# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2019.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""
Pauli Transfer Matrix (PTM) representation of a Quantum Channel.
"""
from __future__ import annotations
import copy
import numpy as np
from qiskit.circuit.quantumcircuit import QuantumCircuit
from qiskit.circuit.instruction import Instruction
from qiskit.exceptions import QiskitError
from qiskit.quantum_info.operators.channel.quantum_channel import QuantumChannel
from qiskit.quantum_info.operators.channel.superop import SuperOp
from qiskit.quantum_info.operators.channel.transformations import _to_ptm
from qiskit.quantum_info.operators.mixins import generate_apidocs
from qiskit.quantum_info.operators.base_operator import BaseOperator
[docs]class PTM(QuantumChannel):
r"""Pauli Transfer Matrix (PTM) representation of a Quantum Channel.
The PTM representation of an :math:`n`-qubit quantum channel
:math:`\mathcal{E}` is an :math:`n`-qubit :class:`SuperOp` :math:`R`
defined with respect to vectorization in the Pauli basis instead of
column-vectorization. The elements of the PTM :math:`R` are
given by
.. math::
R_{i,j} = \frac{1}{2^n} \mbox{Tr}\left[P_i \mathcal{E}(P_j) \right]
where :math:`[P_0, P_1, ..., P_{4^{n}-1}]` is the :math:`n`-qubit Pauli basis in
lexicographic order.
Evolution of a :class:`~qiskit.quantum_info.DensityMatrix`
:math:`\rho` with respect to the PTM is given by
.. math::
|\mathcal{E}(\rho)\rangle\!\rangle_P = S_P |\rho\rangle\!\rangle_P
where :math:`|A\rangle\!\rangle_P` denotes vectorization in the Pauli basis
:math:`\langle i | A\rangle\!\rangle_P = \sqrt{\frac{1}{2^n}} \mbox{Tr}[P_i A]`.
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] <https://arxiv.org/abs/1111.6950>`_
"""
def __init__(
self,
data: QuantumCircuit | Instruction | BaseOperator | np.ndarray,
input_dims: int | tuple | None = None,
output_dims: int | tuple | None = None,
):
"""Initialize a PTM quantum channel operator.
Args:
data (QuantumCircuit or
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.
"""
# If the input is a raw list or matrix we assume that it is
# already a Chi matrix.
if isinstance(data, (list, np.ndarray)):
# Should we force this to be real?
ptm = np.asarray(data, dtype=complex)
# Determine input and output dimensions
dout, din = ptm.shape
if input_dims:
input_dim = np.prod(input_dims)
else:
input_dim = int(np.sqrt(din))
if output_dims:
output_dim = np.prod(input_dims)
else:
output_dim = int(np.sqrt(dout))
if output_dim**2 != dout or input_dim**2 != din or input_dim != output_dim:
raise QiskitError("Invalid shape for PTM matrix.")
else:
# Otherwise we initialize by conversion from another Qiskit
# object into the QuantumChannel.
if isinstance(data, (QuantumCircuit, Instruction)):
# If the input is a Terra QuantumCircuit or Instruction we
# convert it to a SuperOp
data = SuperOp._init_instruction(data)
else:
# We use the QuantumChannel init transform to initialize
# other objects into a QuantumChannel or Operator object.
data = self._init_transformer(data)
input_dim, output_dim = data.dim
# Now that the input is an operator we convert it to a PTM object
rep = getattr(data, "_channel_rep", "Operator")
ptm = _to_ptm(rep, data._data, input_dim, output_dim)
if input_dims is None:
input_dims = data.input_dims()
if output_dims is None:
output_dims = data.output_dims()
# Check input is N-qubit channel
num_qubits = int(np.log2(input_dim))
if 2**num_qubits != input_dim or input_dim != output_dim:
raise QiskitError("Input is not an n-qubit Pauli transfer matrix.")
super().__init__(ptm, num_qubits=num_qubits)
def __array__(self, dtype=None):
if dtype:
np.asarray(self.data, dtype=dtype)
return self.data
@property
def _bipartite_shape(self):
"""Return the shape for bipartite matrix"""
return (self._output_dim, self._output_dim, self._input_dim, self._input_dim)
def _evolve(self, state, qargs=None):
return SuperOp(self)._evolve(state, qargs)
# ---------------------------------------------------------------------
# BaseOperator methods
# ---------------------------------------------------------------------
[docs] def conjugate(self):
# Since conjugation is basis dependent we transform
# to the SuperOp representation to compute the
# conjugate channel
return PTM(SuperOp(self).conjugate())
[docs] def transpose(self):
return PTM(SuperOp(self).transpose())
[docs] def adjoint(self):
return PTM(SuperOp(self).adjoint())
[docs] def compose(self, other: PTM, qargs: list | None = None, front: bool = False) -> PTM:
if qargs is None:
qargs = getattr(other, "qargs", None)
if qargs is not None:
return PTM(SuperOp(self).compose(other, qargs=qargs, front=front))
# Convert other to PTM
if not isinstance(other, PTM):
other = PTM(other)
new_shape = self._op_shape.compose(other._op_shape, qargs, front)
input_dims = new_shape.dims_r()
output_dims = new_shape.dims_l()
if front:
data = np.dot(self._data, other.data)
else:
data = np.dot(other.data, self._data)
ret = PTM(data, input_dims, output_dims)
ret._op_shape = new_shape
return ret
[docs] def tensor(self, other: PTM) -> PTM:
if not isinstance(other, PTM):
other = PTM(other)
return self._tensor(self, other)
[docs] def expand(self, other: PTM) -> PTM:
if not isinstance(other, PTM):
other = PTM(other)
return self._tensor(other, self)
@classmethod
def _tensor(cls, a, b):
ret = copy.copy(a)
ret._op_shape = a._op_shape.tensor(b._op_shape)
ret._data = np.kron(a._data, b.data)
return ret
# Update docstrings for API docs
generate_apidocs(PTM)