant. Inf. Comp. 15, 0579-0811 (2015). arXiv:1111.6950 [quant-ph]

Initialize a quantum channel Superoperator operator.

Parameters

• (QuantumCircuit or (data) – Instruction or BaseOperator or matrix): data to initialize superoperator.

• input_dims (tuple) – the input subsystem dimensions. [Default: None]

• output_dims (tuple) – the output subsystem dimensions. [Default: None]

Raises

QiskitError – if input data cannot be initialized as a superoperator.

Additional Information:

If the input or output dimensions are None, they will be automatically determined from the input data. If the input data is a Numpy array of shape (4**N, 4**N) qubit systems will be used. If the input operator is not an N-qubit operator, it will assign a single subsystem with dimension specified by the shape of the input.

__init__(data, input_dims=None, output_dims=None)

Initialize a quantum channel Superoperator operator.

Parameters

• (QuantumCircuit or (data) – Instruction or BaseOperator or matrix): data to initialize superoperator.

• input_dims (tuple) – the input subsystem dimensions. [Default: None]

• output_dims (tuple) – the output subsystem dimensions. [Default: None]

Raises

QiskitError – if input data cannot be initialized as a superoperator.

Additional Information:

If the input or output dimensions are None, they will be automatically determined from the input data. If the input data is a Numpy array of shape (4**N, 4**N) qubit systems will be used. If the input operator is not an N-qubit operator, it will assign a single subsystem with dimension specified by the shape of the input.

Methods

__init__(data[, input_dims, output_dims])	Initialize a quantum channel Superoperator operator.
adjoint()	Return the adjoint quantum channel.
compose(other[, qargs, front])	Return the operator composition with another SuperOp.
conjugate()	Return the conjugate quantum channel.
copy()	Make a deep copy of current operator.
dot(other[, qargs])	Return the right multiplied operator self * other.
expand(other)	Return the reverse-order tensor product with another SuperOp.
input_dims([qargs])	Return tuple of input dimension for specified subsystems.
is_cp([atol, rtol])	Test if Choi-matrix is completely-positive (CP)
is_cptp([atol, rtol])	Return True if completely-positive trace-preserving (CPTP).
is_tp([atol, rtol])	Test if a channel is trace-preserving (TP)
is_unitary([atol, rtol])	Return True if QuantumChannel is a unitary channel.
output_dims([qargs])	Return tuple of output dimension for specified subsystems.
power(n)	Return the power of the quantum channel.
reshape([input_dims, output_dims, num_qubits])	Return a shallow copy with reshaped input and output subsystem dimensions.
tensor(other)	Return the tensor product with another SuperOp.
to_instruction()	Convert to a Kraus or UnitaryGate circuit instruction.
to_operator()	Try to convert channel to a unitary representation Operator.
transpose()	Return the transpose quantum channel.

Attributes

atol	Default absolute tolerance parameter for float comparisons.
data	Return data.
dim	Return tuple (input_shape, output_shape).
num_qubits	Return the number of qubits if a N-qubit operator or None otherwise.
qargs	Return the qargs for the operator.
rtol	Default relative tolerance parameter for float comparisons.

adjoint()

Return the adjoint quantum channel.

Note

This is equivalent to the matrix Hermitian conjugate in the SuperOp representation ie. for a channel \(\mathcal{E}\), the SuperOp of the adjoint channel \(\mathcal{{E}}^\dagger\) is \(S_{\mathcal{E}^\dagger} = S_{\mathcal{E}}^\dagger\).

property atol

Default absolute tolerance parameter for float comparisons.

compose(other, qargs=None, front=False)

Return the operator composition with another SuperOp.

Parameters

• other (SuperOp) – a SuperOp object.

• qargs (list or None) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).

• front (bool) – If True compose using right operator multiplication, instead of left multiplication [default: False].

Returns

The composed SuperOp.

Return type

SuperOp

Raises

QiskitError – if other cannot be converted to an operator, or has incompatible dimensions for specified subsystems.

Note

Composition (&) by default is defined as left matrix multiplication for matrix operators, while dot() is defined as right matrix multiplication. That is that A & B == A.compose(B) is equivalent to B.dot(A) when A and B are of the same type.

Setting the front=True kwarg changes this to right matrix multiplication and is equivalent to the dot() method A.dot(B) == A.compose(B, front=True).

conjugate()

Return the conjugate quantum channel.

Note

This is equivalent to the matrix complex conjugate in the SuperOp representation ie. for a channel \(\mathcal{E}\), the SuperOp of the conjugate channel \(\overline{{\mathcal{{E}}}}\) is \(S_{\overline{\mathcal{E}^\dagger}} = \overline{S_{\mathcal{E}}}\).

copy()

Make a deep copy of current operator.

property data

Return data.

property dim

Return tuple (input_shape, output_shape).

dot(other, qargs=None)

Return the right multiplied operator self * other.

Parameters

• other (Operator) – an operator object.

• qargs (list or None) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).

Returns

The right matrix multiplied Operator.

Return type

Operator

expand(other)

Return the reverse-order tensor product with another SuperOp.

Parameters

other (SuperOp) – a SuperOp object.

Returns

the tensor product \(b \otimes a\), where \(a\)

is the current SuperOp, and \(b\) is the other SuperOp.

Return type

SuperOp

input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

is_cp(atol=None, rtol=None)

Test if Choi-matrix is completely-positive (CP)

is_cptp(atol=None, rtol=None)

Return True if completely-positive trace-preserving (CPTP).

is_tp(atol=None, rtol=None)

Test if a channel is trace-preserving (TP)

is_unitary(atol=None, rtol=None)

Return True if QuantumChannel is a unitary channel.

property num_qubits

Return the number of qubits if a N-qubit operator or None otherwise.

output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

power(n)

Return the power of the quantum channel.

Parameters

n (float) – the power exponent.

Returns

the channel \(\mathcal{{E}} ^n\).

Return type

SuperOp

Raises

QiskitError – if the input and output dimensions of the SuperOp are not equal.

Note

For non-positive or non-integer exponents the power is defined as the matrix power of the SuperOp representation ie. for a channel \(\mathcal{{E}}\), the SuperOp of the powered channel \(\mathcal{{E}}^\n\) is \(S_{{\mathcal{{E}}^n}} = S_{{\mathcal{{E}}}}^n\).

property qargs

Return the qargs for the operator.

reshape(input_dims=None, output_dims=None, num_qubits=None)

Return a shallow copy with reshaped input and output subsystem dimensions.

Parameters

• input_dims (None or tuple) – new subsystem input dimensions. If None the original input dims will be preserved [Default: None].

• output_dims (None or tuple) – new subsystem output dimensions. If None the original output dims will be preserved [Default: None].

• num_qubits (None or int) – reshape to an N-qubit operator [Default: None].

Returns

returns self with reshaped input and output dimensions.

Return type

BaseOperator

Raises

QiskitError – if combined size of all subsystem input dimension or subsystem output dimensions is not constant.

property rtol

Default relative tolerance parameter for float comparisons.

tensor(other)

Return the tensor product with another SuperOp.

Parameters

other (SuperOp) – a SuperOp object.

Returns

the tensor product \(a \otimes b\), where \(a\)

is the current SuperOp, and \(b\) is the other SuperOp.

Return type

SuperOp

Note

The tensor product can be obtained using the ^ binary operator. Hence a.tensor(b) is equivalent to a ^ b.

to_instruction()

Convert to a Kraus or UnitaryGate circuit instruction.

If the channel is unitary it will be added as a unitary gate, otherwise it will be added as a kraus simulator instruction.

Returns

A kraus instruction for the channel.

Return type

qiskit.circuit.Instruction

Raises

QiskitError – if input data is not an N-qubit CPTP quantum channel.

to_operator()

Try to convert channel to a unitary representation Operator.

transpose()

Return the transpose quantum channel.

Note

This is equivalent to the matrix transpose in the SuperOp representation, ie. for a channel \(\mathcal{E}\), the SuperOp of the transpose channel \(\mathcal{{E}}^T\) is \(S_{mathcal{E}^T} = S_{\mathcal{E}}^T\).