SciPost Submission Page
All you need is spin: SU(2) equivariant variational quantum circuits based on spin networks
by Richard David Petrak East, Guillermo Alonso-Linaje, Chae-Yeun Park
Submission summary
| Authors (as registered SciPost users): | Chae-Yeun Park |
| Submission information | |
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| Preprint Link: | scipost_202401_00014v1 (pdf) |
| Code repository: | https://github.com/XanaduAI/all-you-need-is-spin |
| Date submitted: | 2024-01-15 16:02 |
| Submitted by: | Park, Chae-Yeun |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Variational algorithms require architectures that naturally constrain the optimisation space to run efficiently. In geometric quantum machine learning, one achieves this by encoding group structure into parameterised quantum circuits to include the symmetries of a problem as an inductive bias. However, constructing such circuits is challenging as a concrete guiding principle has yet to emerge. In this paper, we propose the use of spin networks, a form of directed tensor network invariant under a group transformation, to devise SU(2) equivariant quantum circuit ansätze – circuits possessing spin rotation symmetry. By changing to the basis that block diagonalises SU(2) group action, these networks provide a natural building block for constructing parameterised equivariant quantum circuits. We prove that our construction is mathematically equivalent to other known constructions, such as those based on twirling and generalised permutations, but more direct to implement on quantum hardware. The efficacy of our constructed circuits is tested by solving the ground state problem of SU(2) symmetric Heisenberg models on the one-dimensional triangular lattice and on the Kagome lattice. Our results highlight that our equivariant circuits boost the performance of quantum variational algorithms, indicating broader applicability to other real-world problems.
Current status:
Reports on this Submission
Strengths
1) Well-written and very pedagogical
2) Clarity of the results
Weaknesses
1) Lack of comparison with other methods/discussion of complexity scaling
2) Lack of original content (or lack of visibility of what is truly original)
Report
Summary
This article proposes to use SU(2) symmetric (SU(2) equivariant) quantum circuits to describe the groundstates of SU(2) Hamiltonians on a quantum computer, using the variational quantum eigensolver algorithm to optimize the circuit parameters.
The authors first provide a detailed introduction to SU(2) invariant tensor networks/quantum circuits and their representation. They then present an equivalence between permutation gates and the SU(2) equivariant unitary gates. Finally, they benchmark their algorithm on the one-dimensional J1-J2 Heisenberg chain and on the antiferromagnetic Kagome system (through classical simulations).
Report
First, I would like to praise the authors for the quality of their pedagogical introduction. Generally, the paper is very well-written and easy to follow. Nonetheless, I do not think I can recommend the publication of the paper in SciPost Physics in its current form.
Main objection
My main objection for publication is that I do not believe that this paper provides a significant enough breakthrough to deserve publication in SciPost Physics.
Secs 2 and 3 are mostly introductory/paraphrase previous results (note that the authors are honest about this fact). It is fairly straightforward SU(2) representation theory applied to tensor networks, the only caveat being that the tensors are unitary.
Sec. 4 deals with the relation between irreps of the symmetric group and SU(2) irreps. While interesting, it appears to me to only be a marginal generalization of the results of Ref. 20 and 22.
Sec. 5 does present original numerical results, but only as a short benchmark (see also questions below) and without attempting to solve any outstanding issue (understandably given the proposed algorithms are for a quantum computer).
Sec. 6 is an interesting but purely conceptual discussion.
Questions and comments
Methods
1) What is the difference in representability between unitary quantum circuits with finite depth/range and more standard tensor networks such as PEPS? I understand the speed up for a quantum computer.
2) In your algorithm, you use up to 3 vertex gates (due to complexity of physical implementation?). How would you expect your results to change if you were able to directly optimize a gate over the hexagonal cell in your kagome lattice?
If I am not mistaken, two sites gate are not enough to describe all possible circuits. Can you comment on that and your use of 2 tensors for the 1d model?
3) What is the fundamental difference between your circuit and the one used in Ref. 60 for example, beyond your form being more systematic and generic?
Numerical results
1) There are no comparisons with classical methods and no discussions of the scaling of the number of circuit parameters with the system size. A reminder (just a formula) of the complexity of the vQE per sweep would be appreciated. A discussion/numerical evidence of the required number of sweeps (and its scaling with system/parameter) and of the barren plateaus would also be needed in general to properly assess the efficiency of the representation.
2) The one-dimensional Heisenberg system is exactly solvable for $J_2 = 0$ (by Bethe Ansatz) and J2 = 0.5 (Majumdar Gosh point). In particular, $J_2 = 0.44$ is close to the MG point which can be written as a product of independent singlets. The entanglement content of the MG state is therefore very low, and it is representable by an exact matrix product states. Correspondingly, at $J_2 = 0.44$, the groundstate is gapped and has a very short correlation length.
As such, it is surprising that your algorithm struggles more to converge for $J_2 = 0.44$ than for $ J_2 = 0$ (gapless). Could you comment on that?
Additionally, the energy appears to saturate with the number of parameters (number of layers) before jumping closer to the exact value and with a large spread for the deepest circuit. Could you explain why? Is it related to barren plateaus/an absence of solutions?
3) The Kagome antiferromagnet is a notoriously difficult problem, with very small gaps in finite systems (see e.g. https://journals.aps.org/prb/abstract/10.1103/PhysRevB.100.155142 for a not so recent ED work). As such, a discussion on the scaling with system sizes and comparison with existing methods is sorely lacking here. I have also similar concerns as in 2) given the spread of the obtained energy, and the previous observed saturation in the 1d model.
Requested changes
Major changes
- Clarify the original contribution of the paper
- If the numerical results are to be important, proper discussions on scalings and barren plateaus are needed, as well as comparison with other techniques.
Minor changes
- Some notations could be clarified (the Schur gate and the symmetric group are denoted by $S_n$, the generator of the SU(2) equivariant gates is in one section H and in the following $\mathcal{T}$, etc).
- Better bibliography on the physics side of the chosen benchmark problems
- Minor typo p.19. Above 4.3 "exponetiated"