PTM

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

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

Pauli Transfer Matrix (PTM) representation of a Quantum Channel.

The PTM representation of an $n$ -qubit quantum channel $\mathcal{E}$ is an $n$ -qubit SuperOp $R$ defined with respect to vectorization in the Pauli basis instead of column-vectorization. The elements of the PTM $R$ are given by

R_{i,j} = \text{Tr}\left[P_i \mathcal{E}(P_j) \right]

where $[P_0, P_1, ..., P_{4^{n}-1}]$ is the $n$ -qubit Pauli basis in lexicographic order.

Evolution of a DensityMatrix $\rho$ with respect to the PTM is given by

|\mathcal{E}(\rho)\rangle\!\rangle_P = S_P |\rho\rangle\!\rangle_P

where $|A\rangle\!\rangle_P$ denotes vectorization in the Pauli basis $\langle i | A\rangle\!\rangle_P = \text{Tr}[P_i A]$ .

See reference [1] for further details.

References

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 PTM quantum channel 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 PTM.

Additional Information:

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

Methods

adjoint

PTM.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 $S_{\mathcal{E}^\dagger} = S_{\mathcal{E}}^\dagger$ .

compose

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

Return the operator composition with another PTM.

Parameters

other (PTM) – a PTM 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 PTM.

Return type

PTM

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

PTM.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

PTM.copy()

Make a deep copy of current operator.

dot

PTM.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

PTM.expand(other)

Return the reverse-order tensor product with another PTM.

Parameters

other (PTM) – a PTM object.

Returns

the tensor product $b \otimes a$ , where $a$

is the current PTM, and $b$ is the other PTM.

Return type

PTM

input_dims

PTM.input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

is_cp

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

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

is_cptp

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

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

is_tp

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

Test if a channel is trace-preserving (TP)

is_unitary

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

Return True if QuantumChannel is a unitary channel.

output_dims

PTM.output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

power

PTM.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 $S_{{\mathcal{{E}}^n}} = S_{{\mathcal{{E}}}}^n$ .

reshape

PTM.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

PTM.tensor(other)

Return the tensor product with another PTM.

Parameters

other (PTM) – a PTM object.

Returns

the tensor product $a \otimes b$ , where $a$

is the current PTM, and $b$ is the other PTM.

Return type

PTM

Note

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

to_instruction

PTM.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

PTM.to_operator()

Try to convert channel to a unitary representation Operator.

transpose

PTM.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 $S_{mathcal{E}^T} = S_{\mathcal{E}}^T$ .

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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