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Complexity of frustration: a new source of non-local non-stabilizerness

by J. Odavić, T. Haug, G. Torre, A. Hamma, F. Franchini, S. M. Giampaolo

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Submission summary

Authors (as registered SciPost users): Fabio Franchini · Salvatore Marco Giampaolo · Jovan Odavić
Submission information
Preprint Link:  (pdf)
Date submitted: 2022-10-11 11:05
Submitted by: Giampaolo, Salvatore Marco
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


We advance the characterization of complexity in quantum many-body systems by examining $W$-states embedded in a spin chain. Such states show an amount of non-stabilizerness or "magic" (measured as the Stabilizer R\'enyi Entropy -SRE-) that grows logarithmic with the number of qubits/spins. We focus on systems whose Hamiltonian admits a classical point with an extensive degeneracy. Near these points, a Clifford circuit can convert the ground state into a $W$-state, while in the rest of the phase to which the classic point belongs, it is dressed with local quantum correlations. Topological frustrated quantum spin-chains host phases with the desired phenomenology, and we show that their ground state's SRE is the sum of that of the $W$-states plus an extensive local contribution. Our work reveals that $W$-states/frustrated ground states display a non-local degree of complexity that can be harvested as a quantum resource and has no counterpart in GHZ states/non-frustrated systems.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2023-2-24 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2209.10541v1, delivered 2023-02-24, doi: 10.21468/SciPost.Report.6789


The authors consider the relation between frustration in finite size many-body quantum systems and the degree of "non-stabilizerness", a concept which emerges from the framework of quantum computation and is related to the simulability of a system with classical devices using only Clifford resources. The authors consider a broad class of Hamiltonians which exhibit an extensive degeneracy at their classical point in parameter space (in this case the authors essentially focus on an Ising model). By considering odd, finite size systems and periodic boundary conditions, the authors discuss that when a symmetry breaking term is added the ground states of these models are effectively W-type states. The implications of this are then considered for two exemplary cases: the quantum Ising model and the cluster-Ising model.

The article is well written and I think a relevant addition to the literature and I feel merits publication in SciPost. While the analysis itself is relatively straightforward and deals with moderate size systems, I believe the impact of the present submission comes from the connections draw between disparate communities. There are a few points I would invite the authors to consider:

1. In Fig 2 the authors show the behavior of the SRE as a function of system size for various fixed values of \lambda. Does the SRE reveal features of the underlying critical point? Looking at the behavior of the plots it seems that the curves are indeed non-monotonic as a function of \lambda. I would be interested in seeing the behavior of \mathcal{M}_2 as a function of lambda for a moderate value of L. Indeed, we see from lower right panel of Fig 2 that there is a sizeable gap between values of SRE observed depending on the phase the model is in.

2. The models the authors considered host a 2nd order QPT. Does the nature of the phase transition play an important role in the results?

3. There is a curious omission of the results for L=9 for the cluster-Ising model. Considering the small system sizes the authors are exploring and the conclusions being drawn based on them, I feel the including of this data point is important.

Some minor points for the authors to consider:
1. In the abstract I believe there are a few grammatical errors: "grows logarithmic with" -> "grows logarithmically with" and "which the classic point belongs" -> "which the classical point belongs"
2. Below Eq. 1 the set for P_j includes \sigma_k^x. Should this be \sigma_j^x?
3. Above Eq. 5 the authors explain the degeneracy of the models. I find the use of parentheses to caveat the sentence jarring and difficult to follow, i.e. "the CIM is equal to 1 (3) if and only if L is odd and is not (is) and interger...." I feel this can simply be written out explicitly and will be clearer to follow. Similarly around equation 7 the use of parentheses could be alleviated and things written out explicitly for clarity.

  • validity: high
  • significance: ok
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: good

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