# Copyright 2018-2021 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.
"""
This submodule contains functionality for running Variational Quantum Eigensolver (VQE)
computations using PennyLane.
"""
# pylint: disable=too-many-arguments, too-few-public-methods
from collections.abc import Sequence
import itertools
import warnings
import pennylane as qml
from pennylane import numpy as np
from pennylane.operation import Observable, Tensor
from pennylane.queuing import QueuingError
from pennylane.wires import Wires
OBS_MAP = {"PauliX": "X", "PauliY": "Y", "PauliZ": "Z", "Hadamard": "H", "Identity": "I"}
[docs]class Hamiltonian:
r"""Lightweight class for representing Hamiltonians for Variational Quantum
Eigensolver problems.
Hamiltonians can be expressed as linear combinations of observables, e.g.,
:math:`\sum_{k=0}^{N-1} c_k O_k`.
This class keeps track of the terms (coefficients and observables) separately.
Args:
coeffs (Iterable[float]): coefficients of the Hamiltonian expression
observables (Iterable[Observable]): observables in the Hamiltonian expression
simplify (bool): Specifies whether the Hamiltonian is simplified upon initialization
(like-terms are combined). The default value is `False`.
.. seealso:: :class:`~.ExpvalCost`, :func:`~.molecular_hamiltonian`
**Example:**
A Hamiltonian can be created by simply passing the list of coefficients
as well as the list of observables:
>>> coeffs = [0.2, -0.543]
>>> obs = [qml.PauliX(0) @ qml.PauliZ(1), qml.PauliZ(0) @ qml.Hadamard(2)]
>>> H = qml.Hamiltonian(coeffs, obs)
>>> print(H)
(-0.543) [Z0 H2]
+ (0.2) [X0 Z1]
The user can also provide custom observables:
>>> obs_matrix = np.array([[0.5, 1.0j, 0.0, -3j],
[-1.0j, -1.1, 0.0, -0.1],
[0.0, 0.0, -0.9, 12.0],
[3j, -0.1, 12.0, 0.0]])
>>> obs = qml.Hermitian(obs_matrix, wires=[0, 1])
>>> H = qml.Hamiltonian((0.8, ), (obs, ))
>>> print(H)
(0.8) [Hermitian0,1]
Alternatively, the :func:`~.molecular_hamiltonian` function from the
:doc:`/introduction/chemistry` module can be used to generate a molecular
Hamiltonian.
.. Warning::
Hamiltonians can be constructed using Pythonic arithmetic operations. For example:
>>> qml.PauliX(0) + 2 * qml.PauliZ(0) @ qml.PauliZ(1)
is equivalent to the following Hamiltonian:
>>> qml.Hamiltonian([1, 2], [qml.PauliX(0), qml.PauliZ(0) @ qml.PauliZ(1)])
When Hamiltonians are defined using arithmetic operations **inside of QNodes**, constituent observables
may be queued as operations/an error may be thrown. Thus, Hamiltonians must be defined either outside of QNodes,
or inside of QNodes using the conventional method.
Note that this issue also arises when calling the ``simplify()`` method.
"""
# Todo: this is a temporary solution to make the circuit drawer work
num_params = 0
def __init__(self, coeffs, observables, simplify=False):
if len(coeffs) != len(observables):
raise ValueError(
"Could not create valid Hamiltonian; "
"number of coefficients and operators does not match."
)
if any(np.imag(coeffs) != 0):
raise ValueError(
"Could not create valid Hamiltonian; " "coefficients are not real-valued."
)
for obs in observables:
if not isinstance(obs, Observable):
raise ValueError(
"Could not create circuits. Some or all observables are not valid."
)
self._coeffs = list(coeffs)
self._ops = list(observables)
self.data = []
self.return_type = None
if simplify:
self.simplify()
self.queue()
@property
def coeffs(self):
"""Return the coefficients defining the Hamiltonian.
Returns:
Iterable[float]): coefficients in the Hamiltonian expression
"""
return self._coeffs
@property
def ops(self):
"""Return the operators defining the Hamiltonian.
Returns:
Iterable[Observable]): observables in the Hamiltonian expression
"""
return self._ops
@property
def terms(self):
r"""The terms of the Hamiltonian expression :math:`\sum_{k=0}^{N-1} c_k O_k`
Returns:
(tuple, tuple): tuples of coefficients and operations, each of length N
"""
return self.coeffs, self.ops
@property
def wires(self):
r"""The sorted union of wires from all operators.
Returns:
(Wires): Combined wires present in all terms, sorted.
"""
return qml.wires.Wires.all_wires([op.wires for op in self.ops], sort=True)
@property
def name(self):
return "Hamiltonian"
[docs] def simplify(self):
r"""Simplifies the Hamiltonian by combining like-terms.
**Example**
>>> ops = [qml.PauliY(2), qml.PauliX(0) @ qml.Identity(1), qml.PauliX(0)]
>>> H = qml.Hamiltonian([1, 1, -2], ops)
>>> H.simplify()
>>> print(H)
(-1) [X0]
+ (1) [Y2]
"""
coeffs = []
ops = []
for c, op in zip(self.coeffs, self.ops):
op = op if isinstance(op, Tensor) else Tensor(op)
ind = None
for i, other in enumerate(ops):
if op.compare(other):
ind = i
break
if ind is not None:
coeffs[ind] += c
if np.allclose([coeffs[ind]], [0]):
del coeffs[ind]
del ops[ind]
else:
ops.append(op.prune())
coeffs.append(c)
self._coeffs = coeffs
self._ops = ops
def __str__(self):
# Lambda function that formats the wires
wires_print = lambda ob: ",".join(map(str, ob.wires.tolist()))
paired_coeff_obs = list(zip(self.coeffs, self.ops))
paired_coeff_obs.sort(key=lambda pair: (len(pair[1].wires), pair[0]))
terms_ls = []
for coeff, obs in paired_coeff_obs:
if isinstance(obs, Tensor):
obs_strs = [f"{OBS_MAP.get(ob.name, ob.name)}{wires_print(ob)}" for ob in obs.obs]
ob_str = " ".join(obs_strs)
elif isinstance(obs, Observable):
ob_str = f"{OBS_MAP.get(obs.name, obs.name)}{wires_print(obs)}"
term_str = f"({coeff}) [{ob_str}]"
terms_ls.append(term_str)
return " " + "\n+ ".join(terms_ls)
def __repr__(self):
# Constructor-call-like representation
return f"<Hamiltonian: terms={len(self.coeffs)}, wires={self.wires.tolist()}>"
def _obs_data(self):
r"""Extracts the data from a Hamiltonian and serializes it in an order-independent fashion.
This allows for comparison between Hamiltonians that are equivalent, but are defined with terms and tensors
expressed in different orders. For example, `qml.PauliX(0) @ qml.PauliZ(1)` and
`qml.PauliZ(1) @ qml.PauliX(0)` are equivalent observables with different orderings.
.. Note::
In order to store the data from each term of the Hamiltonian in an order-independent serialization,
we make use of sets. Note that all data contained within each term must be immutable, hence the use of
strings and frozensets.
**Example**
>>> H = qml.Hamiltonian([1, 1], [qml.PauliX(0) @ qml.PauliX(1), qml.PauliZ(0)])
>>> print(H._obs_data())
{(1, frozenset({('PauliZ', <Wires = [1]>, ())})),
(1, frozenset({('PauliX', <Wires = [1]>, ()), ('PauliX', <Wires = [0]>, ())}))}
"""
data = set()
for co, op in zip(*self.terms):
obs = op.non_identity_obs if isinstance(op, Tensor) else [op]
tensor = []
for ob in obs:
parameters = tuple(
param.tostring() for param in ob.parameters
) # Converts params into immutable type
tensor.append((ob.name, ob.wires, parameters))
data.add((co, frozenset(tensor)))
return data
[docs] def compare(self, H):
r"""Compares with another :class:`~Hamiltonian`, :class:`~.Observable`, or :class:`~.Tensor`,
to determine if they are equivalent.
Hamiltonians/observables are equivalent if they represent the same operator
(their matrix representations are equal), and they are defined on the same wires.
.. Warning::
The compare method does **not** check if the matrix representation
of a :class:`~.Hermitian` observable is equal to an equivalent
observable expressed in terms of Pauli matrices, or as a
linear combination of Hermitians.
To do so would require the matrix form of Hamiltonians and Tensors
be calculated, which would drastically increase runtime.
Returns:
(bool): True if equivalent.
**Examples**
>>> A = np.array([[1, 0], [0, -1]])
>>> H = qml.Hamiltonian(
... [0.5, 0.5],
... [qml.Hermitian(A, 0) @ qml.PauliY(1), qml.PauliY(1) @ qml.Hermitian(A, 0) @ qml.Identity("a")]
... )
>>> obs = qml.Hermitian(A, 0) @ qml.PauliY(1)
>>> print(H.compare(obs))
True
>>> H1 = qml.Hamiltonian([1, 1], [qml.PauliX(0), qml.PauliZ(1)])
>>> H2 = qml.Hamiltonian([1, 1], [qml.PauliZ(0), qml.PauliX(1)])
>>> H1.compare(H2)
False
>>> ob1 = qml.Hamiltonian([1], [qml.PauliX(0)])
>>> ob2 = qml.Hermitian(np.array([[0, 1], [1, 0]]), 0)
>>> ob1.compare(ob2)
False
"""
if isinstance(H, Hamiltonian):
self.simplify()
H.simplify()
return self._obs_data() == H._obs_data() # pylint: disable=protected-access
if isinstance(H, (Tensor, Observable)):
self.simplify()
return self._obs_data() == {
(1, frozenset(H._obs_data())) # pylint: disable=protected-access
}
raise ValueError("Can only compare a Hamiltonian, and a Hamiltonian/Observable/Tensor.")
def __matmul__(self, H):
r"""The tensor product operation between a Hamiltonian and a Hamiltonian/Tensor/Observable."""
coeffs1 = self.coeffs.copy()
terms1 = self.ops.copy()
if isinstance(H, Hamiltonian):
shared_wires = Wires.shared_wires([self.wires, H.wires])
if len(shared_wires) > 0:
raise ValueError(
"Hamiltonians can only be multiplied together if they act on "
"different sets of wires"
)
coeffs2 = H.coeffs
terms2 = H.ops
coeffs = [c[0] * c[1] for c in itertools.product(coeffs1, coeffs2)]
term_list = itertools.product(terms1, terms2)
terms = [qml.operation.Tensor(t[0], t[1]) for t in term_list]
return qml.Hamiltonian(coeffs, terms, simplify=True)
if isinstance(H, (Tensor, Observable)):
coeffs = coeffs1
terms = [term @ H for term in terms1]
return qml.Hamiltonian(coeffs, terms, simplify=True)
raise ValueError(f"Cannot tensor product Hamiltonian and {type(H)}")
def __add__(self, H):
r"""The addition operation between a Hamiltonian and a Hamiltonian/Tensor/Observable."""
coeffs = self.coeffs.copy()
ops = self.ops.copy()
if isinstance(H, Hamiltonian):
coeffs.extend(H.coeffs.copy())
ops.extend(H.ops.copy())
return qml.Hamiltonian(coeffs, ops, simplify=True)
if isinstance(H, (Tensor, Observable)):
coeffs.append(1)
ops.append(H)
return qml.Hamiltonian(coeffs, ops, simplify=True)
raise ValueError(f"Cannot add Hamiltonian and {type(H)}")
def __mul__(self, a):
r"""The scalar multiplication operation between a scalar and a Hamiltonian."""
if isinstance(a, (int, float)):
coeffs = [a * c for c in self.coeffs.copy()]
return qml.Hamiltonian(coeffs, self.ops.copy())
raise ValueError(f"Cannot multiply Hamiltonian by {type(a)}")
__rmul__ = __mul__
def __sub__(self, H):
r"""The subtraction operation between a Hamiltonian and a Hamiltonian/Tensor/Observable."""
if isinstance(H, (Hamiltonian, Tensor, Observable)):
return self.__add__(H.__mul__(-1))
raise ValueError(f"Cannot subtract {type(H)} from Hamiltonian")
def __iadd__(self, H):
r"""The inplace addition operation between a Hamiltonian and a Hamiltonian/Tensor/Observable."""
if isinstance(H, Hamiltonian):
self._coeffs.extend(H.coeffs.copy())
self._ops.extend(H.ops.copy())
self.simplify()
return self
if isinstance(H, (Tensor, Observable)):
self._coeffs.append(1)
self._ops.append(H)
self.simplify()
return self
raise ValueError(f"Cannot add Hamiltonian and {type(H)}")
def __imul__(self, a):
r"""The inplace scalar multiplication operation between a scalar and a Hamiltonian."""
if isinstance(a, (int, float)):
self._coeffs = [a * c for c in self.coeffs]
return self
raise ValueError(f"Cannot multiply Hamiltonian by {type(a)}")
def __isub__(self, H):
r"""The inplace subtraction operation between a Hamiltonian and a Hamiltonian/Tensor/Observable."""
if isinstance(H, (Hamiltonian, Tensor, Observable)):
self.__iadd__(H.__mul__(-1))
return self
raise ValueError(f"Cannot subtract {type(H)} from Hamiltonian")
[docs] def queue(self, context=qml.QueuingContext):
"""Queues a qml.Hamiltonian instance"""
for o in self.ops:
try:
context.update_info(o, owner=self)
except QueuingError:
o.queue(context=context)
context.update_info(o, owner=self)
except NotImplementedError:
pass
context.append(self, owns=tuple(self.ops))
return self
[docs]class ExpvalCost:
"""Create a cost function that gives the expectation value of an input Hamiltonian.
This cost function is useful for a range of problems including VQE and QAOA.
Args:
ansatz (callable): The ansatz for the circuit before the final measurement step.
Note that the ansatz **must** have the following signature:
.. code-block:: python
ansatz(params, **kwargs)
where ``params`` are the trainable weights of the variational circuit, and
``kwargs`` are any additional keyword arguments that need to be passed
to the template.
hamiltonian (~.Hamiltonian): Hamiltonian operator whose expectation value should be measured
device (Device, Sequence[Device]): Corresponding device(s) where the resulting
cost function should be executed. This can either be a single device, or a list
of devices of length matching the number of terms in the Hamiltonian.
interface (str, None): Which interface to use.
This affects the types of objects that can be passed to/returned to the cost function.
Supports all interfaces supported by the :func:`~.qnode` decorator.
diff_method (str, None): The method of differentiation to use with the created cost function.
Supports all differentiation methods supported by the :func:`~.qnode` decorator.
optimize (bool): Whether to optimize the observables composing the Hamiltonian by
separating them into qubit-wise commuting groups. Each group can then be executed
within a single QNode, resulting in fewer QNodes to evaluate.
Returns:
callable: a cost function with signature ``cost_fn(params, **kwargs)`` that evaluates
the expectation of the Hamiltonian on the provided device(s)
.. seealso:: :class:`~.Hamiltonian`, :func:`~.molecular_hamiltonian`, :func:`~.map`, :func:`~.dot`
**Example:**
To construct an ``ExpvalCost`` cost function, we require a Hamiltonian to measure, and an ansatz
for our variational circuit.
We can construct a Hamiltonian manually,
.. code-block:: python
coeffs = [0.2, -0.543]
obs = [
qml.PauliX(0) @ qml.PauliZ(1) @ qml.PauliY(3),
qml.PauliZ(0) @ qml.Hadamard(2)
]
H = qml.vqe.Hamiltonian(coeffs, obs)
Alternatively, the :func:`~.molecular_hamiltonian` function from the
:doc:`/introduction/chemistry` module can be used to generate a molecular Hamiltonian.
Once we have our Hamiltonian, we can select an ansatz and construct
the cost function.
>>> ansatz = qml.templates.StronglyEntanglingLayers
>>> dev = qml.device("default.qubit", wires=4)
>>> cost = qml.ExpvalCost(ansatz, H, dev, interface="torch")
>>> params = torch.rand([2, 4, 3])
>>> cost(params)
tensor(-0.2316, dtype=torch.float64)
The cost function can then be minimized using any gradient descent-based
:doc:`optimizer </introduction/optimizers>`.
.. UsageDetails::
**Optimizing observables:**
Setting ``optimize=True`` can be used to decrease the number of device executions. The
observables composing the Hamiltonian can be separated into groups that are qubit-wise
commuting using the :mod:`~.grouping` module. These groups can be executed together on a
*single* qnode, resulting in a lower device overhead:
.. code-block:: python
commuting_obs = [qml.PauliX(0), qml.PauliX(0) @ qml.PauliZ(1)]
H = qml.vqe.Hamiltonian([1, 1], commuting_obs)
dev = qml.device("default.qubit", wires=2)
ansatz = qml.templates.StronglyEntanglingLayers
cost_opt = qml.ExpvalCost(ansatz, H, dev, optimize=True)
cost_no_opt = qml.ExpvalCost(ansatz, H, dev, optimize=False)
params = qml.init.strong_ent_layers_uniform(3, 2)
Grouping these commuting observables leads to fewer device executions:
>>> cost_opt(params)
>>> ex_opt = dev.num_executions
>>> cost_no_opt(params)
>>> ex_no_opt = dev.num_executions - ex_opt
>>> print("Number of executions:", ex_no_opt)
Number of executions: 2
>>> print("Number of executions (optimized):", ex_opt)
Number of executions (optimized): 1
"""
def __init__(
self,
ansatz,
hamiltonian,
device,
interface="autograd",
diff_method="best",
optimize=False,
**kwargs,
):
if kwargs.get("measure", "expval") != "expval":
raise ValueError("ExpvalCost can only be used to construct sums of expectation values.")
coeffs, observables = hamiltonian.terms
self.hamiltonian = hamiltonian
"""Hamiltonian: the input Hamiltonian."""
self.qnodes = None
"""QNodeCollection: The QNodes to be evaluated. Each QNode corresponds to the expectation
value of each observable term after applying the circuit ansatz."""
self._multiple_devices = isinstance(device, Sequence)
"""Bool: Records if multiple devices are input"""
if all(c == 0 for c in coeffs) or not coeffs:
self.cost_fn = lambda *args, **kwargs: np.array(0)
return
self._optimize = optimize
self.qnodes = qml.map(
ansatz, observables, device, interface=interface, diff_method=diff_method, **kwargs
)
if self._optimize:
if self._multiple_devices:
raise ValueError("Using multiple devices is not supported when optimize=True")
obs_groupings, coeffs_groupings = qml.grouping.group_observables(observables, coeffs)
d = device[0] if self._multiple_devices else device
w = d.wires.tolist()
@qml.qnode(device, interface=interface, diff_method=diff_method, **kwargs)
def circuit(*qnode_args, obs, **qnode_kwargs):
"""Converting ansatz into a full circuit including measurements"""
ansatz(*qnode_args, wires=w, **qnode_kwargs)
return [qml.expval(o) for o in obs]
def cost_fn(*qnode_args, **qnode_kwargs):
"""Combine results from grouped QNode executions with grouped coefficients"""
total = 0
for o, c in zip(obs_groupings, coeffs_groupings):
res = circuit(*qnode_args, obs=o, **qnode_kwargs)
total += sum([r * c_ for r, c_ in zip(res, c)])
return total
self.cost_fn = cost_fn
else:
self.cost_fn = qml.dot(coeffs, self.qnodes)
[docs] def __call__(self, *args, **kwargs):
return self.cost_fn(*args, **kwargs)
[docs]class VQECost(ExpvalCost):
"""Create a cost function that gives the expectation value of an input Hamiltonian.
.. warning::
Use of :class:`~.VQECost` is deprecated and should be replaced with
:class:`~.ExpvalCost`.
"""
def __init__(self, *args, **kwargs):
warnings.warn(
"Use of VQECost is deprecated and should be replaced with ExpvalCost",
UserWarning,
2,
)
super().__init__(*args, **kwargs)
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