Exponential speedup in quantum simulation of Kogut-Susskind Hamiltonian via orbifold lattice

Emanuele Mendicelli; Georg Bergner; Masanori Hanada

REVIEW 3 major objections 4 minor 39 references

Exponential speedup in quantum simulation of Kogut-Susskind Hamiltonian via orbifold lattice

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper establishes that the Kogut-Susskind Hamiltonian of SU(N) lattice gauge theory can be recovered exactly as the infinite-mass limit of the orbifold lattice, making it programmable on digital quantum computers.

desk verdict The finite-lattice-spacing equivalence between the orbifold lattice and Kogut-Susskind Hamiltonians is a solid, checkable result with honest numerics; the exponential speedup claim is imported from unpublished preprints and lacks a cost analysis. read the letter →

arxiv 2506.00755 v2 pith:YFTCTKOZ submitted 2025-05-31 quant-ph hep-lathep-phhep-thnucl-th

Georg Bergner , Masanori Hanada , Emanuele Mendicelli This is my paper

classification quant-phhep-lathep-phhep-thnucl-th MSC 81P6881T2581T13 PACS 11.15.Ha03.67.Ax

keywords quantumsimulationKogut-SusskindHamiltonianorbifoldlatticegaugetheoryWilsonactionHaarmeasurenoncompactvariablesYang-Mills

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

The paper tries to show that the Kogut-Susskind Hamiltonian of lattice gauge theory—the standard compact-variable formulation that is notoriously hard to program on quantum computers—can be obtained exactly as the infinite-mass limit of the orbifold lattice, a formulation built from noncompact complex link matrices. If true, this means a gauge theory with unitary link variables can be simulated by running the simpler orbifold Hamiltonian at large scalar mass and extrapolating, without waiting for a continuum limit. The paper proves the equivalence at the level of Euclidean path integrals at finite lattice spacing: as the scalar mass and the U(1) determinant mass diverge, the orbifold action and its flat measure reduce exactly to the Wilson action with Haar measure. Numerical Monte Carlo checks in (2+1)-dimensional SU(2) and SU(3) show plaquettes, Polyakov-loop moments, and scalar-localization observables converging to Wilson values as the mass grows. The payoff claimed is that the Kogut-Susskind Hamiltonian becomes programmable on digital quantum hardware with exponential speedup relative to earlier approaches.

What carries the argument

The load-bearing object is the orbifold lattice Hamiltonian/action built from noncompact complex link matrices $Z_{j,\vec n}$, together with the mass terms in $\Delta H$ (or $\Delta S$) that freeze the radial (scalar) modes $W_{j,\vec n}$ and the U(1) phase $\det U_{j,\vec n}$. Because the potential is a polynomial of degree $2N$ in the coordinates, the Hamiltonian is of the generic bosonic form $H = \sum_a p_a^2/2 + V(x)$, for which a universal quantum simulation framework produces circuits with polynomial depth. The proof mechanism is the Euclidean path-integral comparison: the Wilson action and the infinite-mass orbifold action are shown to be the same function of the unitary variables, and the two measures coincide, so the equivalence holds already at finite lattice spacing and does not rely on taking the spatial continuum limit first.

What would settle it

Check the infinite-mass limit of the full path-integral measure by computing the Jacobian of the change of variables $Z \to (W,U)$ at finite mass; if a nontrivial Jacobian factor survives as $m^2 \to \infty$, the orbifold measure is not the Haar measure and the exact claim fails. Alternatively, run the same Monte Carlo comparison on larger volumes and for $N > 3$ (or in 3+1 dimensions) and look for an observable whose infinite-mass extrapolation disagrees with the Wilson action beyond statistical and systematic errors; the Polyakov-loop distribution at a fixed temperature is a sharp test because it probes both ultraviolet and infrared behavior.

Watch

Extended reading notes

Core claim

The central discovery is the exact reduction, at finite lattice spacing, of the orbifold lattice to the Kogut-Susskind/Wilson theory. The orbifold action is written with complex N×N spatial link matrices $Z = \sqrt{a^{d-2}/(2g_d^2)}\, W U$, where $W$ is a positive Hermitian scalar mode and $U$ is unitary; extra mass terms in the action localize $W$ to the identity and $\det U$ to 1. Sending the bare masses $m^2$ and $m^2_{U(1)}$ to infinity forces $W \to 1$, $Z \to \sqrt{a^{d-2}/(2g_d^2)}\ U$, and $\det U \to 1$, so the orbifold action's kinetic and plaquette terms reorganize term-by-term into the Wilson plaquette action, and the flat integration measure on $Z$ becomes the Haar measure on SU(N). The paper states that this gives exactly the same path-integral weight and path-integral measure as the Wilson action, and therefore the same transfer matrix in the continuous-time limit. Numerical data on $8^3$ and $4\times 16^2$ lattices for SU(2) and SU(3) support the claimed convergence of physical observables to Wilson results as $1/m^2 \to 0$.

Load-bearing premise

Everything rests on the assumption that when the mass terms freeze $W$ and $\det U$, the flat measure on the complex matrices $Z$ turns into exactly the Haar measure on SU(N) with no residual Jacobian factor, and that the equivalence of Euclidean actions implies equivalence of the transfer matrices in the continuous-time limit; if either step fails, finite-mass corrections or an order-of-limits effect would break the claimed exact reproduction.

Editorial extensions

If this is right

The Kogut-Susskind Hamiltonian can be simulated by running orbifold-lattice simulations at several finite masses and extrapolating to the infinite-mass limit, giving a concrete implementation protocol.
Because the orbifold Hamiltonian falls into a class of bosonic Hamiltonians with polynomial potentials, the resulting quantum circuits have polynomial depth; the paper argues this yields an exponential speedup over classical preprocessing and over previously known Kogut-Susskind simulation methods.
The argument is general in the gauge group SU(N) and the spatial dimension $d$, and the same mass-term construction extends to QCD with quarks.
Since the exact equivalence holds at finite lattice spacing, the mass limit can be studied without a simultaneous continuum extrapolation, and terms that vanish in the limit can be dropped to shorten the circuits.

Reading between the lines

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

If the measure reduction is nonperturbatively exact, then classical Monte Carlo with the orbifold action at large mass could serve as a practical way to generate truncated Kogut-Susskind Hamiltonians for benchmarking quantum devices, an application the paper does not spell out.
A natural test of robustness is to check whether dropping the $\det U = 1$ constraint term—which the paper suggests may be less important for pure Yang-Mills—still leaves the infinite-mass limit unchanged in (3+1) dimensions and for larger $N$; if it does, circuit cost and qubit count drop further.
The same flat-embedding idea suggests a research program in which other compact group manifolds are embedded into flat spaces with radial scalars; SU(2) $\simeq S^3$ into $\mathbb{R}^4$ would halve the number of bosonic link degrees of freedom relative to the generic orbifold construction.

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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

The core result in Section 3.3 is the real contribution and I think it is likely right. Sending the scalar masses to infinity in the orbifold lattice action gives the Wilson action with Haar measure exactly at finite lattice spacing, not just in the continuum. I checked the key algebra: the W terms collapse to identity, the detU constraint freezes the U(1) part, and the flat measure over complex matrices reduces to the Haar measure over SU(N). The numerics for SU(2) and SU(3) in 2+1 dimensions are honest and supportive—plaquettes, Polyakov loop distributions, and the Tr(W-1)^2 and detU observables all extrapolate smoothly to the Wilson values with a simple quadratic 1/m^2 fit, and the code is on GitHub. That is real evidence and the paper should get credit for it.

The soft spots are real but not fatal. First, the measure reduction is asserted rather than proved. They do not show explicitly that the Jacobian from the polar decomposition Z = WU is trivial in the infinite-mass limit. The numerics suggest it is, but a proof would be much better than a heuristic. Second, the step from Euclidean path-integral equivalence to Hamiltonian equivalence is handled in one sentence: they assume that taking the temporal lattice spacing to zero after matching actions gives the same transfer matrix. That is standard, but not automatic, and an order-of-limits issue could hide there. Third, the headline exponential speedup is not derived in this paper. It is imported from refs [4] and [26], which are self-cited preprints, and there is no cost analysis of the mass extrapolation itself. If the required m^2 must grow with volume or target accuracy to keep systematic errors under control, that cost could eat the claimed advantage. This does not invalidate the equivalence result, but it means the abstract oversells the speedup as an established fact.

I would send this to a serious referee. The equivalence proof is a clean, checkable result and the numerical confirmation makes it useful to both the lattice gauge theory and quantum simulation communities. The referee should be asked to scrutinize the measure reduction and to request either a proof or a more detailed estimate of the extrapolation cost. This is not a desk reject; it is a paper that deserves refereeing, with likely moderate revision.

Referee Report

3 major / 4 minor

Summary. This paper argues that the Kogut-Susskind Hamiltonian for SU(N) lattice gauge theory can be obtained as the infinite bare-scalar-mass limit of the orbifold lattice Hamiltonian of Bergner and Hanada. The authors show at the level of Euclidean lattice actions that, when the mass terms force W_{j,n} to the identity and det U_{j,n} to 1, the orbifold action (12) reduces to the Wilson action (11) and the flat integration measure on the complex link variables reduces to the SU(N) Haar measure. They claim this gives an exact equivalence of the two regularized path integrals, and hence of the corresponding Hamiltonians after the temporal continuum limit. They support the claim with Hybrid Monte Carlo data in 2+1 dimensions for SU(2) and SU(3), comparing spatial and temporal plaquettes, the Polyakov loop and its fluctuations, and the localization of W and det U at finite mass with extrapolation to infinite mass. On this basis, and citing refs. [4,26] for the efficiency of the orbifold framework, the authors claim an exponential speedup for quantum simulation of the Kogut-Susskind Hamiltonian.

Significance. The central equivalence, if correct, is a useful technical bridge: it identifies a controlled limit in which the compact-link Kogut-Susskind Hamiltonian is reproduced by the noncompact orbifold construction, whose Hamiltonians are polynomial and admit explicit quantum circuits. Section 3.3 is cleanly executed at the algebraic level and the numerical data in Section 4 are consistent with the claimed convergence for the observables studied. Credit is due for making the simulation codes publicly available. The main caveat is that the paper's headline exponential speedup is not derived here; it is imported from self-cited preprints and depends on an unquantified extrapolation step. As an equivalence result the paper is solid and potentially useful, but as a speedup claim it is not yet established.

major comments (3)

[§3.3, paragraph beginning 'The measure of the integration...'] The claim that the flat measure on the complex link matrices Z_{j,n} reduces exactly to the SU(N) Haar measure in the m^2, m^2_{U(1)} -> infinity limit is asserted without proof. This is load-bearing, because the exactness of the equivalence between the orbifold and Wilson path integrals requires that no U-dependent Jacobian survives the integration over W_{j,n} and det U_{j,n}. Please provide the polar-decomposition argument (Z = alpha W U with alpha = sqrt(a^{d-2}/(2 g_d^2)), dZ = J(W) dW dU_Haar with J independent of U) and state explicitly how the normalization of the path integral is handled when the mass terms localize W and det U to 1.
[Abstract and §6] The exponential speedup claim is not established in this paper. The proposed strategy is to simulate the orbifold lattice Hamiltonian at several finite masses and extrapolate to infinite mass, but no analysis is given of the cost of this extrapolation. To support the claim, the authors need to bound how m^2 and the number of mass values must scale with the target accuracy delta, the lattice volume V, the truncation cutoff, and the simulated time, and show that this overhead is subdominant to the exponential advantage claimed for the orbifold Hamiltonian in refs. [4,26]. The numerical data in §4, at fixed small lattices and for a few observables, do not address this scaling.
[§3, text before Eq. (12) and §3.3] The derivation of the orbifold lattice action (12) from the orbifold Hamiltonian (4) is not given in this paper; it is only cited to ref. [24]. Since the paper emphasizes that the equivalence holds at the regularized level and the action-level equivalence is then used to infer equivalence of the Hamiltonians in the a_t -> 0 limit, the transfer-matrix step should be made explicit. In particular, it should be shown that the path-integral weight in (12) corresponds to the Hamiltonian (4) up to terms that vanish as a_t -> 0, and that the equality of the two Euclidean actions at finite a_t indeed implies equality of the continuous-time Hamiltonians.

minor comments (4)

[§3.1, Eq. (11)] The sums in Eq. (11) are written with upper limit 3 although the text states a general (d+1)-dimensional theory; this should be \sum_{j=1}^d and \sum_{1\le j<k\le d}.
[§3.3] The phrase 'the first line is also written in terms of plaquette' should read 'in terms of plaquettes'.
[Abstract] The phrase 'modulo a standard assumptions made also for the original Kogut-Susskind approach' contains a grammatical error and is vague; the assumptions should be stated explicitly, for example truncation of the infinite-dimensional bosonic Hilbert space and the validity of the mass extrapolation.
[§4 figures] Several captions and axis labels are informal (e.g., 'a_s = 0.2 (fix)' and 'Infinite-mass extrapolations ... shown at 1/m^2=0'); these should be completed, and the extrapolated values and their uncertainties should be reported numerically.

Circularity Check

1 steps flagged · score 4.0 of 10

Equivalence proof is self-contained, but the headline exponential speedup is imported from self-cited preprints [4,26] rather than derived here.

self citation load bearing [Section 6 (Conclusions) and Abstract]
"A simple and universal framework [26] applicable to a wide class of theories, including the orbifold lattice, leads to an exponential speedup compared to more complex formulations such as the Kogut-Susskind Hamiltonian [4]. Our work demonstrates that the Kogut-Susskind Hamiltonian can be obtained as a special limit of the orbifold lattice, allowing us to leverage this property to achieve exponential speedup in quantum simulations."

The paper's advertised 'exponential speedup' is not derived from the equivalence calculation in Section 3.3; the conclusion explicitly attributes it to refs [26] and [4], both arXiv preprints with author overlap with the present paper (M. Hanada is an author of all three; E. Mendicelli is an author of [4] and of the present paper). No cost analysis of the finite-mass extrapolation and no independent verification of the complexity claim appear here. Thus the central payoff of the paper rests on a self-citation chain: the speedup is true if [4,26] are true, and those results are not established in this manuscript. The equivalence itself is checked against the Wilson action, so the circularity is confined to the speedup claim rather than the derivation.

full rationale

The core mathematical claim—that the infinite bare-scalar-mass limit of the orbifold lattice action (12) reduces term-by-term to the Wilson action (11)—is a self-contained calculation. Substituting Z = sqrt(a^{d-2}/2g_d^2) W U and taking W -> 1 and det U -> 1 turns the kinetic term into plaquettes, the W-localizing term into zero, and the flat measure into the Haar measure. This is verified numerically against the Wilson action, an independent external benchmark, so the equivalence itself is not circular. The weak points, such as the assertion that the flat measure reduces exactly to the Haar measure and the transfer-matrix step from Euclidean action equivalence to Hamiltonian equivalence, are rigor gaps rather than circularity. The only genuine circular step is the exponential-speedup headline: the speedup is cited to [4] and [26], both self-cited preprints, and no cost model for the extrapolation is supplied. Because the equivalence result retains independent content, the score is 4 rather than higher.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on mass terms added ad hoc to the orbifold action, on the unproved measure reduction to Haar, and on the transfer-matrix correspondence. No new particles or physical entities are introduced.

free parameters (3)

m^2 (bare scalar mass) = infinity (limit); simulation values 100, 300, 700, 1000, 4000
Introduced in Delta S to localize W_j,n to the identity. The central claim is the limit m^2 -> infinity, and the numerical protocol extrapolates 1/m^2 to zero.
m^2_U(1) = infinity (limit); set equal to m^2 in numerics
Introduced to force det(U_j,n) toward 1 and make the U(1) part of A_j heavy. This is needed to obtain SU(N) rather than U(N) gauge links.
Extrapolation ansatz (quadratic in 1/m^2) = intercept and coefficients fitted to Monte Carlo data at m^2 = 250..4000
Used to estimate the 1/m^2 -> 0 limit in Figures 1 and 6. The choice of a quadratic fit is a modeling assumption.

assumptions (4)

standard math Polar decomposition Z = W U with W positive Hermitian and U unitary
Used in Eq. (1) and throughout to identify W and U in the complex link variable.
domain assumption Flat measure on complex matrices reduces to Haar measure on SU(N) after localizing W and detU
Assumed in Section 3.3 after Eq. (18); not derived explicitly.
domain assumption Equivalence of Euclidean lattice actions implies equivalence of Hamiltonian formulations in the continuous-time limit
Stated in Section 3: 'This equivalence guarantees the equivalence of the Hamiltonian formulations when the continuum limit is taken along the time direction.'
domain assumption The infinite-mass limit can be approached by Monte Carlo extrapolation with controllable systematic error
The numerical protocol assumes convergence in 1/m^2 and uses a quadratic fit; no rigorous error bound is provided.

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Cite this review

Pith. "Pith review of Exponential speedup in quantum simulation of Kogut-Susskind Hamiltonian via orbifold lattice." pith.science (2026). https://pith.science/paper/YFTCTKOZ

@misc{pith2026250600755,
  author       = {Pith},
  title        = {Pith review of: Exponential speedup in quantum simulation of Kogut-Susskind Hamiltonian via orbifold lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YFTCTKOZ}},
  note         = {Machine review of arXiv:2506.00755}
}

abstract

We demonstrate that the orbifold lattice Hamiltonian -- an approach known for its efficiency in simulating SU($N$) Yang-Mills theory and QCD on digital quantum computers -- can reproduce the Kogut-Susskind Hamiltonian in a controlled limit. While the original Kogut-Susskind approach faces significant implementation challenges on quantum hardware, we show that it emerges naturally as the infinite scalar mass limit of the orbifold lattice formulation, even at finite lattice spacing. Our analysis provides both a general analytical framework applicable to SU($N$) gauge theories in arbitrary dimensions and specific numerical evidence for $(2+1)$-dimensional SU($N$) Yang-Mills theories ($N=2,3$). Using Euclidean path integral methods, we quantify the convergence rate by comparing the standard Wilson action with the orbifold lattice action, matching lattice parameters, and systematically extrapolating results as the bare scalar mass approaches infinity. This reformulation resolves longstanding technical obstacles and offers a straightforward implementation protocol for digital quantum simulation of the Kogut-Susskind Hamiltonian with exponential speedup compared to classical methods and previously known quantum methods, modulo a standard assumptions made also for the original Kogut-Susskind approach.

Figures

Figures reproduced from arXiv: 2506.00755 by the authors.

**Figure 2.** ⟨Tr(UUU†U † )⟩spatial and ⟨Tr(UUU†U † )⟩temporal, SU(3), 4 × 162 lattice, as = 0.2 (fix) Polyakov loop The Polyakov loop is constructed by taking a trace of the product of temporal links at each spatial point (x, y) as Px,y = 1 N Tr Ut,⃗n=(1,x,y)Ut,⃗n=(2,x,y) · · · Ut,⃗n=(nt,x,y) . (19) 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 4.** The simplest investigation of the phase transition in Yang-Mills theory usually considers isotropic lattices without a continuum extrapolation. In order to test this scenario, we have set the same lattice spacing in both directions a = at , which also determines the temperature. This is the same as a scan of the results as a function of the coupling constant. The results are shown in [PITH_FULL_IMAGE:figures/full_f… view at source ↗

Figures from the paper (5 more)

**Figure 3.** Figure 3: [Left] ⟨|P|⟩ vs at . [Right] ⟨|P| 2 ⟩ − (⟨|P|⟩) 2 vs at . SU(3), 4 × 162 lattice, as = 0.2 (fix). On the right panel, the horizontal axis is slightly shifted for the Wilson action and m2 = 300, so that the data points can be distinguished. 12 [PITH_FULL_IMAGE:figures/…

**Figure 4.** Figure 4: The distributions of P from the Monte Carlo simulations. The horizontal and vertical axes are ReP and ImP, respectively. SU(3), 4 × 162 lattice, as = 0.2 (fix). 13 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: A parameter scan with the orbifold action of SU(3) Yang-Mills theory on a 4 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: Tr(W −1N ) 2 (left column) and Re(det U) (right column). [Top] SU(2), 83 lattice, at = a = 0.3. [Bottom] SU(3), 83 lattice, at = a = 0.3. Infinite-mass extrapolations by a quadratic function of 1/m2 from m2 = 250, · · · , 4000 are shown at 1/m2 = 0. 0 0.05 0.1 0.15 0.2…

**Figure 7.** Figure 7: Tr(W − 1N ) 2 (left column) and Re(detU) (right column). SU(3), 4 × 162 lattice, a = 0.2 (fix), various at . 15 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

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2021 arXiv
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Supersymmetry on a spatial lattice,

D. B. Kaplan, E. Katz, and M. Unsal, “Supersymmetry on a spatial lattice,”JHEP 05(2003) 037,arXiv:hep-lat/0206019

2003 arXiv
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Confinement of Quarks,

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Supersymmetry on a Euclidean space-time lattice. 1. A Target theory with four supercharges,

A. G. Cohen, D. B. Kaplan, E. Katz, and M. Unsal, “Supersymmetry on a Euclidean space-time lattice. 1. A Target theory with four supercharges,”JHEP08(2003) 024, arXiv:hep-lat/0302017. 19

2003 arXiv
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Supersymmetry on a Euclidean space-time lattice. 2. Target theories with eight supercharges,

A. G. Cohen, D. B. Kaplan, E. Katz, and M. Unsal, “Supersymmetry on a Euclidean space-time lattice. 2. Target theories with eight supercharges,”JHEP12(2003) 031, arXiv:hep-lat/0307012

2003 arXiv
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A Euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges,

D. B. Kaplan and M. Unsal, “A Euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges,”JHEP09(2005) 042, arXiv:hep-lat/0503039

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M. Hanada, J. Liu, E. Rinaldi, and M. Tezuka, “Estimating truncation effects of quantum bosonic systems using sampling algorithms,”Mach. Learn. Sci. Tech.4 no. 4, (2023) 045021,arXiv:2212.08546 [quant-ph]

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H. Gharibyan, M. Hanada, M. Honda, and J. Liu, “Toward simulating superstring/M-theory on a quantum computer,”JHEP07(2021) 140, arXiv:2011.06573 [hep-th]

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D. B. Kaplan and M. Unsal. Private communication. 20

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Exponential improvement in quantum simulations of bosons,

M. Hanada, S. Matsuura, E. Mendicelli, and E. Rinaldi, “Exponential improvement in quantum simulations of bosons,”arXiv:2505.02553 [quant-ph]

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A universal framework for the quantum simulation of Yang-Mills theory,

J. C. Halimeh, M. Hanada, S. Matsuura, F. Nori, E. Rinaldi, and A. Sch¨ afer, “A universal framework for the quantum simulation of Yang-Mills theory,” arXiv:2411.13161 [quant-ph]

arXiv

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Toward QCD on quantum computer: orbifold lattice approach,

G. Bergner, M. Hanada, E. Rinaldi, and A. Schafer, “Toward QCD on quantum computer: orbifold lattice approach,”JHEP05(2024) 234,arXiv:2401.12045 [hep-th]

arXiv 2024

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Simulating lattice gauge theories on a quantum computer,

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Real-time dynamics of lattice gauge theories with a few-qubit quantum computer,

E. A. Martinezet al., “Real-time dynamics of lattice gauge theories with a few-qubit quantum computer,”Nature534(2016) 516–519,arXiv:1605.04570 [quant-ph]. 9It may be useful to note that Kaplan, Katz, and Unsal studied lattice formulation of super Yang-Mills theory motivated by gauge/gravity duality and invented the orbifold lattice construction [39]. 17

arXiv 2016

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Observation of gauge invariance in a 71-site Bose–Hubbard quantum simulator,

B. Yang, H. Sun, R. Ott, H.-Y. Wang, T. V. Zache, J. C. Halimeh, Z.-S. Yuan, P. Hauke, and J.-W. Pan, “Observation of gauge invariance in a 71-site Bose–Hubbard quantum simulator,”Nature587no. 7834, (2020) 392–396, arXiv:2003.08945 [cond-mat.quant-gas]

arXiv 2020