Update QAOA-Maxcut demo with a few good practices

[dwierichs]

Title:

Summary:

Just updating the MaxCut demo with a few tiny changes that incorporate good practices in Python/PennyLane, which have changed since the creation back in 2019.

Relevant references:

Possible Drawbacks:

Related GitHub Issues:

---

If you are writing a demonstration, please answer these questions to facilitate the marketing process.

• GOALS — Why are we working on this now?

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• AUDIENCE — Who is this for?

  Eg. Chemistry researchers, PL educators, beginners in quantum computing.

• KEYWORDS — What words should be included in the marketing post?

• Which of the following types of documentation is most similar to your file?
  (more details here)

• Tutorial
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[github-actions]

👋 Hey, looks like you've updated some demos!

🐘 Don't forget to update the dateOfLastModification in the associated metadata files so your changes are reflected in Glass Onion (search and recommendations).

Please hide this comment once the field(s) are updated. Thanks!

[github-actions]

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Deployment Info:

• Pull Request ID: 1251
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[mudit2812]

@ikurecic could you confirm that this should be removed? I'm not particularly familiar with how qml metadata is stored. I know about the JSON files, but I'm not aware if any other metadata is used or not.

[mudit2812]

Isn't this just an IsingZZ?

[dwierichs]

Yes, up to a global phase. I was considering to change it but the text explicitly describes this and has a circuit diagram, so I thought it could be fine without change.

But I'm all for reduction if we don't consider that too invasive

[dwierichs]

So should I change it ? :D

[dwierichs]

I think I won't because I'm not the original author, and Angus put effort into making that figure etc. I'll just add a comment in a note environment.

[mudit2812]

Sorry for not responding, I think it's good to leave it as is based on the context you gave. Thanks :)

[mudit2812]

The graphs with the distribution for the bit-strings looks great! However, the actual numerical values in the demo output are dramatically different:

Published version:

p=1
Objective after step     5:  2.0000000
Objective after step    10:  2.0000000
Objective after step    15:  4.0000000
Objective after step    20:  3.0000000
Objective after step    25:  1.0000000
Objective after step    30:  2.0000000
Optimized (gamma, beta) vectors:
[[-0.82383311]
 [-1.13584295]]
Most frequently sampled bit string is: 0101

p=2
Objective after step     5:  4.0000000
Objective after step    10:  3.0000000
Objective after step    15:  4.0000000
Objective after step    20:  3.0000000
Objective after step    25:  4.0000000
Objective after step    30:  3.0000000
Optimized (gamma, beta) vectors:
[[ 0.87696449  1.31508883]
 [ 0.90138291 -0.34955715]]
Most frequently sampled bit string is: 0101

After this PR:

p=1
Objective after step     5:  0.5000000
Objective after step    10:  0.5000000
Objective after step    15:  0.5000000
Objective after step    20:  2.5000000
Objective after step    25:  0.5000000
Objective after step    30:  2.5000000
Optimized (gamma, beta) vectors:
[[-0.90759251]
 [-1.2352846 ]]
Most frequently sampled bit string is: 1010

p=2
Objective after step     5:  0.5000000
Objective after step    10:  2.5000000
Objective after step    15:  2.5000000
Objective after step    20:  2.5000000
Objective after step    25:  2.5000000
Objective after step    30:  2.5000000
Optimized (gamma, beta) vectors:
[[ 1.54174446 -0.7153231 ]
 [ 0.35679571 -0.72652338]]
Most frequently sampled bit string is: 0101

I think this is happening because the device only has one shot. Before, that would give 4 samples due to 4 executions, but now there's only one sample. It's surprising to see that the distribution still looks as good as it does 😆. I'm guessing changing this line to the following will fix it:

dev = qml.device("lightning.qubit", wires=n_wires, shots=len(graph))

[dwierichs]

Thanks for the analysis @mudit2812 , that's a good catch!

• Overall, this demo is stochastic and seeds don't seem to be fixed. So with single-shot expectation values, even the original version varies wildly (The optimized parameters, for example, look almost unrelated between runs 😅 )
• The variance for the previous approach is $V_1=\sum_{(a,b)\in\mathcal{E}}1-\langle Z_aZ_b\rangle^2$ where $\mathcal{E}$ is the set of edges. For the new approach it is

$$V_2=|\mathcal{E}|+\sum_{(a,b)\in\mathcal{E}'} \langle Z_aZ_b\rangle-\left(\sum_{(a,b)\in\mathcal{E}}\langle Z_aZ_b\rangle\right)^2,$$ $$=V_1+\sum_{(a,b)\in\mathcal{E}'} \langle Z_aZ_b\rangle-\sum_{(a,b)\neq(c,d)\in\mathcal{E}}\langle Z_aZ_b\rangle\langle Z_cZ_d\rangle,$$

where $\mathcal{E}'$ are pairs of next-nearest neighbours. So you are right that the variances differ a lot, and this actually makes me concerned about the "simplification" I'm proposing here. It is more of a switch in measurement strategy, and one that is highly non-trivial to see through (in particular, while allocating len(graph) shots makes sense coming from the current version, I'm not sure whether it can be motivated from the measurement problem itself? 🤔 )

Mhm.

[mudit2812]

Thanks @dwierichs . It sounds like a non-trivial change. Maybe it makes sense to go over the demo body as well just to make sure that the text is not inconsistent with the new implementation. Otherwise, happy to approve.

[dwierichs]

@mudit2812 The variance discussion stands, but I actually noticed a small bug I encoded with the change:
I basically changed the objective to $C=\frac{1}{2}(1-\sum_{(j,k)} \langle Z_j Z_k\rangle)$ but the correct thing of course is
$C=\frac{1}{2}\sum_{(j,k)} (1-\langle Z_j Z_k\rangle) = \frac{1}{2}( |\mathcal{E}| - \sum_{(j,k)} \langle Z_j Z_k\rangle)$. I fixed that and everything works as expected as as before now :) In particular, perfect bitstrings are being sampled in the end, as anticipated by the main text.
I also went through the main text and adapted where required for this update.

[mudit2812]

LGTM!