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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_00043v2  (pdf)
Date accepted: 2023-08-18
Date submitted: 2023-08-04 18:08
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.

Published as SciPost Phys. 15, 131 (2023)

Author comments upon resubmission

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

Dear editor

We have carefully read the report of the second referee. First, let us thank her/him for her/his positive evaluation of our work and for her/his suggestions which allowed us to further improve our paper.
In the following you can find a detailed answer to all the points she/he raised


> 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.

We thank the referee for reading our work carefully and for her/his positive opinion.

> 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.

We agree with the referee that in some points we have taken for granted the reader's knowledge of some concepts of quantum information, and that this choice of ours may limit the number of researchers who approach the reading of our paper. Therefore, accepting her/his suggestion, both in the introduction and in the conclusions, we have added some sentences which, even without going into the details of the concepts introduced, can help to better understand their role in our work.

> 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?
> As the referee correctly observes, and as highlighted also in the article (page 7), the non-frustrated models show a maximum at the
> phase transition which becomes more and more evident as the size of the system increases. This phenomenon is consistent with
> the results shown in figure 3 where it is observed that the SRE for the non-frustrated models depends linearly on $L$ with a
> proportionality coefficient that increases as the critical point is approached. In agreement with the conclusions of the article in ref.
> [42] this behavior is related to the local nature of magic in non-frustrated systems. On the contrary, as highlighted in our work, the
> SRE for frustrated systems has two different origins.

As for the lower resolution of panels B and C, the reason is quite simple. Our first evaluations were made for the Ising model only. Then we decided to extend our analysis to the second model as well. Since the evaluation of the points take a long time, and the trend of the SRE of the Ising model seemed clear even with fewer points, we decided to reduce them.
There is no other physical or numerical reason behind this choice.
To prove this, while we were working on the text, we have evaluated other points and increased the resolution in the two panels.

>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}=|−⟩,A_{0,1}=\sigma z|−⟩/\sqrt{L} A_{1,1}=|−⟩
> 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?

As the referee correctly points out, there is a MPS representation of the W-state with boundary tensors, however this a non-translational invariant MPS. Translational invariant MPS representation with constant bond dimension do not exist (e.g. see arXiv:2306.16456 or arXiv:quant-ph/0608197). Note that one can find a translational invariant MPS for the W-state with linearly increasing bond dimension. The W-state (and any other state with logarithmically scaling stabilizer entropy) cannot have a translational invariant MPS representation (with constant bond dimension) as this requires an extensive stabilizer entropy. To make our point clearer, we have modified the text at the end of sec. 4.

> 3) While most of the text and the plots are focused 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?

As we have shown throughout our paper, there are many differences between the behavior of topologically frustrated and non-frustrated systems. The latter show a strictly local magic (except at the critical point) and a linear dependence of the SRE on the system size. Conversely, the magic for the former cannot be considered local and the dependence on the size of the system is not linear in the frustrated phase. This is the keystone of the article around which, as rightly observed by the referee, we have focused our analysis. However, in the conclusions, in order to further underline the difference between the two behaviors, the point indicated by the referee seemed useful to us. In fact, the difference between the two behaviors is even more evident near the critical point, where the SRE for the non-frustrated models vanishes, while the frustrated models show an excess of magic sufficient to simulate a T-gate.

> 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?

We would like to thank the referee for this comment, which makes us understand that he/she read our paper with interest. However, before entering the merit of our reply, we need to clarify a point. The cluster-Ising model analyzed by us in the paper, has a more complex symmetry (Z_2 X Z_2, see Smacchia et al. in ref.[62]) and not the simple Z_2 of the Ising model. Coming back to the original question, there are several possible generalizations that we are currently working on. Among these, the generalization of the problem to 2D systems is one of the most interesting ones. The technical problem that we are facing right now, is that the size of the ground state manifold at the classical point grows a lot as the size of the system increases. But the first results, obtained so far on very small systems, are very promising.


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

We thank the referee for her/his careful reading. We have fixed the sentence

> 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.

Re-reading the part of our paper concerning points ii and iii raised by the referee, we noticed that the text was not very clear. We have rewritten it, removing the footnote.

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

We thank the referee for her/his suggestion that we have gladly accepted

> v) why does the quantity R(L,λ) which plays a central role in the analysis of the results, does not carry a name?

Following the suggestion of the referee, we have named the quanity:" relative frustrated SRE correction"

List of changes

- We have changed fig.2 increasing the number of the points used in Panels B and C

- In the first and last section of the paper, we have added some sentences to better introduce some concepts of quantum information and quantum computation.

- At the end of section 4 we have changed the text to make the relation between MPS and W-state clearer.

- We have added other 6 references.

- Minor text corrections.

- All changes made in the text are in blue

Reports on this Submission

Anonymous Report 1 on 2023-8-8 (Invited Report)


All previous points have been taken into consideration, and the manuscript has been noticeably improved with respect to all of them. It should be published without further revisions.

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

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