Graph Theoretic Approach to Quantum Nonstabilizerness

Albert Gasull; Flavio Baccari; Jordi Tura; Mengyao Hu; Ruiyun Zhang; Yingjian Liu

Graph Theoretic Approach to Quantum Nonstabilizerness

T0 review · 0 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper proves that, when a set of measured Pauli operators has no active dependencies and a perfect frustration graph, the reduced robustness of magic equals the largest sum of absolute Pauli expectation values over a pairwise anticommu

desk verdict Solid paper: exact reduction of reduced-RoM to a maximum-weight clique problem on perfect frustration graphs; main theorems hold up, numerics need work. read the letter →

arxiv 2607.26154 v1 pith:DLZ5CBFW submitted 2026-07-28 quant-ph

Yingjian Liu , Albert Gasull , Mengyao Hu , Ruiyun Zhang , Flavio Baccari , Jordi Tura This is my paper · ORCID

classification quant-ph

keywords nonstabilizernessmagicreducedrobustnessoffrustrationgraphperfectgraphsmaximum-weightcliquePaulimeasurementsCliffordcovariance

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

To certify that a quantum state has 'magic'—the resource behind universal quantum computation—one normally needs full tomography and an optimization over exponentially many stabilizer states. Using a limited set of Pauli measurements promises efficiency, but the resulting reduced stabilizer polytope is hard to characterize. This paper shows the difficulty decomposes into two graph-theoretic obstructions: the frustration graph, which records which measured Paulis anticommute and hence which can be measured together, and the sign dependencies imposed by Pauli product relations. The authors prove that when no 'active dependency' forces a parity constraint inside a commuting set, those sign dependencies can be discarded exactly; if the frustration graph is also perfect, the reduced robustness of magic collapses to a closed-form sum over pairwise anticommuting cliques, bounded by the square root of the largest anticommuting set. This yields a polynomial-time magic witness, a capacity-versus-measurability tradeoff, and a graph-based design principle for scalable certification.

What carries the argument

The load-bearing object is the frustration graph G_M: nodes are the measured Pauli operators, and edges join anticommuting pairs. Its independent sets are exactly the commuting contexts that can be measured in a single setting, and its cliques are pairwise anticommuting sets. The paper's technical move is to relax the reduced stabilizer polytope by retaining all sign patterns on every maximal independent set instead of only the parity-consistent ones; this sign-relaxed polytope is exact precisely when M has no active dependency (a minimal commuting subset whose Paulis multiply to ±1). In the dual of the relaxed monotone, feasibility becomes a maximum-weight independent set computation on G_M

What would settle it

Take any Pauli measurement set M satisfying the theorem's hypotheses—no active dependencies and a perfect frustration graph—and any n-qubit state ρ; compute RoM_M(ρ) by solving the exact linear program over the reduced stabilizer polytope (enumerating its vertices via Eq. (3)) and compare it to max(1, max_Q Σ_{P∈Q}|tr(Pρ)|). If they differ for any such instance, or if the value exceeds √cl(G_M), the theorem is false. A smaller-scale check: for the single-qubit set M={X,Y,Z} (the complete graph on three nodes, perfect, no active dependencies), RoM_M(ρ) must equal max(1, |tr(Xρ)|+|tr(Yρ)|+|tr(Zρ

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Extended reading notes

Core claim

The central discovery is Theorem 5: for an n-qubit state ρ and a Pauli measurement set M with no active dependencies and a perfect frustration graph G_M, the reduced robustness of magic equals RoM_M(ρ) = max(1, max_Q Σ_{P∈Q} |tr(Pρ)|), where Q ranges over cliques of G_M, i.e., over pairwise anticommuting subsets of measured Pauli operators. This value is at most √cl(G_M), the square root of the maximum size of such a set. As a corollary, the reduced robustness—and hence a certificate of nonstabilizerness (a value exceeding 1)—is computable in polynomial time, since it reduces to a maximum-weight clique problem on a perfect graph. The bound is tight: it is attained by any state supported on t

Load-bearing premise

The central claim rests on the exact vertex description of the reduced stabilizer polytope: every extreme point must be indexed by a maximal commuting set of measured Paulis together with an admissible sign pattern; if that description missed any extreme points, the sign relaxation would not be exact and the closed-form clique formula would fail.

Editorial extensions

If this is right

Evaluating RoM_M(ρ) for any state becomes a maximum-weight clique problem with weights |tr(Pρ)|, solvable in polynomial time when M is dependency-free and G_M is perfect—no optimization over exponentially many stabilizer states.
The witness capacity of a measurement set is exactly √cl(G_M), so a larger capacity requires a larger pairwise-anticommuting set; since at most 2n+1 mutually anticommuting Paulis exist on n qubits, the universal ceiling is √(2n+1), attained by the Jordan–Wigner Majorana set.
The condition χ(G_M)=cl(G_M) for perfect graphs means all needed expectation values can be collected in at most cl(G_M) commuting measurement settings, so the closed form also reduces experimental overhead.
Clifford rotations of a solvable measurement set preserve both the closed form and the capacity while covering different Pauli directions, so rotating the measurement ensemble enlarges the set of states detected without raising the bound; numerics confirm higher detection rates for Haar-random and variational states.
The same graph-theoretic reference (independent-set supports of stabilizer states) underlies quadratic magic witnesses and stabilizer Rényi entropy, giving a unified perspective on magic diagnostics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the closed form needs only |tr(Pρ)| for P in a clique, an experimentalist could certify magic from a handful of Pauli expectation values without any tomography: measure each anticommuting clique (requiring at most cl(G_M) commuting settings by perfectness) and check whether any clique sum exceeds 1.
The dependency-free condition may be achievable by adding auxiliary Pauli operators to a measurement set, since adding operators can break active dependencies; if so, the exact formula could cover far more practical measurement sets than the four families constructed here.
The paper leaves the mixed-state quadratic profile as an algebraic identity, not a magic measure; a natural next step would be to find a stabilizer upper bound on the dephased purity, which would yield a graph-theoretic mixed-state magic criterion.
The weighted Lovász bound is proposed as a sound polynomial-time lower bound for imperfect graphs; one could test numerically which imperfect frustration graphs make this bound tight, potentially extending the closed form beyond perfect graphs.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: the central claim is sound and it is a real step forward. Theorems 3 and 5 give a clean, polynomial-time computable closed form for reduced robustness of magic whenever the measurement set has no active dependencies and the frustration graph is perfect. The proof chain is clear: sign relaxation is exact iff no active dependencies; the dual separation becomes MWIS; perfect graphs collapse the antiblocker to clique incidence; then the closed form follows from Cauchy-Schwarz and anticommutation. I checked the key steps and I do not see a load-bearing gap. The Clifford covariance observation is simple but useful for coverage.

What is genuinely new: the sign-relaxed polytope and the exactness criterion, the MWIS dual, and the perfect-graph closed form with the sqrt(clique number) capacity bound. That is more than a repackaging of the reduced-polytope framework from Varela et al. and the MWIS energy-witness idea.

Soft spots, in order of importance. First, the numerical section is not reproducible as reported. No code, no data, no error bars, no details on how the Clifford rotations were sampled or how the MWIS was solved at n=8. For a paper whose practical pitch is scalable certification, that is a real omission, but it does not affect the theorems. Second, the 'no active dependencies' condition is characterized but not efficiently verifiable as far as the paper shows; the authors leave that for future work, which is honest, but it means the class of measurement sets for which the closed form applies is not yet certified in practice. Third, minor gaps: the cross-polytope claim for K_{2n+1} and the chordal claim for the tree set are asserted without proof, and polynomial-time MWIS on perfect graphs is cited rather than implemented. The latter is standard, so I do not weight it much. Fourth, the paper leans on Eq. (3), the V-representation of the reduced stabilizer polytope from Ref. [20], without proving it. I believe it is correct — the argument sketched in the stress test works — but it is an external dependency and worth making explicit. Also, Appendix C honestly states that a graph-theoretic mixed-state resource criterion is still missing; the main results are about linear reduced RoM, not the full magic resource theory for mixed states.

Who this is for: anyone working on magic certification with limited measurement sets, or on the geometry of stabilizer polytopes. It deserves a serious referee. I would send it to review with the request that the numerical section be brought up to a reproducible standard and the dependency verification question addressed, but the theory part is ready.

Referee Report

0 major / 5 minor

Summary. The paper studies the reduced stabilizer polytope STAB(M) obtained by projecting n-qubit stabilizer states onto a small set M of Pauli expectation values. It introduces a sign-relaxed polytope ^STAB(M), proves (Theorem 3) that the relaxation is exact iff M has no active dependencies, shows (Proposition 4) that the dual separation oracle for the relaxed robustness is a maximum-weight independent set problem on the frustration graph G_M, and derives (Theorem 5) a closed form RoM_M(ρ)=max(1, max_{Q clique} ∑_{P∈Q}|tr(Pρ)|) ≤ √cl(G_M) when G_M is perfect. It further proves the universal ceiling √(2n+1), attained by the Jordan-Wigner set, establishes Clifford covariance, and reports numerical detection rates for four measurement sets.

Significance. The central result, if correct, identifies a nontrivial family of measurement-limited magic witnesses that are evaluable in polynomial time with no fitting parameters: the closed form in Theorem 5 is a parameter-free consequence of LP duality and perfect-graph antiblocker duality. The no-active-dependencies condition is a crisp, checkable graph-theoretic criterion, and the bound √cl(G_M) gives a transparent trade-off between witness capacity and simultaneous measurability. The Appendix B proofs are careful and complete. I find no load-bearing flaw in the central derivation; the stress-test concern about Eq. (3) is a completeness issue rather than a correctness issue. The paper's main value is theoretical; the numerical figures are illustrative and would benefit from reproducibility details.

minor comments (5)

[Appendix A, Eq. (A1) [main text Eq. (3)]] The exact V-representation STAB(M)=conv{v_{S,f}: S∈I_max(G_M), f∈B_S} is imported from the unreviewed preprint Ref. [20], and Theorems 3 and 5 rest on it. I believe the representation is correct (commuting support plus sign-averaging over a maximal extension gives the convex hull), but the manuscript should include a self-contained derivation or explicitly state this as a lemma with proof, so the central claim is not conditional on an external source.
[Results, JW set paragraph and Proposition 9] The assertions that STAB(K_{2n+1}) is a cross-polytope and that the JW set has a unique nontrivial dependency ∏γ_i∝1 are stated without proof. The latter is load-bearing for the no-active-dependencies check and deserves a one-line proof; the former is illustrative and can be labelled as such.
[Results, numerical paragraph and Fig. 2] The detection rates are reported without error bars, confidence intervals, or information about independent seeds/randomness. No code or data repository is mentioned. Please provide these reproducibility details, or state clearly that the figures are schematic.
[Theorem 5, paragraph after Eq. (11)] The polynomial-time claim for evaluating Eq. (11) invokes the ellipsoid method for MWIS on perfect graphs [30,31]. This is a theoretical polynomial-time result, not a practical algorithm. For the specific graphs used (complete graphs, incidence graphs of paths, chordal trees), much simpler combinatorial algorithms exist; the text should make this distinction explicit.
[General presentation] Minor typographical and clarity issues: 'addtion' in the JW-set paragraph; the phrase 'Euclidean ball' in the same paragraph is not derived and should carry a reference; the notation ^STAB(M)\STAB(M) in Fig. 1 should be defined (it is clear from context but a formal definition would help).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central Theorem 5 derivation is self-contained given standard LP-duality and perfect-graph results, with no fitted inputs or self-citation load-bearing steps.

full rationale

The paper's main claim, Theorem 5, is a mathematical derivation rather than a fitted prediction. The reduced stabilizer polytope V-representation in Eq. (3) is imported from Ref. [20], but that is an external prior result, not a self-citation, and it does not already contain the closed form RoM_M(rho)=max(1, max_Q sum_{P in Q} |tr(P rho)|). The derivation proceeds by an explicit sign relaxation (Eq. 5), an exactness criterion (Definition 1, Lemma 2, Theorem 3), an LP-dual reformulation (Proposition 4, Eqs. 10/B1-B5), and the standard perfect-graph antiblocker identity (Lemmas 7-8). Each step is proved in the appendix, and the final closed form follows by convex-duality algebra, not by assuming the conclusion. The no-active-dependency condition is characterized independently in Lemma 2, and the clique-number ceiling is proved from an anticommutation rank argument and a variance bound (Eqs. B17-B20), not imported from the claim. Clifford covariance (Proposition 6) is a coordinate identity under conjugation, not a relabeling that manufactures predictions. There are no fitted parameters, no renamed empirical pattern presented as a derivation, and no load-bearing self-citation: the cited graph-theoretic facts (Lovász sandwich, perfect graph collapse, GLS polynomial-time MWIS) are standard external results. The paper even flags its own remaining open problems and missing steps (e.g., imperfect graphs without active dependencies, and the mixed-state quadratic bound in Appendix C), which further supports that the central solvable-regime result is not circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central claim rests on the imported V-representation of the reduced stabilizer polytope from Ref. [20], on standard LP duality, and on the perfect-graph antiblocker identity. There are no fitted free parameters and no invented entities. The 'no active dependencies' condition is a defined graph-theoretic property, not a fitted constant.

assumptions (5)

domain assumption The reduced stabilizer polytope has the exact V-representation STAB(M) = conv{v_{S,f}: S ∈ I_max(G_M), f ∈ B_S} (Eq. 3).
Imported from Ref. [20]; all vertex-based arguments in Theorems 3 and 5 rest on this characterization.
standard math Strong LP duality holds for the primal/dual pairs defining reduced robustness (Eqs. (2)/(4) and (6)/(B2)).
Used in Proposition 4 and Theorem 5; the primal is feasible because the relaxed vertices affinely span R^m and the objective is bounded below.
standard math Perfect graphs satisfy α_w(G)=ϑ(G,w)=χ̄_w(G) and the antiblocker identity {z≥0: α_z(G)≤1}=conv{1_Q: Q clique} (Lemma 8).
A consequence of the strong perfect graph theorem and GLS; used to collapse the dual MWIS constraint to clique sums in Theorem 5.
domain assumption Pairwise commuting Pauli operators are simultaneously measurable, and the frustration graph's independent sets exactly capture such jointly measurable contexts.
Standard Pauli algebra; used throughout for the vertex indexing and the coloring argument for joint measurements.
standard math A pairwise anticommuting set of n-qubit Pauli operators has cardinality at most 2n+1.
Rank bound on the symplectic Gram matrix (Proposition 9); used to derive the universal ceiling sqrt(2n+1).

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Cite this review

Pith. "Pith review of Graph Theoretic Approach to Quantum Nonstabilizerness." pith.science (2026). https://pith.science/paper/DLZ5CBFW

@misc{pith2026260726154,
  author       = {Pith},
  title        = {Pith review of: Graph Theoretic Approach to Quantum Nonstabilizerness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DLZ5CBFW}},
  note         = {Machine review of arXiv:2607.26154}
}

read the original abstract

Detecting nonstabilizerness requires full tomography and an optimization over exponentially many stabilizer states. A limited Pauli measurement set promises resource-efficient magic certification, yet the resulting reduced stabilizer polytope is generally difficult to characterize. We trace this difficulty into two coupled obstructions: the simultaneous measurability of measurements captured by their frustration graph structure, and the consistency of sign dependencies from stabilizer formalism. We show that the sign dependencies can be discarded exactly whenever active dependencies are absent, and that perfect frustration graphs then make this reduced polytope efficiently solvable. This solvable regime derives a closed form bounded by the clique number of the frustration graph, revealing a tradeoff between witness capacity and simultaneous measurability. Clifford covariance allows rotated measurement sets to enlarge the detectable state space without raising the capacity. Graph structure therefore emerges as both a certificate of tractability and a design principle for scalable magic resource detection.

Figures

Figures reproduced from arXiv: 2607.26154 by the authors.

**Figure 2.** FIG. 2. 8-qubit magic-detection rates for the Local, Ising, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Representative frustration graph structures for four solvable Pauli measurement sets. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Works this paper leans on

51 extracted references · 2 linked inside Pith

[1]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010)

2010
[2]

Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D

D. Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D. thesis, California Institute of Technology (1997)

1997
[3]

P. W. Shor, Fault-tolerant quantum computation, inPro- ceedings of the 37th Annual Symposium on Foundations of Computer Science(IEEE, 1996) pp. 56–65

1996
[4]

Gottesman and I

D. Gottesman and I. L. Chuang, Demonstrating the vi- ability of universal quantum computation using telepor- tation and single-qubit operations, Nature402, 390–393 (1999)

1999
[5]

Bravyi and A

S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal Clifford gates and noisy ancillas, Physical Review A71, 022316 (2005)

2005
[6]

Wills, M.-H

A. Wills, M.-H. Hsieh, and H. Yamasaki, Constant- overhead magic state distillation, Nature Physics21, 1842–1846 (2025)

2025
[7]

X. Wang, M. M. Wilde, and Y. Su, Quantifying the magic of quantum channels, New Journal of Physics21, 103002 (2019)

2019
[8]

X. Wang, M. M. Wilde, and Y. Su, Efficiently computable bounds for magic state distillation, Physical Review Let- ters124, 090505 (2020)

2020

Show all 51 references

[9]

Veitch, C

V. Veitch, C. Ferrie, D. Gross, and J. Emerson, Negative quasi-probability as a resource for quantum computation, New Journal of Physics14, 113011 (2012)

2012
[10]

Howard and E

M. Howard and E. Campbell, Application of a resource theory for magic states to fault-tolerant quantum com- puting, Physical Review Letters118, 090501 (2017)

2017
[11]

Leone, S

L. Leone, S. F. E. Oliviero, and A. Hamma, Stabi- lizer R´ enyi entropy, Physical Review Letters128, 050402 (2022)

2022
[12]

Leone and L

L. Leone and L. Bittel, Stabilizer entropies are monotones for magic-state resource theory, Physical Review A110, L040403 (2024)

2024
[13]

Haug and L

T. Haug and L. Piroli, Stabilizer entropies and nonstabi- lizerness monotones, Quantum7, 1092 (2023)

2023
[14]

Veitch, S

V. Veitch, S. A. H. Mousavian, D. Gottesman, and J. Emerson, The resource theory of stabilizer quantum computation, New Journal of Physics16, 013009 (2014). 6

2014
[15]

H. J. Garc ´ ıa, I. L. Markov, and A. W. Cross, On the geometry of stabilizer states, Quantum Information and Computation14, 683–720 (2014)

2014
[16]

Heinrich and D

M. Heinrich and D. Gross, Robustness of magic and symmetries of the stabiliser polytope, Quantum3, 132 (2019)

2019
[18]

A. B. P. Junior, S. Zamora, R. A. Macˆ edo, T. S. Sarubi, J. M. Varela, G. W. C. Rocha, D. A. Moreira, and R. Chaves, A trace distance-based geometric analysis of the stabilizer polytope for few-qubit systems, Physics Letters A576, 131417 (2026)

2026
[19]

Leone, J

L. Leone, J. Eisert, and S. F. E. Oliviero, The unbear- able hardness of deciding about magic, arXiv:2602.22330 (2026)

2026
[21]

Chapman and S

A. Chapman and S. T. Flammia, Characterization of solvable spin models via graph invariants, Quantum4, 278 (2020)

2020
[22]

Cabello, S

A. Cabello, S. Severini, and A. Winter, Graph-theoretic approach to quantum correlations, Physical Review Let- ters112, 040401 (2014)

2014
[23]

Z.-P. Xu, R. Schwonnek, and A. Winter, Bounding the joint numerical range of Pauli strings by graph parame- ters, PRX Quantum5, 020318 (2024)

2024
[24]

R. M. Karp, Reducibility among combinatorial prob- lems, inComplexity of Computer Computations, edited by R. E. Miller, J. W. Thatcher, and J. D. Bohlinger (Springer, 1972) pp. 85–103

1972
[25]

R. A. Macˆ edo, A. de Oliveira Junior, N. E. Comar, L. L. Keller, J. B. Brask, L. C. C´ eleri, and R. Chaves, Every little thing heat does is magic, arXiv:2604.08663 (2026)

2026 arXiv
[26]

Chudnovsky, N

M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Annals of Mathematics164, 51–229 (2006)

2006
[30]

Gr¨ otschel, L

M. Gr¨ otschel, L. Lov´ asz, and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimiza- tion, Combinatorica1, 169–197 (1981)

1981
[31]

Gr¨ otschel, L

M. Gr¨ otschel, L. Lov´ asz, and A. Schrijver,Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics, Vol. 2 (Springer, 1988)

1988
[33]

Sarkar and E

R. Sarkar and E. van den Berg, On sets of maxi- mally commuting and anticommuting Pauli operators, Research in the Mathematical Sciences8, 14 (2021)

2021
[34]

Brauer and H

R. Brauer and H. Weyl, Spinors inndimensions, Amer- ican Journal of Mathematics57, 425–449 (1935)

1935
[35]

S. Ipek, A. T. Yucel, F. Shahi, C. Ozdemir, and C. Okay, Phase-space tableau simulation for quantum computa- tion, Physical Review A113, 032409 (2026)

2026
[37]

Zurel, L

M. Zurel, L. Z. Cohen, and R. Raussendorf, Simulation of quantum computation with magic states via Jordan– Wigner transformations, Physical Review A112, 042602 (2025)

2025
[38]

Nation, R

C. Nation, R. P. A. Simon, S. Banerjee, F. Martini, A. Ri- cottone, F. Cerisola, and L. Dellantonio, Clifford symme- tries in quantum many-body systems, arXiv:2605.18966 (2026)

2026 arXiv
[39]

A. E. Brouwer and W. H. Haemers,Spectra of Graphs, Universitext (Springer, 2012). 7 Appendix A: Reduced polytope structure For a Pauli measurement setM, the reduced stabilizer polytope STAB(M) of Eq. (1) is the projection of the full stabilizer polytope onto the measured coord...

2012
[40]

J. M. Varela, L. L. Keller, A. de Oliveira Junior, D. A. Moreira, R. Chaves, and R. A. Macˆ edo, Predicting magic from very few measurements, arXiv:2602.18939 (2026). 14

2026
[41]

Hamaguchi, K

H. Hamaguchi, K. Hamada, and N. Yoshioka, Handbook for quantifying robustness of magic, Quantum8, 1461 (2024)

2024
[42]

Lov´ asz, On the Shannon capacity of a graph, IEEE Transactions on Information Theory25, 1–7 (1979)

L. Lov´ asz, On the Shannon capacity of a graph, IEEE Transactions on Information Theory25, 1–7 (1979)

1979
[43]

Gr¨ otschel, L

M. Gr¨ otschel, L. Lov´ asz, and A. Schrijver,Geometric Algorithms and Combinatorial Optimization, Algorithms and Com- binatorics, Vol. 2 (Springer, 1988)

1988
[44]

Gr¨ otschel, L

M. Gr¨ otschel, L. Lov´ asz, and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica1, 169–197 (1981)

1981
[45]

Lov´ asz, Normal hypergraphs and the perfect graph conjecture, Discrete Mathematics2, 253–267 (1972)

L. Lov´ asz, Normal hypergraphs and the perfect graph conjecture, Discrete Mathematics2, 253–267 (1972)

1972
[46]

Lov´ asz, A characterization of perfect graphs, Journal of Combinatorial Theory, Series B13, 95–98 (1972)

L. Lov´ asz, A characterization of perfect graphs, Journal of Combinatorial Theory, Series B13, 95–98 (1972)

1972
[47]

Chv´ atal, On certain polytopes associated with graphs, Journal of Combinatorial Theory, Series B18, 138–154 (1975)

V. Chv´ atal, On certain polytopes associated with graphs, Journal of Combinatorial Theory, Series B18, 138–154 (1975)

1975
[48]

D. R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Mathematical Programming1, 168–194 (1971)

1971
[49]

D. R. Fulkerson, Anti-blocking polyhedra, Journal of Combinatorial Theory, Series B12, 50–71 (1972)

1972
[50]

Cabello, S

A. Cabello, S. Severini, and A. Winter, Graph-theoretic approach to quantum correlations, Physical Review Letters112, 040401 (2014)

2014
[51]

Z.-P. Xu, R. Schwonnek, and A. Winter, Bounding the joint numerical range of Pauli strings by graph parameters, PRX Quantum5, 020318 (2024)

2024
[52]

Sarkar and E

R. Sarkar and E. van den Berg, On sets of maximally commuting and anticommuting Pauli operators, Research in the Mathematical Sciences8, 14 (2021)

2021
[53]

Brauer and H

R. Brauer and H. Weyl, Spinors inndimensions, American Journal of Mathematics57, 425–449 (1935)

1935
[54]

S. Ipek, A. T. Yucel, F. Shahi, C. Ozdemir, and C. Okay, Phase-space tableau simulation for quantum computation, Physical Review A113, 032409 (2026)

2026
[55]

Z.-P. Xu, J. Wang, Q. Ye, G. Koßmann, R. Schwonnek, and A. Winter, Simultaneous variances of Pauli strings, weighted independence numbers, and a new kind of perfection of graphs, arXiv:2511.13531 (2025)

2025
[56]

Howard, J

M. Howard, J. Wallman, V. Veitch, and J. Emerson, Contextuality supplies the ‘magic’ for quantum computation, Nature 510, 351–355 (2014)

2014
[57]

A. E. Brouwer and W. H. Haemers,Spectra of Graphs, Universitext (Springer, 2012)

2012
[58]

Leone, S

L. Leone, S. F. E. Oliviero, and A. Hamma, Stabilizer R´ enyi entropy, Physical Review Letters128, 050402 (2022)

2022

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010)

2010

[2] [2]

Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D

D. Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D. thesis, California Institute of Technology (1997)

1997

[3] [3]

P. W. Shor, Fault-tolerant quantum computation, inPro- ceedings of the 37th Annual Symposium on Foundations of Computer Science(IEEE, 1996) pp. 56–65

1996

[4] [4]

Gottesman and I

D. Gottesman and I. L. Chuang, Demonstrating the vi- ability of universal quantum computation using telepor- tation and single-qubit operations, Nature402, 390–393 (1999)

1999

[5] [5]

Bravyi and A

S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal Clifford gates and noisy ancillas, Physical Review A71, 022316 (2005)

2005

[6] [6]

Wills, M.-H

A. Wills, M.-H. Hsieh, and H. Yamasaki, Constant- overhead magic state distillation, Nature Physics21, 1842–1846 (2025)

2025

[7] [7]

X. Wang, M. M. Wilde, and Y. Su, Quantifying the magic of quantum channels, New Journal of Physics21, 103002 (2019)

2019

[8] [8]

X. Wang, M. M. Wilde, and Y. Su, Efficiently computable bounds for magic state distillation, Physical Review Let- ters124, 090505 (2020)

2020