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Chi

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

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Bases: qiskit.quantum_info.operators.channel.quantum_channel.QuantumChannel

Pauli basis Chi-matrix representation of a quantum channel.

The Chi-matrix representation of an nn-qubit quantum channel E\mathcal{E} is a matrix χ\chi such that the evolution of a DensityMatrix ρ\rho is given by

E(ρ)=i,jχi,jPiρPj\mathcal{E}(ρ) = \sum_{i, j} \chi_{i,j} P_i ρ P_j

where [P0,P1,...,P4n1][P_0, P_1, ..., P_{4^{n}-1}] is the nn-qubit Pauli basis in lexicographic order. It is related to the Choi representation by a change of basis of the Choi-matrix into the Pauli basis.

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](opens in a new tab)

Initialize a quantum channel Chi-matrix operator.

Parameters

  • or (data (QuantumCircuit) – 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 is not an N-qubit channel or cannot be initialized as a Chi-matrix.

Additional Information:

If the input or output dimensions are None, they will be automatically determined from the input data. The Chi matrix representation is only valid for N-qubit channels.


Methods

adjoint

Chi.adjoint()

Return the adjoint quantum channel.

Note

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

compose

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

Return the operator composition with another Chi.

Parameters

  • other (Chi) – a Chi 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 Chi.

Return type

Chi

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 @ (equivalent to dot()) is defined as right matrix multiplication. That is that A & B == A.compose(B) is equivalent to B @ A == 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

Chi.conjugate()

Return the conjugate quantum channel.

Note

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

copy

Chi.copy()

Make a deep copy of current operator.

dot

Chi.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

Note

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

expand

Chi.expand(other)

Return the reverse-order tensor product with another Chi.

Parameters

other (Chi) – a Chi object.

Returns

the tensor product bab \otimes a, where aa

is the current Chi, and bb is the other Chi.

Return type

Chi

input_dims

Chi.input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

is_cp

Chi.is_cp(atol=None, rtol=None)

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

is_cptp

Chi.is_cptp(atol=None, rtol=None)

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

is_tp

Chi.is_tp(atol=None, rtol=None)

Test if a channel is trace-preserving (TP)

is_unitary

Chi.is_unitary(atol=None, rtol=None)

Return True if QuantumChannel is a unitary channel.

output_dims

Chi.output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

power

Chi.power(n)

Return the power of the quantum channel.

Parameters

n (float) – the power exponent.

Returns

the channel En\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 E\mathcal{{E}}, the SuperOp of the powered channel En\mathcal{{E}}^n is SEn=SEnS_{{\mathcal{{E}}^n}} = S_{{\mathcal{{E}}}}^n.

reshape

Chi.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.

tensor

Chi.tensor(other)

Return the tensor product with another Chi.

Parameters

other (Chi) – a Chi object.

Returns

the tensor product aba \otimes b, where aa

is the current Chi, and bb is the other Chi.

Return type

Chi

Note

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

to_instruction

Chi.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

Chi.to_operator()

Try to convert channel to a unitary representation Operator.

transpose

Chi.transpose()

Return the transpose quantum channel.

Note

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


Attributes

atol

Default value: 1e-08

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 value: 1e-05

settings

Return settings.

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