Construct two-site observables to evaluate two-site correlations

[LieBUPT]

I'm working on simulating a spin-1/2 system and need to compute the expectation value of the observable:
$$\ J_x^2 = \left( \sum_{i=1}^N \sigma_i^x \right)^2 $$,
Expanding this observable requires calculating the two-site spin correlations $\langle \sigma_i^x\sigma_j^x\rangle$ for arbitrary $(i,j)$-pairs (not just nearest neighbors). I initially tried using the XX gate from the mqt.yaqs library (imported via from mqt.yaqs.core.libraries.gate_library import XX) to compute these correlations, but encountered the following error:

AssertionError: Only nearest-neighbor observables are currently implemented.

I’m wondering:

1. Are there alternative methods in mqt.yaqs (or complementary approaches) to compute two-site correlations for non-nearest-neighbor pairs?
2. If no such methods exist, is direct tensor contraction the recommended way to calculate these correlations?
3. It would be more convenient if the method MPS.expect supported MPOs as input.

[aaronleesander]

Hey!

1. Right now there is no way to do long-range observables in YAQS, but this is mostly because we have not needed it yet during development.
   If I have time soon, I may try to implement it since it is not too major of a change, otherwise if you would like to contribute, the relevant function is here:

Do you have any reference values we could use as a small easy test?

2. Direct tensor contraction is the correct way to do it, yes, just be aware that it will scale with the distance of those correlations. Depending on your problem, there may be ways to speed it up by only calculating observables where i - j < some distance threshold. In general, you probably won't find any methods or tricks that are better than direct tensor contraction and tailoring it to your problem.

3. Sure this is probably true. We could change observables to being MPOs in all cases. This may just take some work. Thanks for pointing this out.

[LieBUPT]

Hi! Sorry for the late reply. Since doing tensor contractions directly requires optimizing the index ordering, I found it easier to convert the YAQS-format MPS into a TeNPy-format MPS and then use TeNPy’s built-in functions to compute expectation values.

To make this work, I made a few adjustments to the wrapper functions in YAQS. This introduces some additional overhead, but it’s still small compared to the runtime of step_through. Thanks to your tools, I was able to finish a noisy simulation of a long-range spin-1/2 lattice model (size 6×6) within a few hours.

By the way, I noticed that from_pauli_sum becomes very slow when constructing the MPO for systems larger than 4×4. I’m not sure what’s causing the slowdown, so for now I’m constructing the MPO in TeNPy and then converting it back into the required format.

Overall, your tools are really amazing and helped me a lot. Thanks again!

[aaronleesander]

@LieBUPT I updated the code for from_pauli_sum to use a more tenpy-like implementation that should be significantly faster now. This is currently on the developer branch if you are comfortable using that still, otherwise its hould be released soon.

Thanks for pointing it out!