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REVIEW 3 major objections 4 minor 48 references

Dense Hamiltonians at the Parseval Limit: The Noncommutative BH Constant is Exponential and the Quantum FEI Conjecture is False

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper constructs explicit degree-d Hamiltonians that saturate Parseval's bound with exp(Θ(d²)) equal coefficients, proving the noncommutative Bohnenblust–Hille constant is exponential and the quantum FEI conjecture is false.

desk verdict A genuinely new construction that likely proves exponential noncommutative BH and refutes quantum FEI, but the draft has two load-bearing typos that make the theorem false as written. read the letter →

arxiv 2608.01424 v1 pith:GIXIUAOP submitted 2026-08-02 math.FA math-phmath.MPquant-ph

Joseph Slote This is my paper · ORCID
classification math.FAmath-phmath.MPquant-ph
keywords noncommutativeBohnenblust–HillePaulidecompositionParsevalidentityquantumBooleanfunctionsFourierEntropy–InfluenceconjectureanticommutingmonomialsGolaycomplementarysequencesHamiltonian
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

This paper constructs, for every degree d, a norm-1 Hermitian operator (a Hamiltonian) whose Pauli expansion contains N(d)=exp(Θ(d²)) nonzero coefficients, all of magnitude exactly 1/√N(d)—the largest magnitude allowed by Parseval's identity. Such 'Parseval-limit' coefficient sets are impossible for bounded diagonal operators (Boolean functions), whose flat spectra have size at most exp(O($d^{{1.5}}$)). The construction immediately gives a lower bound on the noncommutative Bohnenblust–Hille constant that matches existing upper bounds, settling BH_{M_2}^{≤d}=exp(Θ(d)). Because the operators are unitary, they are quantum Boolean functions, and their flat spectra refute the quantum Fourier Entropy–Influence conjecture; the ratio of Fourier entropy to influence reaches Ω(log n), matching the best known upper bound.

What carries the argument

Degree is split d=r+k with r=⌈d/2⌉. The construction takes L (largest power of 2 below 3^r) pairwise anticommuting degree-r Pauli monomials P_ℓ and attaches commuting diagonal operators B_ℓ of degree k defined by B_ℓ=Σ_{a_1,...,a_k} H_{ℓ,a_1}H_{a_1,a_2}⋯H_{a_{k-1},a_k} Z_{a_1}⋯Z_{a_k} for the L×L Sylvester Hadamard matrix H. The identity Σ_ℓ B_ℓ²=$L^{{k+1}}$I makes M_d=Σ_ℓ P_ℓ⊗B_ℓ satisfy M_d²=$L^{{k+1}}$I, so normalization yields a unitary; each B_ℓ contributes L^k unit coefficients, giving the flat spectrum.

What would settle it

For r=2, (3r−1)/2=2.5 is not an integer, so Fact 6 cannot hold as written. A direct search over all Pauli monomials on n=4 qubits for 9 pairwise anticommuting degree-2 monomials would settle whether such a family exists; if it does not (or if the maximal size is smaller than 3^r for any r), the construction collapses for even d.

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Extended reading notes

Core claim

Theorems 1–5 establish that, for every degree d, there is a Hermitian unitary operator A_d with N(d)=$L^{{⌊d/2⌋+1}}$=$3^{{d²/4+Θ(d)}}$ nonzero Pauli coefficients, all of magnitude 1/√N(d), where L is the largest power of 2 below $3^{{⌈d/2⌉}}$. Because the coefficients are flat, the $ℓ^{{2d/(d+1)}}$ norm is $N^{{1/(2d)}}$≥$3^{{d/8}}$, so BH_{M_2}^{≤d}≥$3^{{d/8}}$; with the upper bounds in [VZ23, BSVZ26] this gives BH_{M_2}^{≤d}=exp(Θ(d)). The same flatness gives Fourier entropy H=Θ(d²) and influence I=d, so H/I=Ω(log n) on n=Θ(d·$3^{{d/2}}$) qubits, contradicting Conjecture 4 and matching the bound H=O(log(2n)I).

Load-bearing premise

The construction rests on Fact 6, the existence (claimed from [JKMN20]) of 3^r pairwise anticommuting degree-r Pauli monomials on (3r−1)/2 qubits; if such families do not exist as stated, the dense operators and all corollaries fail.

Editorial extensions

If this is right

  • BH_{M_2}^{≤d}=exp(Θ(d)), settling the asymptotics of the noncommutative Bohnenblust–Hille constant.
  • The noncommutative BH constant is exponentially larger than the classical hypercube BH constant, which is at most C^{√(d log d)}.
  • The quantum Fourier Entropy–Influence conjecture is false: H[²]/I[A] can be Ω(log n) for n-qubit quantum Boolean functions.
  • The upper bound H=O(log(2n)I) from [BGJ+24] is tight up to constants, so the correct quantum FEI relationship is now known.
  • The construction yields explicit degree-d Hamiltonians with exp(Θ(d²)) equal coefficients, providing test functions for quantum learning and operator norm estimation.

Reading between the lines

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

  • The same anticommuting-core plus Golay-dressing recipe may transfer to other operator algebras (fermions, larger local dimensions), possibly giving dense unitary operators with different exponent bases.
  • The flat-spectrum operators are natural extremal candidates for other quantum inequalities that favor uniform coefficient distributions, such as hypercontractive or Poincaré-type estimates on the unitary group.
  • If larger families of pairwise anticommuting degree-r Pauli monomials exist than 3^r, the exponent base 3 in N(d) could be improved, potentially closing the gap between the new lower bound and the known upper bound constant.
  • The sharp constant C = limsup_{d} (BH_{M_2}^{≤d})^{1/d} is left between 3^{1/8} and 3^{1/2}; numerical evaluation of the explicit A_d for small d could indicate whether the true value sits near one endpoint.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper constructs, for each d≥1, a Hermitian operator A_d of operator norm one, homogeneous Pauli degree d, whose Pauli expansion has N(d) flat nonzero coefficients of magnitude 1/√N(d), with N(d) claimed to be exp(Ω(d^2))—the largest scale allowed by Parseval. The construction combines a maximal pairwise-anticommuting family of degree-r Pauli monomials (attributed to a fermion-to-qubit construction) with a Hadamard/Golay-type diagonal operator of degree k. The resulting operators are unitary, hence quantum Boolean functions. The paper derives two consequences: a lower bound BH^{≤d}_{M_2} ≥ 3^{d/8}, which together with prior upper bounds gives BH^{≤d}_{M_2} = exp(Θ(d)); and a refutation of the quantum Fourier Entropy-Influence conjecture, with the ratio H/I shown to grow like log n. It also proves an upper bound N(d)≤exp(O(d^2)) for any bounded degree-d Hamiltonian, yielding exp(Θ(d^2)) for the coefficient-packing problem. As written, however, the statement of the core Fact 6 and the definition of L contain errors that affect the validity of the construction.

Significance. If the construction is repaired, this is a substantial result. It gives an explicit, parameter-free dense Hamiltonian at the Parseval limit, settles the growth of the noncommutative BH constant as exponential up to the base of the exponent, and disproves a published conjecture. The core algebraic mechanism—an anticommuting core extended by commuting Golay/Hadamard suffixes—is elegant, and the cross-term cancellation and flatness are clean. The use of prior upper bounds [VZ23, BSVZ26, BGJ+24] is external and not circular; the lower-bound construction is self-contained modulo Fact 6. The main caveats are definitional but load-bearing: the printed qubit count in Fact 6 and the printed value of L are inconsistent with the exponential claims, and without correction the construction and all corollaries fail.

major comments (3)
  1. [Section 4, Fact 6] As printed, Fact 6 states that F_r contains 3^r pairwise anticommuting degree-r Pauli monomials on (3r-1)/2 qubits. For even r this is not an integer (e.g. r=2 would require 2.5 qubits), so the fact is false as stated and the proof of Theorem 1 is undefined for d with r=⌈d/2⌉ even (e.g. d=4). The intended statement is almost certainly (3^r-1)/2 qubits; this matches 2n+1=3^r. Because the entire construction chooses the P_ℓ from this family, the corrected existence statement must be stated and proved or cited precisely; otherwise Theorem 1 and both applications fail.
  2. [Theorem 1 and Section 4 (definition of L)] The theorem defines L as the largest power of 2 less than 3⌈d/2⌉, and the proof repeats this as 'less than 3r'. Then L=Θ(d), so N(d)=L^{⌊d/2⌋+1}=exp(O(d log d)), contradicting the claimed N(d)=3^{d^2/4+Θ(d)}. The equality displayed in Theorem 1 is therefore false as written. The construction needs L to be the largest power of 2 less than 3^r = 3^{⌈d/2⌉}; with that correction log L=(log_2 3)r+O(1) and N(d)=exp((log_2 3)r(k+1)+O(k))=3^{d^2/4+Θ(d)}. This is load-bearing: Theorem 2's 3^{d/8} and Theorem 5's H=Θ(d^2) both depend on the exponential-in-d^2 value of N(d).
  3. [Theorem 1, qubit-count bullet] The bullet stating the number of qubits as (3⌈d/2⌉−1)/2 + L⌊d/2⌋ = Θ(d3^{d/2}) is inconsistent with every version above. With the printed L, the count is Θ(d^2), not Θ(d3^{d/2}); with the corrected L and corrected Fact 6 the denominator should be (3^r−1)/2, not (3r−1)/2. The equality to Θ(d3^{d/2}) becomes correct only with both corrections. This needs to be fixed because the lower bound H[Â_d^2]/I[Â_d] ≳ log n in Theorem 5 uses n=Θ(d3^{d/2}).
minor comments (4)
  1. [Section 4, k=0 case] For d=1, k=⌊d/2⌋=0 and the formula for B_ℓ via G has no arguments. The construction should define B_ℓ=1 (or otherwise handle k=0 separately). This is minor because the asymptotic claims concern large d.
  2. [Section 4, indistinctness vs anticommutation] In the computation of M_d^2, the phrase 'because each P_ℓ is distinct' explains the coefficient count, but the cancellation of cross terms comes from anticommutation. Please clarify that distinction.
  3. [Theorem 2 proof] The calculation ∥Â_d∥_p = N^{1/(2d)} is correct for flat coefficients but should be shown in one line: (N·N^{-p/2})^{1/p}=N^{1/p-1/2}=N^{1/(2d)}.
  4. [Notation] The notation BH^{≤d}_{M_2} in the abstract and BH≤d M2 in the body should be standardized; also define O_c(d) when first used in the asymptotic sharpness paragraph.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lower-bound Hamiltonian is explicitly constructed from independent ingredients; prior-work citations are external benchmarks, not fitted inputs.

full rationale

The derivation is self-contained. Theorem 1's Hamiltonian is built explicitly: Fact 6 (from the independent [JKMN20] ternary-tree construction) supplies a pairwise-anticommuting family F_r; the paper chooses an L-sized subset and extends with diagonal Golay/Hadamard operators B_ell. The normalization step uses only the identity sum_ell B_ell^2 = L^{k+1} I, so A_d is unitary and flat by direct calculation. No parameter is fitted to the target quantities; N(d), the coefficient magnitudes, the qubit count, and the BH lower bound are read off from the construction. The upper bounds used for comparison ([VZ23, BSVZ26] for BH and [BGJ+24] for the FEI relation) are external theorems, not consequences of this paper's construction. [BSVZ26] shares an author with the present paper, but it is an independent stated upper bound and is not used to derive the central lower bound, so it does not create circularity. A separate non-circular caveat: Fact 6 as printed says the family lives on (3r-1)/2 qubits, which is not an integer for even r; the intended version is presumably (3^r-1)/2, matching the cited construction. This is a correctness/typo concern, not a circularity concern, and does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central construction uses no fitted constants or free parameters; it is explicit. It does rely on (a) the existence of pairwise anticommuting Pauli families from [JKMN20], (b) standard Hadamard matrix identities, and (c) external upper bounds from [BSVZ26] and [BGJ+24] used for tightness and the FEI refutation. The main structural risk is (a), which is cited but not proved in this paper.

assumptions (3)
  • domain assumption Fact 6: for every r, there exists a family F_r of 3^r pairwise anticommuting degree-r Pauli monomials on (in the intended form) approximately 3^r qubits
    Cited from [JKMN20] and used in the core construction of Theorem 1; as stated with (3r-1)/2 qubits the fact is not well-defined for even r.
  • standard math Properties of Sylvester Hadamard matrices: H H^T = L I and entries +/-1
    Used to prove M_d^2 = L^{k+1} I and flatness of coefficients in Section 4.
  • domain assumption Upper bound BH_{M2}^{<=d} <= exp(O(d)) (from [BSVZ26]) and H <= log(2n) I (from [BGJ+24])
    Used to conclude tightness of the lower bound and the FEI refutation; these are external theorems from prior work.

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Pith. "Pith review of Dense Hamiltonians at the Parseval Limit: The Noncommutative BH Constant is Exponential and the Quantum FEI Conjecture is False." pith.science (2026). https://pith.science/paper/GIXIUAOP

@misc{pith2026260801424,
  author       = {Pith},
  title        = {Pith review of: Dense Hamiltonians at the Parseval Limit: The Noncommutative BH Constant is Exponential and the Quantum FEI Conjecture is False},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GIXIUAOP}},
  note         = {Machine review of arXiv:2608.01424}
}
abstract

For each $d\geq 1$ we construct a norm-1 Hermitian operator whose Pauli expansion contains $N(d)=\exp(\Omega(d^2))$ terms, each of degree $d$ and magnitude $1/\sqrt{N(d)}$ - the largest magnitude permitted by Parseval's identity. For comparison, if a bounded diagonal operator (or equivalently, a bounded degree-$d$ function on the Boolean cube) has $N(d)$ Pauli coefficients, all of magnitude $\Omega(1/\sqrt{N(d)})$, then $N(d)\leq \exp(\widetilde{O}(d^{1.5}))$. This construction implies the noncommutative Bohnenblust--Hille (BH) constant satisfies $\mathrm{BH}_{M_2}(d)\geq\exp(\Omega(d))$. Together with the upper bounds proved in prior work, this settles the asymptotic growth of $\mathrm{BH}_{M_2}(d)$ as exponential. Our lower bound also asymptotically separates $\mathrm{BH}_{M_2}(d)$ from the (classical) hypercube BH constant $\mathrm{BH}_{\{\pm 1\}}(d)$, which in turn is known to be subexponential: $\mathrm{BH}_{\{\pm 1\}}(d)\leq C^{\sqrt{d \log d}}$. Our Hamiltonians are also unitary and thus quantum Boolean functions in the sense of Montanaro and Osborne (2010). As such they refute the quantum Fourier Entropy-Influence conjecture of Bu et al. (2024), a natural generalization of the classical Fourier Entropy-Influence conjecture due to Friedgut and Kalai (1996).

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Works this paper leans on

48 extracted references · 32 canonical work pages

  1. [1]

    Physical Review Letters , volume =

    Quantum Fingerprinting , author =. Physical Review Letters , volume =. 2001 , doi =. quant-ph/0102001 , archivePrefix =

    arXiv 2001

  2. [2]

    Dimension-free discretizations of the uniform norm by small product sets , volume =

    Becker, Lars and Klein, Ohad and Slote, Joseph and Volberg, Alexander and Zhang, Haonan , year =. Dimension-free discretizations of the uniform norm by small product sets , volume =. Inventiones mathematicae , publisher =. doi:10.1007/s00222-024-01306-9 , number =

    work page doi:10.1007/s00222-024-01306-9

  3. [3]

    2026 , eprint=

    Approximating the operator norm of local Hamiltonians via few quantum states , author=. 2026 , eprint=

    work page 2026

  4. [4]

    Noncommutative Bohnenblust–Hille inequalities , volume =

    Volberg, Alexander and Zhang, Haonan , year =. Noncommutative Bohnenblust–Hille inequalities , volume =. Mathematische Annalen , publisher =. doi:10.1007/s00208-023-02680-0 , number =

    work page doi:10.1007/s00208-023-02680-0

  5. [5]

    Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning , volume =

    Jiang, Zhang and Kalev, Amir and Mruczkiewicz, Wojciech and Neven, Hartmut , year =. Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning , volume =. doi:10.22331/q-2020-06-04-276 , journal =

    doi:10.22331/q-2020-06-04-276 2020

  6. [6]

    Friedgut, Ehud and Kalai, Gil , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1996 , NUMBER =. doi:10.1090/S0002-9939-96-03732-X , URL =

    doi:10.1090/s0002-9939-96-03732-x 1996

  7. [7]

    and Jaffe, Arthur and Koh, Dax Enshan and Li, Lu , year =

    Bu, Kaifeng and Garcia, Roy J. and Jaffe, Arthur and Koh, Dax Enshan and Li, Lu , year =. Complexity of Quantum Circuits via Sensitivity, Magic, and Coherence , volume =. Communications in Mathematical Physics , publisher =. doi:10.1007/s00220-024-05030-6 , number =

    doi:10.1007/s00220-024-05030-6

  8. [8]

    QMA/qpoly Is Contained In PSPACE/poly: De-Merlinizing Quantum Protocols

    Scott Aaronson , title =. Proceedings of the 21st Annual IEEE Conference on Computational Complexity (CCC 2006) , pages =. 2006 , doi =. quant-ph/0510230 , archivePrefix =

    work page Pith review arXiv 2006

Show all 48 references
  1. [9]

    Theory of Computing , publisher =

    The need for structure in quantum speed-up , author =. Theory of Computing , publisher =. doi:10.4086/toc.2014.v010a006 , url =

    2014 doi

  2. [10]

    Heisenberg-

    Asadian, Ali and Erker, Paul and Huber, Marcus and Kl\"ockl, Claude , year = 2016, month =. Heisenberg-. Phys. Rev. A , publisher =. doi:10.1103/PhysRevA.94.010301 , url =

    2016 doi

  3. [11]

    Fr\'. The. Advances in Mathematics , volume = 264, pages =. doi:https://doi.org/10.1016/j.aim.2014.07.029 , issn =

    2014 doi

  4. [12]

    Journal of Physics A: Mathematical and Theoretical , publisher =

    Bloch vectors for qudits , author =. Journal of Physics A: Mathematical and Theoretical , publisher =. doi:10.1088/1751-8113/41/23/235303 , url =

    doi

  5. [13]

    doi:10.1017/CBO9780511543012 , place =

    Analysis in Integer and Fractional Dimensions , author =. doi:10.1017/CBO9780511543012 , place =

    doi

  6. [14]

    Annals of Mathematics , publisher =

    On the Absolute Convergence of Dirichlet Series , author =. Annals of Mathematics , publisher =

  7. [15]

    Israel Journal of Mathematics , publisher =

    On the distribution of the fourier spectrum of Boolean functions , author =. Israel Journal of Mathematics , publisher =. doi:10.1007/bf02785861 , url =

    doi

  8. [16]

    Proceedings of the American Mathematical Society , publisher =

    The Bohnenblust--Hille inequality combined with an inequality of Helson , author =. Proceedings of the American Mathematical Society , publisher =

  9. [17]

    PRX Quantum , volume =

    Learning to Predict Arbitrary Quantum Processes , author =. PRX Quantum , volume =. 2023 , month =. doi:10.1103/PRXQuantum.4.040337 , url =

    2023 doi

  10. [18]

    Andreas Defant and Leonhard Frerick and Joaquim Ortega-Cerd. The. Annals of Mathematics , publisher =. doi:10.4007/annals.2011.174.1.13 , url =

    2011 doi

  11. [20]

    Andreas Defant and Mieczys. On the. Mathematische Annalen , publisher =. doi:10.1007/s00208-018-1756-y , url =

    doi

  12. [21]

    Irit Dinur and Ehud Friedgut and Guy Kindler and Ryan O'Donnell , year = 2007, month = aug, journal =. On the. doi:10.1007/s11856-007-0068-9 , url =

    2007 doi

  13. [22]

    doi:10.1017/9781108691611 , place =

    Dirichlet Series and Holomorphic Functions in High Dimensions , author =. doi:10.1017/9781108691611 , place =

    doi

  14. [23]

    Ben Efraim and F

    L. Ben Efraim and F. Lust-Piquard , year = 2007, month = aug, journal =. Poincar. doi:10.1007/s00440-007-0094-x , url =

    2007 doi

  15. [24]

    Rouz\'e, Cambyse and Wirth, Melchior and Zhang, Haonan , year = 2022, month = 9, eprint =

    2022

  16. [25]

    Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing , location =

    Learning Low-Degree Functions from a Logarithmic Number of Random Queries , author =. Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing , location =. doi:10.1145/3519935.3519981 , isbn = 9781450392648, url =

  17. [26]

    Defant, Andreas and Sevilla-Peris, Pablo , TITLE =. Funct. Approx. Comment. Math. , FJOURNAL =. 2014 , NUMBER =. doi:10.7169/facm/2014.50.1.2 , URL =

    2014 doi

  18. [27]

    Littlewood, J. E. , title =. The Quarterly Journal of Mathematics , volume =. 1930 , month =. doi:10.1093/qmath/os-1.1.164 , url =

    1930 doi

  19. [28]

    Osborne , title =

    Ashley Montanaro and Tobias J. Osborne , title =. Chicago Journal of Theoretical Computer Science , year =. doi:10.4086/cjtcs.2010.001 , url =

    2010 doi

  20. [29]

    Noncommutative Bohnenblust-Hille inequality in the Heisenberg-Weyl and Gell-Mann bases with applications to fast learning , publisher =

    Slote, Joseph and Volberg, Alexander and Zhang, Haonan , keywords =. Noncommutative Bohnenblust-Hille inequality in the Heisenberg-Weyl and Gell-Mann bases with applications to fast learning , publisher =. 2023 , copyright =. doi:10.48550/ARXIV.2301.01438 , url =

    2023 doi arXiv

  21. [30]

    and Audenaert, K

    Gross, D. and Audenaert, K. and Eisert, J. , title =. Journal of Mathematical Physics , volume =. 2007 , doi =

    2007

  22. [31]

    Analysis of Boolean Functions , DOI=

    O'Donnell, Ryan , year=. Analysis of Boolean Functions , DOI=

  23. [32]

    1993 , issue_date =

    Linial, Nathan and Mansour, Yishay and Nisan, Noam , title =. 1993 , issue_date =. doi:10.1145/174130.174138 , journal =

    1993

  24. [33]

    Electron

    Siddharth Iyer and Anup Rao and Victor Reis and Thomas Rothvoss and Amir Yehudayoff , title =. Electron. Colloquium Comput. Complex. , volume =. 2021 , url =. TR21-102 , timestamp =

    2021

  25. [34]

    1992 , month = feb, publisher =

    Jean Bourgain and Jeff Kahn and Gil Kalai and Yitzhak Katznelson and Nathan Linial , title =. 1992 , month = feb, publisher =. doi:10.1007/bf02808010 , url =

    1992 doi

  26. [35]

    and Kais, Sabre , TITLE=

    Wang, Yuchen and Hu, Zixuan and Sanders, Barry C. and Kais, Sabre , TITLE=. Frontiers in Physics , VOLUME=. 2020 , URL=. doi:10.3389/fphy.2020.589504 , ISSN=

    2020

  27. [36]

    and Carrasco, Jose and Kraus, Barbara and Zoller, Peter , journal =

    Gonz\'alez-Cuadra, Daniel and Zache, Torsten V. and Carrasco, Jose and Kraus, Barbara and Zoller, Peter , journal =. Hardware Efficient Quantum Simulation of Non-Abelian Gauge Theories with Qudits on. 2022 , month =. doi:10.1103/PhysRevLett.129.160501 , url =

    2022 doi

  28. [37]

    Sohaib and Job, Joshua Adam and Li, Andy C

    Kurkcuoglu, Doga Murat and Alam, M. Sohaib and Job, Joshua Adam and Li, Andy C. Y. and Macridin, Alexandru and Perdue, Gabriel N. and Providence, Stephen , keywords =. Quantum simulation of ^4 theories in qudit systems , publisher =. 2021 , copyright =. doi:10.48550/ARXIV.2108...

    2021 doi arXiv

  29. [38]

    Journal of Physics A: Mathematical and General , abstract =

    A J Scott , title =. Journal of Physics A: Mathematical and General , abstract =. 2006 , month =. doi:10.1088/0305-4470/39/43/009 , url =

    2006 doi

  30. [39]

    Efficient Simulation of Random Quantum States and Operators , publisher =

    Dankert, Christoph , keywords =. Efficient Simulation of Random Quantum States and Operators , publisher =. 2005 , copyright =. doi:10.48550/ARXIV.QUANT-PH/0512217 , url =

    2005 doi

  31. [40]

    2020 , month = jun, publisher =

    Hsin-Yuan Huang and Richard Kueng and John Preskill , title =. 2020 , month = jun, publisher =. doi:10.1038/s41567-020-0932-7 , url =

    2020 doi

  32. [41]

    Nazarov and M

    F. Nazarov and M. Sodin and A. Volberg , title =. 2003 , month = dec, publisher =. doi:10.1007/bf02773070 , url =

    2003 doi

  33. [42]

    , year =

    Yomdin, Y. , year =. Remez-type inequality for discrete sets , volume =. Israel Journal of Mathematics , publisher =. doi:10.1007/s11856-011-0131-4 , number =

    doi

  34. [43]

    and Montanaro, Ashley , year =

    Harrow, Aram W. and Montanaro, Ashley , year =. Extremal eigenvalues of local. doi:10.22331/q-2017-04-25-6 , journal =

    2017 doi

  35. [44]

    A cb-Bohnenblust–Hille inequality with constant one and its applications in learning theory , volume =

    Arunachalam, Srinivasan and Dutt, Arkopal and Escudero Gutiérrez, Francisco and Palazuelos, Carlos , year =. A cb-Bohnenblust–Hille inequality with constant one and its applications in learning theory , volume =. Mathematische Annalen , publisher =. doi:10.1007/s00208-025-0314...

    doi

  36. [45]

    Towards a Proof of the Fourier-Entropy Conjecture? , volume =

    Kelman, Esty and Kindler, Guy and Lifshitz, Noam and Minzer, Dor and Safra, Muli , year =. Towards a Proof of the Fourier-Entropy Conjecture? , volume =. Geometric and Functional Analysis , publisher =. doi:10.1007/s00039-020-00544-2 , number =

    doi

  37. [46]

    New Construction of Complementary Sequence (or Array) Sets and Complete Complementary Codes , year=

    Wang, Zilong and Ma, Dongxu and Gong, Guang and Xue, Erzhong , journal=. New Construction of Complementary Sequence (or Array) Sets and Complete Complementary Codes , year=

  38. [47]

    Proceedings of the American Mathematical Society , volume=

    Lower bounds for the constants in the Bohnenblust--Hille inequality: the case of real scalars , author=. Proceedings of the American Mathematical Society , volume=

  39. [48]

    Improved Bounds on Fourier Entropy and Min-entropy , year =

    Arunachalam, Srinivasan and Chakraborty, Sourav and Kouck\'. Improved Bounds on Fourier Entropy and Min-entropy , year =

  40. [49]

    Discrete Analysis , year =

    Joseph Slote and Alexander Volberg and Haonan Zhang , title =. Discrete Analysis , year =. doi:10.19086/da.137968 , url =

    doi

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