Add finite difference function to compute the gradient and second-order derivatives of parametrized observables #1090

agran2018 · 2021-02-17T19:24:35Z

Context:
Adds a finite difference function to evaluate the gradient and second-order derivative of callable functions. This is particular useful to evaluate the derivative of molecular Hamiltonians with respect to the nuclear coordinates in order to perform molecular geometry optimizations.

Description of the Change:
The functions _fd_first_order_centered, _fd_second_order_centered and finite_diff() has been added to the _grad.py module.

Benefits:
Allows users to run VQAs to perform the joint optimization of the circuit and Hamiltonian parameters.

Possible Drawbacks:
The finite difference approximation is susceptible to numerical errors.

Related GitHub Issues:
None

… array with atomic positions

…mbols of the molecule and a numpy array with atomic coordinates

…/pennylane into modify-molecular-hamiltonian

Co-authored-by: Josh Izaac <josh146@gmail.com>

Co-authored-by: ixfoduap <40441298+ixfoduap@users.noreply.github.com>

…Hamiltonian

github-actions · 2021-02-17T19:25:01Z

Hello. You may have forgotten to update the changelog!
Please edit .github/CHANGELOG.md with:

A one-to-two sentence description of the change. You may include a small working example for new features.
A link back to this PR.
Your name (or GitHub username) in the contributors section.

…ians

zeyueN

Looking great! Just left a couple of minor comments.

zeyueN · 2021-03-13T02:32:22Z

pennylane/_grad.py

+            if (_np.array(_idx) > bounds).any():
+                raise ValueError(
+                    "Indices in 'idx' can not be greater than {}; got {}".format(
+                        tuple(bounds), _idx
+                    )


Any reason for letting this function raise this instead of letting numpy raise the error during calculation?

Yep, I'm in favour of removing this and just letting NumPy raise the error

In fact, you could probably remove the entire else: branch.

tests/test_finite_diff.py

ixfoduap

It's funny that implementing a finite difference function has required so many iterations and discussions. I wonder if there is a lesson to be learnt for us?

Code looks good. I suggest clarifying the docstrings further, particularly by adding example usage to the user-facing finite_diff. Some more testing of the various types of input functions and gradients would also be a good idea in my opinion

.github/CHANGELOG.md

ixfoduap · 2021-03-15T18:10:17Z

pennylane/_grad.py

@@ -181,3 +182,206 @@ def _jacobian_function(*args, **kwargs):
        return _np.stack([_jacobian(func, arg)(*args, **kwargs) for arg in argnum]).T

    return _jacobian_function
+
+
+def _fd_first_order_centered(f, argnum, delta, *args, idx=None, **kwargs):


@agran2018 Please avoid using excessive underscores _ when choosing function names. Docstrings can help explain details of the function. Their names should be concise and intuitive.

In this case I suggest _first_order and _second_order, or possibly _fd_first_order and _fd_second_order

Please, see this discussion in the ADR about indicating the finite-difference type in the function names. We can discuss it here and go with the one you like the most.

@ixfoduap since this is a private function (and not user facing), it is generally okay to be a little bit more explicit/verbose.

But if it were a public, user-facing function, I would agree!

ixfoduap · 2021-03-15T18:19:05Z

pennylane/_grad.py

+        argnum (int): which argument to take a gradient with respect to
+        delta (float): step size used to evaluate the finite difference
+        idx (list[int]): if argument ``args[argnum]`` is an array, ``idx`` can


I'm very confused with having both argnum and idx. Is the idea that we can have a function f(x,y,z) and we may want to take a derivative with respect to y[3]? Then we'd call _first_order(f, 1, idx=3)?

If so, I think this needs to be better explained in the docstrings

Yes. You got it correctly.

pennylane/_grad.py

ixfoduap · 2021-03-15T18:47:01Z

pennylane/_grad.py

+        )
+
+    if idx is None:
+        idx = [(), ()]


I'm confused. What exactly happens when the user doesn't pass idx? As far as I can tell, from line 288 when we make the assignment i, j = idx, the rest of the code treats i,j as integers, e.g., shift_i[i]. So what does this function do when i,j = [(), ()]?

Yep, this confused me as well. I think this might be sneakily using a feature of NumPy that array[tuple()] simply returns the whole array.

Yep, this seems to be the case.

I didn't know this @agran2018, you taught me something 😆

Definitely worth commenting this within the code though, so that it is clear what is happening.

I think there is some confusion here. This function gives the second derivative of the Hamiltonian. We don't need the full Hessian of the Hamiltonian as they enter individually in the calculation of the second derivative of the energy. Then,

For multivariable functions idx is a list with the indices i and j involved in the second derivative.

for one variable function idx defaults to None and by having idx= [(), ()] we can compute the second derivative of one variable functions using the code block implementing the diagonal case.

Indeed @josh146, I learned this from the first function you proposed :-).

ixfoduap · 2021-03-15T18:52:57Z

pennylane/_grad.py

+            derivative :math:`\frac{\partial^2 f(x)}{\partial x_i \partial x_j}`.
+        delta (float): step size used to evaluate the finite differences
+
+    Returns:


I think this could really use some example usage, especially given the subtleties in how to use argnum and idx, and the fact that idx takes different forms depending on the order of the derivative. Would eb nice to have an example computing:

Full gradient

Single derivative

Second derivative

for a function with several multi-dimensional arguments

Yes, not finished yet.

Agree, every user-facing function in PL should have an **Example** section after the arguments!

An example of the full hessian might also be nice.

Examples added.

@josh146, if it is OK with you, I would rather have an example for the Hessian in the docstring of the second-order derivative of the energy since this is something users would like to do to compute normal modes.

ixfoduap · 2021-03-15T18:56:17Z

pennylane/_grad.py

+            + f(*args[:argnum], x - shift_i - shift_j, *args[argnum + 1 :], **kwargs)
+        ) * delta ** -2
+
+    return deriv2


If I understand correctly, for a function f(x, y) we cannot take a derivative with respect to x[i] and y[j], right? Only x[i] and x[j]

Yes, that's right.

Should we be supporting x[i] and y[j]? Probably doable.

Not essential for the kind of properties we want to look at (nuclear gradients, force constants, vibrational frequencies).

ixfoduap · 2021-03-15T19:05:11Z

tests/test_finite_diff.py

+
+
+@pytest.mark.parametrize(
+    ("x", "y", "argnum", "idx", "delta", "exp_grad"),


Do you need to parametrize x and y? 🤔

Probably we can give them explicitly in the list but I think will deteriorate readability of the tests ?

tests/test_finite_diff.py

Co-authored-by: ixfoduap <40441298+ixfoduap@users.noreply.github.com>

josh146

looks great @agran2018! Approved 💯 although conditional on:

adding an example to the docstring
considering whether df/dx[i]dy[j] makes sense
exploring if we should remove dtype="O".

josh146 · 2021-03-16T11:30:59Z

pennylane/_grad.py

@@ -14,6 +14,7 @@
 """
 This module contains the autograd wrappers :class:`grad` and :func:`jacobian`
 """
+from functools import partial
 import numpy as _np


I think this is a throwback to previously when we imported both vanilla numpy and autograd numpy:

import numpy as _np from pennylane import numpy as np

At some point, I imagine the second line got deleted.

However, this is going to be fixed in #1131 :)

josh146 · 2021-03-16T11:32:09Z

pennylane/_grad.py

@@ -181,3 +182,206 @@ def _jacobian_function(*args, **kwargs):
        return _np.stack([_jacobian(func, arg)(*args, **kwargs) for arg in argnum]).T

    return _jacobian_function
+
+
+def _fd_first_order_centered(f, argnum, delta, *args, idx=None, **kwargs):


@ixfoduap since this is a private function (and not user facing), it is generally okay to be a little bit more explicit/verbose.

But if it were a public, user-facing function, I would agree!

josh146 · 2021-03-16T11:33:04Z

pennylane/_grad.py

+            if (_np.array(_idx) > bounds).any():
+                raise ValueError(
+                    "Indices in 'idx' can not be greater than {}; got {}".format(
+                        tuple(bounds), _idx
+                    )


Yep, I'm in favour of removing this and just letting NumPy raise the error

josh146 · 2021-03-16T11:35:52Z

pennylane/_grad.py

+        )
+
+    x = _np.array(args[argnum])
+    gradient = _np.zeros_like(x, dtype="O")


Should we make this a list, gradient = []?

Then, further down, we can do

gradient.append( (f(*args[:argnum], x + shift, *args[argnum + 1 :], **kwargs) - f(*args[:argnum], x - shift, *args[argnum + 1 :], **kwargs)) * delta ** -1 ) return _np.array(gradient).reshape(x.shape)

I think this might be beneficial, in that we get a float array or an object array when needed, rather than it always being an object array. @agran2018 maybe worth changing, and seeing if the tests still pass?

@josh146, I tried this yesterday and it did not work for me:

247 # return gradient --> 248 return _np.array(gradient).reshape(x.shape) 249 250 ValueError: cannot reshape array of size 4 into shape (2,)

Also, we need to keep track of the indices corresponding to the zero and non-zero components of the gradients if the list idx is given. This is important for constrained geometry optimization which is very useful.

Oh, of course - the shape output should be the output shape of f 🤦

josh146 · 2021-03-16T11:37:01Z

pennylane/_grad.py

+        f (function): function with signature ``f(*args, **kwargs)``
+        argnum (int): which argument to take the derivative with respect to
+        delta (float): step size used to evaluate the finite differences
+        idx (list[int]): if argument ``args[argnum]`` is an array, `idx`` specifies


Maybe an example in the docstring will help? E.g., for function f(x, y, z), argnum=1, idx=[3, 2] will differentiate f with respect to element (3, 2) of argument y.

pennylane/_grad.py

josh146 · 2021-03-16T11:44:36Z

pennylane/_grad.py

+            derivative :math:`\frac{\partial^2 f(x)}{\partial x_i \partial x_j}`.
+        delta (float): step size used to evaluate the finite differences
+
+    Returns:


Agree, every user-facing function in PL should have an **Example** section after the arguments!

pennylane/_grad.py

josh146 · 2021-03-16T11:46:21Z

pennylane/_grad.py

+            derivative :math:`\frac{\partial^2 f(x)}{\partial x_i \partial x_j}`.
+        delta (float): step size used to evaluate the finite differences
+
+    Returns:


An example of the full hessian might also be nice.

Co-authored-by: Josh Izaac <josh146@gmail.com>

pennylane/_grad.py

agran2018 and others added 23 commits February 8, 2021 18:21

Modify 'read_structure()' to output a list with atomic symbols and an…

29f6062

… array with atomic positions

Update 'test_meanfield()'

a5dd54f

Polish docstrings

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Modify 'molecular_hamiltonian()' to receive a list with the atomic sy…

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…mbols of the molecule and a numpy array with atomic coordinates

Merge branch 'master' into modify-molecular-hamiltonian

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Update 'qchem/CHANGELOG.md'

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Merge branch 'modify-molecular-hamiltonian' of github.com:PennyLaneAI…

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…/pennylane into modify-molecular-hamiltonian

Apply suggestions from code review

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Co-authored-by: Josh Izaac <josh146@gmail.com>

Modify 'CHANGELOG.md'

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Apply suggestions from code review

34db6da

Co-authored-by: ixfoduap <40441298+ixfoduap@users.noreply.github.com>

Merge branch 'master' into modify-molecular-hamiltonian

cd4e301

Make 'name' a kwarg in meanfield and 'molecular_hamiltonian' functions

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Update 'CHANGELOG.md'

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Apply suggestions from code review

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Co-authored-by: ixfoduap <40441298+ixfoduap@users.noreply.github.com>

Merge branch 'master' into modify-molecular-hamiltonian

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Merge branch 'master' into modify-molecular-hamiltonian

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Add function 'derivative' to compute the derivative of the molecular …

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…Hamiltonian

Merge branch 'modify-molecular-hamiltonian' into gradient-hamiltonian

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Add functions 'derivative' and 'gradient'

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Add unit tests for functions 'derivative' and 'gradient'

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Black 'structure.py'

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Merge branch 'master' into modify-molecular-hamiltonian

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Merge branch 'modify-molecular-hamiltonian' into gradient-hamiltonian

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agran2018 added WIP 🚧 Work-in-progress qchem ⚛️ Related to the QChem package labels Feb 17, 2021

agran2018 requested review from antalszava, soranjh and ixfoduap February 17, 2021 19:24

agran2018 self-assigned this Feb 17, 2021

agran2018 added 6 commits March 12, 2021 13:41

Add tests for the second derivative

3aefd69

Generalize unit tests checking the gradient of molecular hamiltonians

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Generalize unit tests for the second derivative of molecular hamilton…

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…ians

Modify unit test

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Update CHANGELOG.md

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Polish docstrings

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agran2018 changed the title ~~Add finite difference function to compute the first derivatives of parametrized observables~~ Add finite difference function to compute the gradient and second-order derivatives of parametrized observables Mar 12, 2021

zeyueN approved these changes Mar 13, 2021

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josh146 approved these changes Mar 16, 2021

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agran2018 commented Mar 16, 2021

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Add check + corresponding test

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agran2018 added review-ready 👌 PRs which are ready for review by someone from the core team. and removed WIP 🚧 Work-in-progress labels Mar 16, 2021

agran2018 added 4 commits March 16, 2021 18:15

Update Examples

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Remove trailing-whitespace

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Modify Examples

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Fix typos in the docstrings

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agran2018 mentioned this pull request Mar 17, 2021

Add function to compute second-order nuclear derivatives #1096

Closed

Merge branch 'master' into gradient-hamiltonian

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agran2018 merged commit a98f252 into master Mar 17, 2021

agran2018 deleted the gradient-hamiltonian branch March 17, 2021 19:30



		@pytest.mark.parametrize(
		("x", "y", "argnum", "idx", "delta", "exp_grad"),

Add finite difference function to compute the gradient and second-order derivatives of parametrized observables #1090

Add finite difference function to compute the gradient and second-order derivatives of parametrized observables #1090

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