Implement Qdrift as ProductFormula

[anedumla]

Summary

This pull request implements the QDrift method of Trotterization as a product formula and fixes Issue #7218.

Details and comments

The implementation is virtually the same as in https://github.com/Qiskit/qiskit-terra/blob/main/qiskit/opflow/evolutions/trotterizations/qdrift.py with the only change being that

self.sampled_ops = algorithm_globals.random.choice(
            np.array(scaled_ops, dtype=object),
            size=(int(N * self.reps),), p=weights / lambd,
        )

becomes

self.sampled_ops = algorithm_globals.random.choice(
            np.array(scaled_ops, dtype=object),
            size=(int(np.ceil(N * self.reps)),),
            p=weights / lambd,
        )

to make sure that the sample size is always at least 1 and because N is chosen to satisfy an error bound, so the former int(N * self.reps) would not have been strictly correct in some instances.

[coveralls]

Pull Request Test Coverage Report for Build 1515324749

• 43 of 46 (93.48%) changed or added relevant lines in 6 files are covered.
• No unchanged relevant lines lost coverage.
• Overall coverage increased (+0.007%) to 82.857%

---

Changes Missing Coverage	Covered Lines	Changed/Added Lines	%
qiskit/synthesis/evolution/lie_trotter.py	3	4	75.0%
qiskit/synthesis/evolution/qdrift.py	29	31	93.55%

Totals
Change from base Build 1509385150:	0.007%
Covered Lines:	50147
Relevant Lines:	60522

---

💛 - Coveralls

[jakelishman]

One super minor comment, but otherwise this looks fine to accept to me.

All changes have been marked resolved by Dariusz

[jakelishman]

I originally had this as automerge (trying to get things merged before the Monday/Tuesday rush), but then I realised that @dlasecki had never actually commented on whether the algorithm does exactly what it's supposed to, so I don't know if his original review was meant to be a complete check of everything he thought needed changing. I don't know enough about this particular expansion to approve that properly, so I'll wait for Dariusz to give his tick as well before merging.

[Cryoris]

This test fails for me locally (the second one only) and I don't quite get why the coefficients of the sampled ops should be different than 1 since we now don't change the coeffs anymore

[anedumla]

Not sure why it fails for you, maybe it's got something to do with the random seed? The coefficients should be the same but not one (N here is num_gates for us):

[anedumla]

ah maybe making num_gates to an int makes the test fail

[Cryoris]

@anedumla I added a functional test to see if it works as expected and had to change the num_gates to an int (I think in the paper they use an int too).

[Cryoris]

LGTM thanks for the implementation!