Source code for pennylane.fermi.conversion
# Copyright 2018-2023 Xanadu Quantum Technologies Inc.
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
# http://www.apache.org/licenses/LICENSE-2.0
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""Functions to convert a fermionic operator to the qubit basis."""
from functools import singledispatch
from typing import Union
import numpy as np
import pennylane as qml
from pennylane.operation import Operator
from pennylane.pauli import PauliSentence, PauliWord
from .fermionic import FermiSentence, FermiWord
# pylint: disable=unexpected-keyword-arg
[docs]def jordan_wigner(
fermi_operator: Union[FermiWord, FermiSentence],
ps: bool = False,
wire_map: dict = None,
tol: float = None,
) -> Union[Operator, PauliSentence]:
r"""Convert a fermionic operator to a qubit operator using the Jordan-Wigner mapping.
The fermionic creation and annihilation operators are mapped to the Pauli operators as
.. math::
a^{\dagger}_0 = \left (\frac{X_0 - iY_0}{2} \right ), \:\: \text{...,} \:\:
a^{\dagger}_n = Z_0 \otimes Z_1 \otimes ... \otimes Z_{n-1} \otimes \left (\frac{X_n - iY_n}{2} \right ),
and
.. math::
a_0 = \left (\frac{X_0 + iY_0}{2} \right ), \:\: \text{...,} \:\:
a_n = Z_0 \otimes Z_1 \otimes ... \otimes Z_{n-1} \otimes \left (\frac{X_n + iY_n}{2} \right ),
where :math:`X`, :math:`Y`, and :math:`Z` are the Pauli operators.
Args:
fermi_operator(FermiWord, FermiSentence): the fermionic operator
ps (bool): whether to return the result as a PauliSentence instead of an
Operator. Defaults to False.
wire_map (dict): a dictionary defining how to map the orbitals of
the Fermi operator to qubit wires. If None, the integers used to
order the orbitals will be used as wire labels. Defaults to None.
tol (float): tolerance for discarding the imaginary part of the coefficients
Returns:
Union[PauliSentence, Operator]: a linear combination of qubit operators
**Example**
>>> w = qml.fermi.from_string('0+ 1-')
>>> jordan_wigner(w)
(
-0.25j * (Y(0) @ X(1))
+ (0.25+0j) * (Y(0) @ Y(1))
+ (0.25+0j) * (X(0) @ X(1))
+ 0.25j * (X(0) @ Y(1))
)
>>> jordan_wigner(w, ps=True)
-0.25j * Y(0) @ X(1)
+ (0.25+0j) * Y(0) @ Y(1)
+ (0.25+0j) * X(0) @ X(1)
+ 0.25j * X(0) @ Y(1)
>>> jordan_wigner(w, ps=True, wire_map={0: 2, 1: 3})
-0.25j * Y(2) @ X(3)
+ (0.25+0j) * Y(2) @ Y(3)
+ (0.25+0j) * X(2) @ X(3)
+ 0.25j * X(2) @ Y(3)
"""
return _jordan_wigner_dispatch(fermi_operator, ps, wire_map, tol)
@singledispatch
def _jordan_wigner_dispatch(fermi_operator, ps, wire_map, tol):
"""Dispatches to appropriate function if fermi_operator is a FermiWord or FermiSentence."""
raise ValueError(f"fermi_operator must be a FermiWord or FermiSentence, got: {fermi_operator}")
@_jordan_wigner_dispatch.register
def _(fermi_operator: FermiWord, ps=False, wire_map=None, tol=None):
wires = list(fermi_operator.wires) or [0]
identity_wire = wires[0]
if len(fermi_operator) == 0:
qubit_operator = PauliSentence({PauliWord({}): 1.0})
else:
coeffs = {"+": -0.5j, "-": 0.5j}
qubit_operator = PauliSentence({PauliWord({}): 1.0}) # Identity PS to multiply PSs with
for item in fermi_operator.items():
(_, wire), sign = item
z_string = dict(zip(range(wire), ["Z"] * wire))
qubit_operator @= PauliSentence(
{
PauliWord({**z_string, **{wire: "X"}}): 0.5,
PauliWord({**z_string, **{wire: "Y"}}): coeffs[sign],
}
)
for pw in qubit_operator:
if tol is not None and abs(qml.math.imag(qubit_operator[pw])) <= tol:
qubit_operator[pw] = qml.math.real(qubit_operator[pw])
if not ps:
# wire_order specifies wires to use for Identity (PauliWord({}))
qubit_operator = qubit_operator.operation(wire_order=[identity_wire])
if wire_map:
return qubit_operator.map_wires(wire_map)
return qubit_operator
@_jordan_wigner_dispatch.register
def _(fermi_operator: FermiSentence, ps=False, wire_map=None, tol=None):
wires = list(fermi_operator.wires) or [0]
identity_wire = wires[0]
qubit_operator = PauliSentence() # Empty PS as 0 operator to add Pws to
for fw, coeff in fermi_operator.items():
fermi_word_as_ps = jordan_wigner(fw, ps=True)
for pw in fermi_word_as_ps:
qubit_operator[pw] = qubit_operator[pw] + fermi_word_as_ps[pw] * coeff
if tol is not None and abs(qml.math.imag(qubit_operator[pw])) <= tol:
qubit_operator[pw] = qml.math.real(qubit_operator[pw])
qubit_operator.simplify(tol=1e-16)
if not ps:
qubit_operator = qubit_operator.operation(wire_order=[identity_wire])
if wire_map:
return qubit_operator.map_wires(wire_map)
return qubit_operator
[docs]def parity_transform(
fermi_operator: Union[FermiWord, FermiSentence],
n: int,
ps: bool = False,
wire_map: dict = None,
tol: float = None,
) -> Union[Operator, PauliSentence]:
r"""Convert a fermionic operator to a qubit operator using the parity mapping.
.. note::
Hamiltonians created with this mapping should be used with operators and states that are
compatible with the parity basis.
In parity mapping, qubit :math:`j` stores the parity of all :math:`j-1` qubits before it.
In comparison, :func:`~.jordan_wigner` simply uses qubit :math:`j` to store the occupation number.
In parity mapping, the fermionic creation and annihilation operators are mapped to the Pauli operators as
.. math::
\begin{align*}
a^{\dagger}_0 &= \left (\frac{X_0 - iY_0}{2} \right )\otimes X_1 \otimes X_2 \otimes ... X_n, \\\\
a^{\dagger}_n &= \left (\frac{Z_{n-1} \otimes X_n - iY_n}{2} \right ) \otimes X_{n+1} \otimes X_{n+2} \otimes ... \otimes X_n
\end{align*}
and
.. math::
\begin{align*}
a_0 &= \left (\frac{X_0 + iY_0}{2} \right )\otimes X_1 \otimes X_2 \otimes ... X_n,\\\\
a_n &= \left (\frac{Z_{n-1} \otimes X_n + iY_n}{2} \right ) \otimes X_{n+1} \otimes X_{n+2} \otimes ... \otimes X_n
\end{align*}
where :math:`X`, :math:`Y`, and :math:`Z` are the Pauli operators and :math:`n` is the number of qubits, i.e., spin orbitals.
Args:
fermi_operator(FermiWord, FermiSentence): the fermionic operator
n (int): number of qubits, i.e., spin orbitals in the system
ps (bool): whether to return the result as a :class:`~.PauliSentence` instead of an
:class:`~.Operator`. Defaults to ``False``.
wire_map (dict): a dictionary defining how to map the orbitals of
the Fermi operator to qubit wires. If ``None``, the integers used to
order the orbitals will be used as wire labels. Defaults to ``None``.
tol (float): tolerance for discarding the imaginary part of the coefficients
Returns:
Union[PauliSentence, Operator]: a linear combination of qubit operators
**Example**
>>> w = qml.fermi.from_string('0+ 1-')
>>> parity_transform(w, n=6)
(
-0.25j * Y(0)
+ (-0.25+0j) * (X(0) @ Z(1))
+ (0.25+0j) * X(0)
+ 0.25j * (Y(0) @ Z(1))
)
>>> parity_transform(w, n=6, ps=True)
-0.25j * Y(0)
+ (-0.25+0j) * X(0) @ Z(1)
+ (0.25+0j) * X(0)
+ 0.25j * Y(0) @ Z(1)
>>> parity_transform(w, n=6, ps=True, wire_map={0: 2, 1: 3})
-0.25j * Y(2)
+ (-0.25+0j) * X(2) @ Z(3)
+ (0.25+0j) * X(2)
+ 0.25j * Y(2) @ Z(3)
"""
return _parity_transform_dispatch(fermi_operator, n, ps, wire_map, tol)
@singledispatch
def _parity_transform_dispatch(fermi_operator, n, ps, wire_map, tol):
"""Dispatches to appropriate function if fermi_operator is a FermiWord or FermiSentence."""
raise ValueError(f"fermi_operator must be a FermiWord or FermiSentence, got: {fermi_operator}")
@_parity_transform_dispatch.register
def _(fermi_operator: FermiWord, n, ps=False, wire_map=None, tol=None):
wires = list(fermi_operator.wires) or [0]
identity_wire = wires[0]
coeffs = {"+": -0.5j, "-": 0.5j}
qubit_operator = PauliSentence({PauliWord({}): 1.0}) # Identity PS to multiply PSs with
for item in fermi_operator.items():
(_, wire), sign = item
if wire >= n:
raise ValueError(
f"Can't create or annihilate a particle on qubit number {wire} for a system with only {n} qubits"
)
x_string = dict(zip(range(wire + 1, n), ["X"] * (n - wire)))
pw1 = (
PauliWord({**{wire: "X"}, **x_string})
if wire == 0
else PauliWord({**{wire - 1: "Z"}, **{wire: "X"}, **x_string})
)
pw2 = PauliWord({**{wire: "Y"}, **x_string})
qubit_operator @= PauliSentence({pw1: 0.5, pw2: coeffs[sign]})
for pw in qubit_operator:
if tol is not None and abs(qml.math.imag(qubit_operator[pw])) <= tol:
qubit_operator[pw] = qml.math.real(qubit_operator[pw])
if not ps:
# wire_order specifies wires to use for Identity (PauliWord({}))
qubit_operator = qubit_operator.operation(wire_order=[identity_wire])
if wire_map:
return qubit_operator.map_wires(wire_map)
return qubit_operator
@_parity_transform_dispatch.register
def _(fermi_operator: FermiSentence, n, ps=False, wire_map=None, tol=None):
wires = list(fermi_operator.wires) or [0]
identity_wire = wires[0]
qubit_operator = PauliSentence() # Empty PS as 0 operator to add Pws to
for fw, coeff in fermi_operator.items():
fermi_word_as_ps = parity_transform(fw, n, ps=True)
for pw in fermi_word_as_ps:
qubit_operator[pw] = qubit_operator[pw] + fermi_word_as_ps[pw] * coeff
if tol is not None and abs(qml.math.imag(qubit_operator[pw])) <= tol:
qubit_operator[pw] = qml.math.real(qubit_operator[pw])
qubit_operator.simplify(tol=1e-16)
if not ps:
qubit_operator = qubit_operator.operation(wire_order=[identity_wire])
if wire_map:
return qubit_operator.map_wires(wire_map)
return qubit_operator
[docs]def bravyi_kitaev(
fermi_operator: Union[FermiWord, FermiSentence],
n: int,
ps: bool = False,
wire_map: dict = None,
tol: float = None,
) -> Union[Operator, PauliSentence]:
r"""Convert a fermionic operator to a qubit operator using the Bravyi-Kitaev mapping.
.. note::
Hamiltonians created with this mapping should be used with operators and states that are
compatible with the Bravyi-Kitaev basis.
In the Bravyi-Kitaev mapping, both occupation number and parity of the orbitals are stored non-locally.
In comparison, :func:`~.jordan_wigner` stores the occupation number locally while storing the parity
non-locally and vice-versa for :func:`~.parity_transform`. In the Bravyi-Kitaev mapping, the
fermionic creation and annihilation operators for even-labelled orbitals are mapped to the Pauli operators as
.. math::
\begin{align*}
a^{\dagger}_0 &= \frac{1}{2} \left ( X_0 -iY_{0} \right ), \\\\
a^{\dagger}_n &= \frac{1}{2} \left ( X_{U(n)} \otimes X_n \otimes Z_{P(n)} -iX_{U(n)} \otimes Y_{n} \otimes Z_{P(n)}\right ), \\\\
\end{align*}
and
.. math::
\begin{align*}
a_0 &= \frac{1}{2} \left ( X_0 + iY_{0} \right ), \\\\
a_n &= \frac{1}{2} \left ( X_{U(n)} \otimes X_n \otimes Z_{P(n)} +iX_{U(n)} \otimes Y_{n} \otimes Z_{P(n)}\right ). \\\\
\end{align*}
Similarly, the fermionic creation and annihilation operators for odd-labelled orbitals are mapped to the Pauli operators as
.. math::
\begin{align*}
a^{\dagger}_n &= \frac{1}{2} \left ( X_{U(n)} \otimes X_n \otimes Z_{P(n)} -iX_{U(n)} \otimes Y_{n} \otimes Z_{R(n)}\right ), \\\\
\end{align*}
and
.. math::
\begin{align*}
a_n &= \frac{1}{2} \left ( X_{U(n)} \otimes X_n \otimes Z_{P(n)} +iX_{U(n)} \otimes Y_{n} \otimes Z_{R(n)}\right ), \\\\
\end{align*}
where :math:`X`, :math:`Y`, and :math:`Z` are the Pauli operators, and :math:`U(n)`, :math:`P(n)` and :math:`R(n)`
represent the update, parity and remainder sets, respectively [`arXiv:1812.02233 <https://arxiv.org/abs/1812.02233>`_].
Args:
fermi_operator(FermiWord, FermiSentence): the fermionic operator
n (int): number of qubits, i.e., spin orbitals in the system
ps (bool): whether to return the result as a PauliSentence instead of an
Operator. Defaults to False.
wire_map (dict): a dictionary defining how to map the orbitals of
the Fermi operator to qubit wires. If None, the integers used to
order the orbitals will be used as wire labels. Defaults to None.
tol (float): tolerance for discarding the imaginary part of the coefficients
Returns:
Union[PauliSentence, Operator]: a linear combination of qubit operators
**Example**
>>> w = qml.fermi.from_string('0+ 1-')
>>> bravyi_kitaev(w, n=6)
(
-0.25j * Y(0)
+ (-0.25+0j) * (X(0) @ Z(1))
+ (0.25+0j) * X(0)
+ 0.25j * (Y(0) @ Z(1))
)
>>> bravyi_kitaev(w, n=6, ps=True)
-0.25j * Y(0)
+ (-0.25+0j) * X(0) @ Z(1)
+ (0.25+0j) * X(0)
+ 0.25j * Y(0) @ Z(1)
>>> bravyi_kitaev(w, n=6, ps=True, wire_map={0: 2, 1: 3})
-0.25j * Y(2)
+ (-0.25+0j) * X(2) @ Z(3)
+ (0.25+0j) * X(2)
+ 0.25j * Y(2) @ Z(3)
"""
return _bravyi_kitaev_dispatch(fermi_operator, n, ps, wire_map, tol)
def _update_set(j, bin_range, n):
"""
Computes the update set of the j-th orbital in n qubits.
Args:
j (int) : the orbital index
bin_range (int) : smallest power of 2 equal to or greater than
given number of qubits, e.g., 8 for 5 qubits
n (int) : number of qubits
Returns:
numpy.ndarray: Array containing the update set
"""
indices = np.array([], dtype=int)
midpoint = int(bin_range / 2)
if bin_range % 2 != 0:
return indices
if j < midpoint:
indices = np.append(indices, np.append(bin_range - 1, _update_set(j, midpoint, n)))
else:
indices = np.append(indices, _update_set(j - midpoint, midpoint, n) + midpoint)
indices = np.array([u for u in indices if u < n])
return indices
def _parity_set(j, bin_range):
"""
Computes the parity set of the j-th orbital in n qubits.
Args:
j (int) : the orbital index
bin_range (int) : smallest power of 2 equal to or greater than
given number of qubits, e.g., 8 for 5 qubits
Returns:
numpy.ndarray: Array of qubits which determine the parity of qubit j
"""
indices = np.array([], dtype=int)
midpoint = int(bin_range / 2)
if bin_range % 2 != 0:
return indices
if j < midpoint:
indices = np.append(indices, _parity_set(j, midpoint))
else:
indices = np.append(
indices,
np.append(
_parity_set(j - midpoint, midpoint) + midpoint,
midpoint - 1,
),
)
return indices
def _flip_set(j, bin_range):
"""
Computes the flip set of the j-th orbital in n qubits.
Args:
j (int) : the orbital index
bin_range (int) : smallest power of 2 equal to or greater than
given number of qubits, e.g., 8 for 5 qubits
Returns:
numpy.ndarray: Array containing information if the phase of orbital j is same as qubit j.
"""
indices = np.array([])
midpoint = int(bin_range / 2)
if bin_range % 2 != 0:
return indices
if j < midpoint:
indices = np.append(indices, _flip_set(j, midpoint))
elif midpoint <= j < bin_range - 1:
indices = np.append(indices, _flip_set(j - midpoint, midpoint) + midpoint)
else:
indices = np.append(
np.append(indices, _flip_set(j - midpoint, midpoint) + midpoint),
midpoint - 1,
)
return indices
@singledispatch
def _bravyi_kitaev_dispatch(fermi_operator, n, ps, wire_map, tol):
"""Dispatches to appropriate function if fermi_operator is a FermiWord or FermiSentence."""
raise ValueError(f"fermi_operator must be a FermiWord or FermiSentence, got: {fermi_operator}")
@_bravyi_kitaev_dispatch.register
def _(fermi_operator: FermiWord, n, ps=False, wire_map=None, tol=None):
wires = list(fermi_operator.wires) or [0]
identity_wire = wires[0]
coeffs = {"+": -0.5j, "-": 0.5j}
qubit_operator = PauliSentence({PauliWord({}): 1.0}) # Identity PS to multiply PSs with
bin_range = int(2 ** np.ceil(np.log2(n)))
for (_, wire), sign in fermi_operator.items():
if wire >= n:
raise ValueError(
f"Can't create or annihilate a particle on qubit index {wire} for a system with only {n} qubits."
)
u_set = _update_set(wire, bin_range, n)
update_string = dict(zip(u_set, ["X"] * len(u_set)))
p_set = _parity_set(wire, bin_range)
parity_string = dict(zip(p_set, ["Z"] * len(p_set)))
if wire % 2 == 0:
qubit_operator @= PauliSentence(
{
PauliWord({**parity_string, **{wire: "X"}, **update_string}): 0.5,
PauliWord({**parity_string, **{wire: "Y"}, **update_string}): coeffs[sign],
}
)
else:
f_set = _flip_set(wire, bin_range)
r_set = np.setdiff1d(p_set, f_set)
remainder_string = dict(zip(r_set, ["Z"] * len(r_set)))
qubit_operator @= PauliSentence(
{
PauliWord({**parity_string, **{wire: "X"}, **update_string}): 0.5,
PauliWord({**remainder_string, **{wire: "Y"}, **update_string}): coeffs[sign],
}
)
for pw in qubit_operator:
if tol is not None and abs(qml.math.imag(qubit_operator[pw])) <= tol:
qubit_operator[pw] = qml.math.real(qubit_operator[pw])
if not ps:
# wire_order specifies wires to use for Identity (PauliWord({}))
qubit_operator = qubit_operator.operation(wire_order=[identity_wire])
if wire_map:
return qubit_operator.map_wires(wire_map)
return qubit_operator
@_bravyi_kitaev_dispatch.register
def _(fermi_operator: FermiSentence, n, ps=False, wire_map=None, tol=None):
wires = list(fermi_operator.wires) or [0]
identity_wire = wires[0]
qubit_operator = PauliSentence() # Empty PS as 0 operator to add Pws to
for fw, coeff in fermi_operator.items():
fermi_word_as_ps = bravyi_kitaev(fw, n, ps=True)
for pw in fermi_word_as_ps:
qubit_operator[pw] = qubit_operator[pw] + fermi_word_as_ps[pw] * coeff
if tol is not None and abs(qml.math.imag(qubit_operator[pw])) <= tol:
qubit_operator[pw] = qml.math.real(qubit_operator[pw])
qubit_operator.simplify(tol=1e-16)
if not ps:
qubit_operator = qubit_operator.operation(wire_order=[identity_wire])
if wire_map:
return qubit_operator.map_wires(wire_map)
return qubit_operator
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