# 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.
"""
Kraus representation of a Quantum Channel.
"""
import copy
from numbers import Number
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.predicates import is_identity_matrix
from qiskit.quantum_info.operators.channel.quantum_channel import QuantumChannel
from qiskit.quantum_info.operators.op_shape import OpShape
from qiskit.quantum_info.operators.channel.choi import Choi
from qiskit.quantum_info.operators.channel.superop import SuperOp
from qiskit.quantum_info.operators.channel.transformations import _to_kraus
from qiskit.quantum_info.operators.mixins import generate_apidocs
[Doku]class Kraus(QuantumChannel):
r"""Kraus representation of a quantum channel.
For a quantum channel :math:`\mathcal{E}`, the Kraus representation is
given by a set of matrices :math:`[A_0,...,A_{K-1}]` such that the
evolution of a :class:`~qiskit.quantum_info.DensityMatrix`
:math:`\rho` is given by
.. math::
\mathcal{E}(\rho) = \sum_{i=0}^{K-1} A_i \rho A_i^\dagger
A general operator map :math:`\mathcal{G}` can also be written using the
generalized Kraus representation which is given by two sets of matrices
:math:`[A_0,...,A_{K-1}]`, :math:`[B_0,...,A_{B-1}]` such that
.. math::
\mathcal{G}(\rho) = \sum_{i=0}^{K-1} A_i \rho B_i^\dagger
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, input_dims=None, output_dims=None):
"""Initialize a quantum channel Kraus 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 cannot be initialized as a
a list of Kraus matrices.
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 list of Numpy arrays of shape (2**N, 2**N) qubit systems will be
used. If the input does not correspond to an N-qubit channel, it
will assign a single subsystem with dimension specified by the
shape of the input.
"""
# If the input is a list or tuple we assume it is a list of Kraus
# matrices, if it is a numpy array we assume that it is a single Kraus
# operator
# TODO properly handle array construction from ragged data (like tuple(np.ndarray, None))
# and document these accepted input cases. See also Qiskit/qiskit-terra#9307.
if isinstance(data, (list, tuple, np.ndarray)):
# Check if it is a single unitary matrix A for channel:
# E(rho) = A * rho * A^\dagger
if _is_matrix(data):
# Convert single Kraus op to general Kraus pair
kraus = ([np.asarray(data, dtype=complex)], None)
shape = kraus[0][0].shape
# Check if single Kraus set [A_i] for channel:
# E(rho) = sum_i A_i * rho * A_i^dagger
elif isinstance(data, list) and len(data) > 0:
# Get dimensions from first Kraus op
kraus = [np.asarray(data[0], dtype=complex)]
shape = kraus[0].shape
# Iterate over remaining ops and check they are same shape
for i in data[1:]:
op = np.asarray(i, dtype=complex)
if op.shape != shape:
raise QiskitError("Kraus operators are different dimensions.")
kraus.append(op)
# Convert single Kraus set to general Kraus pair
kraus = (kraus, None)
# Check if generalized Kraus set ([A_i], [B_i]) for channel:
# E(rho) = sum_i A_i * rho * B_i^dagger
elif isinstance(data, tuple) and len(data) == 2 and len(data[0]) > 0:
kraus_left = [np.asarray(data[0][0], dtype=complex)]
shape = kraus_left[0].shape
for i in data[0][1:]:
op = np.asarray(i, dtype=complex)
if op.shape != shape:
raise QiskitError("Kraus operators are different dimensions.")
kraus_left.append(op)
if data[1] is None:
kraus = (kraus_left, None)
else:
kraus_right = []
for i in data[1]:
op = np.asarray(i, dtype=complex)
if op.shape != shape:
raise QiskitError("Kraus operators are different dimensions.")
kraus_right.append(op)
kraus = (kraus_left, kraus_right)
else:
raise QiskitError("Invalid input for Kraus channel.")
op_shape = OpShape.auto(dims_l=output_dims, dims_r=input_dims, shape=kraus[0][0].shape)
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)
op_shape = data._op_shape
output_dim, input_dim = op_shape.shape
# Now that the input is an operator we convert it to a Kraus
rep = getattr(data, "_channel_rep", "Operator")
kraus = _to_kraus(rep, data._data, input_dim, output_dim)
# Initialize either single or general Kraus
if kraus[1] is None or np.allclose(kraus[0], kraus[1]):
# Standard Kraus map
data = (kraus[0], None)
else:
# General (non-CPTP) Kraus map
data = kraus
super().__init__(data, op_shape=op_shape)
@property
def data(self):
"""Return list of Kraus matrices for channel."""
if self._data[1] is None:
# If only a single Kraus set, don't return the tuple
# Just the fist set
return self._data[0]
else:
# Otherwise return the tuple of both kraus sets
return self._data
[Doku] def is_cptp(self, atol=None, rtol=None):
"""Return True if completely-positive trace-preserving."""
if self._data[1] is not None:
return False
if atol is None:
atol = self.atol
if rtol is None:
rtol = self.rtol
accum = 0j
for op in self._data[0]:
accum += np.dot(np.transpose(np.conj(op)), op)
return is_identity_matrix(accum, rtol=rtol, atol=atol)
def _evolve(self, state, qargs=None):
return SuperOp(self)._evolve(state, qargs)
# ---------------------------------------------------------------------
# BaseOperator methods
# ---------------------------------------------------------------------
[Doku] def conjugate(self):
ret = copy.copy(self)
kraus_l, kraus_r = self._data
kraus_l = [np.conj(k) for k in kraus_l]
if kraus_r is not None:
kraus_r = [k.conj() for k in kraus_r]
ret._data = (kraus_l, kraus_r)
return ret
[Doku] def transpose(self):
ret = copy.copy(self)
ret._op_shape = self._op_shape.transpose()
kraus_l, kraus_r = self._data
kraus_l = [np.transpose(k) for k in kraus_l]
if kraus_r is not None:
kraus_r = [np.transpose(k) for k in kraus_r]
ret._data = (kraus_l, kraus_r)
return ret
[Doku] def adjoint(self):
ret = copy.copy(self)
ret._op_shape = self._op_shape.transpose()
kraus_l, kraus_r = self._data
kraus_l = [np.conj(np.transpose(k)) for k in kraus_l]
if kraus_r is not None:
kraus_r = [np.conj(np.transpose(k)) for k in kraus_r]
ret._data = (kraus_l, kraus_r)
return ret
[Doku] def compose(self, other, qargs=None, front=False):
if qargs is None:
qargs = getattr(other, "qargs", None)
if qargs is not None:
return Kraus(SuperOp(self).compose(other, qargs=qargs, front=front))
if not isinstance(other, Kraus):
other = Kraus(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:
ka_l, ka_r = self._data
kb_l, kb_r = other._data
else:
ka_l, ka_r = other._data
kb_l, kb_r = self._data
kab_l = [np.dot(a, b) for a in ka_l for b in kb_l]
if ka_r is None and kb_r is None:
kab_r = None
elif ka_r is None:
kab_r = [np.dot(a, b) for a in ka_l for b in kb_r]
elif kb_r is None:
kab_r = [np.dot(a, b) for a in ka_r for b in kb_l]
else:
kab_r = [np.dot(a, b) for a in ka_r for b in kb_r]
ret = Kraus((kab_l, kab_r), input_dims, output_dims)
ret._op_shape = new_shape
return ret
[Doku] def tensor(self, other):
if not isinstance(other, Kraus):
other = Kraus(other)
return self._tensor(self, other)
[Doku] def expand(self, other):
if not isinstance(other, Kraus):
other = Kraus(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)
# Get tensor matrix
ka_l, ka_r = a._data
kb_l, kb_r = b._data
kab_l = [np.kron(ka, kb) for ka in ka_l for kb in kb_l]
if ka_r is None and kb_r is None:
kab_r = None
else:
if ka_r is None:
ka_r = ka_l
if kb_r is None:
kb_r = kb_l
kab_r = [np.kron(a, b) for a in ka_r for b in kb_r]
ret._data = (kab_l, kab_r)
return ret
def __add__(self, other):
qargs = getattr(other, "qargs", None)
if not isinstance(other, QuantumChannel):
other = Choi(other)
return self._add(other, qargs=qargs)
def __sub__(self, other):
qargs = getattr(other, "qargs", None)
if not isinstance(other, QuantumChannel):
other = Choi(other)
return self._add(-other, qargs=qargs)
def _add(self, other, qargs=None):
# Since we cannot directly add two channels in the Kraus
# representation we try and use the other channels method
# or convert to the Choi representation
return Kraus(Choi(self)._add(other, qargs=qargs))
def _multiply(self, other):
if not isinstance(other, Number):
raise QiskitError("other is not a number")
ret = copy.copy(self)
# If the number is complex we need to convert to general
# kraus channel so we multiply via Choi representation
if isinstance(other, complex) or other < 0:
# Convert to Choi-matrix
ret._data = Kraus(Choi(self)._multiply(other))._data
return ret
# If the number is real we can update the Kraus operators
# directly
val = np.sqrt(other)
kraus_r = None
kraus_l = [val * k for k in self._data[0]]
if self._data[1] is not None:
kraus_r = [val * k for k in self._data[1]]
ret._data = (kraus_l, kraus_r)
return ret
def _is_matrix(data):
# return True if data is a 2-d array/tuple/list
if not isinstance(data, np.ndarray):
data = np.array(data, dtype=object)
return data.ndim == 2
# Update docstrings for API docs
generate_apidocs(Kraus)