Add functions to find correct sector for qubit tapering #2041

obliviateandsurrender · 2021-12-17T12:54:11Z

Context:
This PR adds functions to find the correct sector for which the tapered Hamiltonian will correspond to the ground state of the system.

Description of the Change:

_sector_energy function returns the energy of the Hamiltonian in a given sector.
find_optimal_sector function returns the correct sector containing the ground state. It offers two strategies to do so - (i) by performing a brute force through all the possible permutations (exponential), and (ii) by building the sector using a relation between Pauli-Z operator and occupation number under JW transform (linear).

Benefits:
Possible Drawbacks:
Related GitHub Issues:

…o opt_symmetry_sector

codecov · 2021-12-20T09:53:48Z

Codecov Report

Merging #2041 (3128fb8) into master (9902cc3) will increase coverage by 0.00%.
The diff coverage is 100.00%.

@@           Coverage Diff           @@
##           master    #2041   +/-   ##
=======================================
  Coverage   99.18%   99.18%           
=======================================
  Files         226      226           
  Lines       17354    17368   +14     
=======================================
+ Hits        17212    17226   +14     
  Misses        142      142

Impacted Files	Coverage Δ
pennylane/hf/__init__.py	`100.00% <ø> (ø)`
pennylane/hf/tapering.py	`100.00% <100.00%> (ø)`

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pennylane/hf/tapering.py

soranjh · 2021-12-20T23:54:12Z

pennylane/hf/tapering.py

+    return energy
+
+
+def find_optimal_sector(qubit_op, generators, paulix_ops, brute_force=True, num_electrons=0):


I think this function should only return the correct sector and not the energy. This means that we should not perform the brute force method here since the user can always call the transform_hamiltonian function with any desired sector. Also, if the user can already obtain the correct sector, then no need to do a brute force at all. Therefore, the args and return might be:

Args: generators, reference HF state
Return: list of eigenvalues

Instead of the reference HF state, we might have mol, core=None, active=None or something similar.

Agree with Soran 💯 Why use an inefficient brute force method when we know a better algorithm?

Agreed. My original intention to include brute force was for the sake of completeness. But now that transform_hamiltonian can already allow users to do it already, I also believe that we can skip doing it here.

soranjh · 2021-12-20T23:54:36Z

pennylane/hf/tapering.py

+    To obtain the optimal sector one can brute force through all the permutation or by utilize the following
+    relation between the Pauli-Z qubit operator and the occupation number under Jordan-Wigner transform.
+
+    .. math::
+
+        \sigma_z^{i} = a_{i}^{\dagger}a_{i} - 1
+
+    This relation allows us to figure out whether the orbital is occupied or unoccupied in a given symmetry sector,
+    to build the correct eigensector.


The theory can be explained a bit better.

Agree. How exactly does knowing if an orbital is occupied or not help us build the correct eigensector?

tests/hf/test_tapering.py

soranjh

Thanks @obliviateandsurrender. I have two major suggestions:

Focus this PR (and the function) to obtain the sector, no need to compute the energy here which means that we do not need to perform the brute force method.
It will be great if we have a slightly more transparent explanation of how the sector is found..

soranjh · 2021-12-21T00:35:19Z

pennylane/hf/tapering.py

+
+    .. math::
+
+        \sigma_z^{i} = a_{i}^{\dagger}a_{i} - 1


Shouldn't this be \sigma_z^{i} = 1 - 2 a_{i}^{\dagger}a_{i} or maybe I am missing something?

Number operator is [[0, 0], [0, 1]] so indeed the formula Soran is quoting should be the correct one. Also, I think it's better to express the number operator in terms of the Pauli operators. I suggest changing to

n_i=a_{i}^{\dagger}a_{i}=I-2\sigma_z^i

But please check that we are using labels on the Pauli operators consistently. I recall in other PRs we were using \sigma_i^z, though I may be wrong

Updated. Even I'd gotten the same result when I was calculating it by myself. But I wonder why Setia's paper that we are following used this definition (Eq 17).

ixfoduap

The code performing the main calculation looks good, but the rest of the PR needs additional work.

pennylane/hf/tapering.py

ixfoduap · 2021-12-21T14:18:38Z

pennylane/hf/tapering.py

+    """Computes energy of the given tapered Hamiltonian.
+
+    Args:
+        sector (pennylane.Hamiltonian): tapered Hamiltonian whose energy has to be calculated


I'm still very confused by what we mean by "sector". I thought it was a list of eigenvalues? But here it is a Hamiltonian?

I am sorry for the confusion. I somehow used "sector" here to represent the tapered Hamiltonian. I guess while implementing the function I more or less remembered quite a literal definition of the word sector (as in part of the whole). Didn't realize the conflict.

ixfoduap · 2021-12-21T14:20:55Z

pennylane/hf/tapering.py

+    """
+
+    if len(sector.wires) < 2:
+        energy = min(sp.linalg.eigvals(qml.utils.sparse_hamiltonian(sector).toarray()))


Won't this be very expensive for more complicated Hamiltonians? Thinking long term, if there is ever an experiment attempting to simulate a large system, it may be prohibitive to directly diagonalize the Hamiltonian. Isn't that a problem here?

Yes. The energy calculation was being done only to support the brute force method. Now that I have removed it, this is not needed either.

ixfoduap · 2021-12-21T14:22:05Z

pennylane/hf/tapering.py

+    return energy
+
+
+def find_optimal_sector(qubit_op, generators, paulix_ops, brute_force=True, num_electrons=0):


Agree with Soran 💯 Why use an inefficient brute force method when we know a better algorithm?

pennylane/hf/tapering.py

ixfoduap · 2021-12-21T14:47:05Z

pennylane/hf/tapering.py

+                f" got 'orbitals'={num_orbitals} < 'electrons'={num_electrons}."
+            )
+
+        hf_str = np.where(np.arange(num_orbitals) < num_electrons, 1, 0)


Clever implementation!

pennylane/hf/tapering.py

ixfoduap · 2021-12-21T14:57:54Z

tests/hf/test_tapering.py

@@ -349,10 +303,13 @@ def test_generate_paulis(generators, num_qubits, result):
        ),
    ],
 )
-def test_generate_symmetries(hamiltonian, num_qubits, res_generators, res_pauli_ops):
+def test_generate_symmetries(symbols, geometry, num_qubits, res_generators, res_pauli_ops):


Why is this change being made?

To avoid hardcoding the input Hamiltonian which could be huge for any molecule other than H2.

soranjh · 2021-12-23T00:14:19Z

pennylane/hf/tapering.py

+def optimal_sector(qubit_op, generators, active_electrons):
+    r"""Get the optimal sector which contains the ground state.
+
+    To obtain the optimal sector, we need to choose the right eigenvalue for the symmetries :math:`\vec{\tau}`. We can do


It seems to me that the optimal eigenvalue sector is obtained by applying each \tau to the referee HF state. Assuming that \taus are composed of only I and Pauli-Z, then finding the eigenvalue is trivial by just looking at the wires of the Pauli-Z ops and the occupation state of the qubits.

Thanks for the confirmation on this! I have updated the docstring to make it more concise.

soranjh · 2021-12-23T00:17:10Z

pennylane/hf/tapering.py

@@ -464,3 +465,65 @@ def transform_hamiltonian(h, generators, paulix_ops, paulix_sector):
    c = qml.math.stack(c)

    return _simplify(qml.Hamiltonian(c, o))
+
+
+def optimal_sector(qubit_op, generators, active_electrons):


Why not just use h instead of qubit_op?

I am just a bit more inclined towards using descriptive variable names.

tests/hf/test_tapering.py

pennylane/hf/tapering.py

soranjh

Looks good to me, thanks for the changes.

tests/hf/test_tapering.py

…pennylane into opt_symmetry_sector

pennylane/hf/tapering.py

Jaybsoni · 2022-01-04T17:59:17Z

pennylane/hf/tapering.py

+    perm = []
+    for tau in generators:
+        symmstr = np.array([1 if wire in tau.ops[0].wires else 0 for wire in qubit_op.wires])
+        coeff = -1 if numpy.logical_xor.reduce(numpy.logical_and(symmstr, hf_str)) else 1


I like that this is a 1-liner

Jaybsoni · 2022-01-04T18:02:25Z

Looks good to me. Sorry for the delay!

implement find_optimal_sector for qubit tapering

981c1a0

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

obliviateandsurrender changed the title ~~Add functions to find correct sector for qubit tapering~~ [WIP] Add functions to find correct sector for qubit tapering Dec 17, 2021

obliviateandsurrender added 2 commits December 20, 2021 14:27

Merge branch 'master' of https://github.com/PennyLaneAI/pennylane int…

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…o opt_symmetry_sector

fix docstring and remove qchem dependence

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obliviateandsurrender changed the title ~~[WIP] Add functions to find correct sector for qubit tapering~~ Add functions to find correct sector for qubit tapering Dec 20, 2021

obliviateandsurrender removed the WIP 🚧 Work-in-progress label Dec 20, 2021

obliviateandsurrender marked this pull request as ready for review December 20, 2021 10:37

obliviateandsurrender requested review from ixfoduap, Jaybsoni and soranjh December 20, 2021 10:38

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obliviateandsurrender added 4 commits December 22, 2021 20:05

remove _sector_energy and update docstring

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update changelog, doctring and a testcase

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obliviateandsurrender requested review from ixfoduap and soranjh December 23, 2021 03:28

Merge branch 'master' into opt_symmetry_sector

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soranjh mentioned this pull request Dec 23, 2021

Add function to taper Hartree Fock state for qubit tapering #2042

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obliviateandsurrender added 4 commits December 24, 2021 03:10

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Merge branch 'opt_symmetry_sector' of https://github.com/PennyLaneAI/…

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…pennylane into opt_symmetry_sector

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Merge branch 'master' into opt_symmetry_sector

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obliviateandsurrender merged commit 49b94fa into master Jan 6, 2022

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Add functions to find correct sector for qubit tapering #2041

Add functions to find correct sector for qubit tapering #2041

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Jaybsoni commented Jan 4, 2022

		return energy


		def find_optimal_sector(qubit_op, generators, paulix_ops, brute_force=True, num_electrons=0):

Add functions to find correct sector for qubit tapering #2041

Add functions to find correct sector for qubit tapering #2041

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