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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: scipost_202305_00043v1  (pdf)
Date submitted: 2023-05-28 11:23
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

Author comments upon resubmission

Errors in user-supplied markup (flagged; corrections coming soon)

Dear Editor,

We thank the referees for her/his positive judgment of our work and his/her suggestions and requests that have helped us to prepare a clearer version of the text. We are now confident that this revised version of our work will match all the requirements to be published in a journal as relevant as SciPost. Together with the changes suggested by the referee, we have also modified the format of our paper, adding a table of contents and the DOI of all the cited works to better adhere to the style of the journal.
In the following, you can find a detailed reply to the referee's remarks


Referee -> 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:

Authors -> We thank the referee for her/his positive judgment on our paper.

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 the 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.

Authors -> We agree with the referee that showing the behavior of the SRE as function of \lambda at fixed size of the chain can be a relevant piece of information that allows to better highlight the differences between frustrated and unfrustrated models. Hence, in the new version, we have added it for all the models considered in the paper, for dimensions of the systems up to L=15. In the new Fig.2 we see a clear difference between frustrated and unfrustrated models. In the last ones, the critical point at $\lambda=1$ is always characterized by a local maximum of the SRE. This peak is associated with the non-local behavior of magic shown in ref[42].
On the other hand, the frustrated model does feature such delocalization of magic also in the gapped phase (due to frustration).

Referee -> 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?

Authors -> We thank the referee for her/his very relevant questions. To be honest, it is very hard to provide a precise answer to it. As we can see from figure 2 that the unfrustrated models shows a peak in correspondence of the QPT that are connected to the divergence of the correlation length that is a standard phenomenology of the second order QPT. Usually such a divergence is absent in the first order QPT, and hence we can expect that the local maximum of the SRE that we see in the new Fig.2 will be absent in a similar plot for a system showing a first order QPT. But this is very speculative and, at this moment, we cannot provide a general behavior. Overall, the point raised by the referee is of extreme interest, and we wish to focus on it in a future analysis.

Referee -> 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.

Authors -> For both the frustrated and the unfrustrated versions of the Cluster-Ising model, the systems with L integer multiple of three are very peculiar. In the second case, this difference was underlined in the seminal paper of Smacchia (ref. [62] of the paper), while in the frustrated counterpart, we have that the ground state of the system shows a three-fold degeneracy, which is completely absent for other odd values of $L$. For these reasons, in the first version of the paper we have decided to focus on the case with $L$ not an integer multiple of 3. Unfortunately, for our mistake, in the submitted version of the paper, the paragraph in which we explain this point was canceled. However, accepting the suggestion of the referee, we have decided to add to the new version also the case with the CIM with L an integer multiple of 3. But to underline its peculiar behavior, we have decided to keep the discussion about the SRE in the CIM separated in the two cases, i.e. L equal or different from an odd integer multiple of 3. All the plots were redrawn to take into account this change.

Referee -> 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"

Authors -> We thank the referee for her/his carefully reading our paper. We have fixed these two points in the abstract.

Referee -> 2. Below Eq. 1 the set for P_j includes \sigma_k^x. Should this be \sigma_j^x?

Authors -> We thank again the referee for her/his carefully reading our paper. We have fixed the notation in the text.

Referee -> 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.

Authors -> We agree with the referee that the two parts in the text were not clear. We have re-written the two parts making them clearer and easier to read.

Best regards
To behalf of the authors
Salvatore Marco Giampaolo

List of changes

Here is the list of changes we made to respond to the referee's remarks

- We add a plot (fig 2 in the new version), and the relative discussion, showing the behavior of the SRE as function of \lambda at fixed size of the chain for all models analyzed;

- We modified fig 3 (fig 2 in the old version) separating the CIM in two subcases: L integer multiple of 3 and L non integer multiple of 3;

- We modified fig 4 (fig 3 in the old version) separating the CIM in two subcases: L integer multiple of 3 and L non integer multiple of 3;

- We added a table of contents at the beginning of the paper

- We added the DOI of all the cited papers

- We made minor text corrections

All the changes in the paper are in blue.

Reports on this Submission

Anonymous Report 2 on 2023-7-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202305_00043v1, delivered 2023-07-26, doi: 10.21468/SciPost.Report.7565


The Authors consider the finite-size scaling of the degree of “non-stabilizerness” of many-body systems, a quantity that in quantum computation determines the simulability with Clifford resources only. The analysis revolves around the case of Ising-like spin chains which exhibit a (linearly) extensive degeneracy at their classical point, due to some sort of (topological) frustration, i.e., one ferromagnetic defect in an otherwise anti-ferromagnetic background. Once an additional field is added (e.g., the common transverse magnetic field, but not only), the degeneracy is lifted and the ground-state becomes a translationally-invariant superposition of the classical ones. These can be interpreted as W-states, and some of the conclusions seem to hint at the fact that this constitutes a novel recipe to realize useful resource states for quantum computation purposes.

Per se, the topic is interesting and timely, and some of the results potentially useful across the fields of quantum computation/simulation and condensed matter physics.
There are however some points to be clarified and some issues with the presentation to be solved, before publication on SciPost.


0) The nomenclature of “stabilizers”, “clifford gates”, and so on, is given for granted and not even briefly recalled, which could be desirable for self-containedness of the paper. The same applies to the paragraphs in the Conclusions where the Authors suddenly discuss about resources for fault-tolerant quantum computation, T-gates and so on, without having put the things in too much context / having recalled the basics.

1) In Fig. 2, the unfrustrated curves clearly display a peak (most probably even diverging) at the quantum critical point, while the unfrustrated curves do not: why exactly is that the case? From the analysis presented by the Authors in Figs. 3 & 4, and the main text, I was not able to reconstruct it, unless I overlooked something.
By the way, why are panels B & C at such lower resolution (# points) than panel A? Any technical reason worth to be mentioned?

2) On page 10, end of Sec. 4, I do not understand the statement “In contrast, the W-state has no efficient representation as a translational invariant MPS [13]”, since it is rather easy to write down a MPS with bond-dimension 2, namely (to use the notation of Eq. (2)):
$A_{0,0} = \left|-\right\rangle, \quad A_{0,1} = \sigma_z \left|-\right\rangle / \sqrt{L}, \quad A_{1,1} = \left|-\right\rangle$
and boundary vectors that force to start with index 0 and terminate with index 1.
Since it seems that the Authors give a lot of meaning to this “non-representability”, could they clarify what they exactly mean? Does my objection compromise their conclusions?

3) While most of the text and the plots are focussed on the scaling of the different entropic quantities (in primis the “non-stabilizerness”) with the size of the system, the last paragraphs of the Conclusions seem instead to highlight that the size does not really matter for the usefulness of a state for quantum computational purposes, stating explicitly that “a single three qubit W-state, and therefore any topological frustrated one-dimensional system, could provide an amount of non-stabilizer resources sufficient for the realization of a T-gate.” Are these two aspects complementary, or is there one that should be regarded more as the element of novelty and core message of the work?

4) One even more general question, maybe an outlook for future studies, is how much of the picture outlined by the Authors in this work generalizes to other kind of spin chains (e.g., with symmetries different/larger than Z_2), and — more intriguingly — to spatial dimension larger than one, where frustrated classical Ising models may exhibit _exponential_ degeneracy of the ground-state manifold, and quantum terms lift it by the mechanism called “order-by-disorder”… Can the Author provide any comment/outlook here?



i) at page 4, around Eq.(2) it should read “W-state” or the verbs should be in their plural form…

ii) same page, footnote 1: N should be L, right?

iii) same footnote: formulated like this, the statement is true only for states of the computational basis and the Pauli matrix is \sigma_z (or anyway the one that defines the computational basis) — I guess the Authors mean instead that, if the state is factorized, one can always locally rotate the basis site-by-site, etc.

iv) page 5: “an invariant under spatial translation Hamiltonian” —> “a translationally-invariant Hamiltonian”

v) why does the quantity $\mathcal{R}(L, \lambda)$, which plays a central role in the analysis of the results, does not carry a name?

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

Anonymous Report 1 on 2023-5-30 (Invited Report)


The authors have made suitable modifications to their manuscript following the first round of review. I recommend publication in its current form.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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