Entanglement renormalization circuits for $2d$ Gaussian Fermion States

Andrew C. Potter; Sing Lam Wong

REVIEW 3 major objections 5 minor 54 references

Entanglement renormalization circuits for $2d$ Gaussian Fermion States

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper introduces a circuit-compression algorithm that prepares 2D Gaussian fermion ground states with logarithmic-depth circuits and error that falls exponentially with block size.

desk verdict A genuinely new 2d GMERA construction with a clever Wannierization fix and an expanding Z2 encoding, but the critical-point exponential scaling that drives the complexity claim is shakier than the text admits. read the letter →

arxiv 2506.04200 v1 pith:5WVDSRQH submitted 2025-06-04 quant-ph cond-mat.str-el

Sing Lam Wong , Andrew C. Potter This is my paper

classification quant-phcond-mat.str-el

keywords GaussianfermionstatesGMERAcorrelationmatrixcompressionWannierizationHaldanemodelCherninsulatorDiracsemimetalfermion-to-qubitencoding

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 claims that a two-dimensional version of Gaussian multiscale entanglement renormalization (GMERA) can compress the ground states of a broad class of 2D free-fermion systems, including topologically trivial insulators, Chern insulators, and critical Dirac semimetals, into a quantum circuit of logarithmic depth. The key numerical finding, demonstrated on the Haldane model on a honeycomb lattice, is that the approximation error per site decays exponentially in the block radius $r$ and is essentially independent of system size $L$. If correct, this gives explicit upper bounds on the qubit and gate resources required to prepare area-law-entangled fermionic states, including topological and critical states that cannot be captured efficiently by shallow PEPS-type circuits. A second contribution is a fermion-to-qubit encoding based on an expanding $\mathbb{Z}_2$ topological order, which keeps the Pauli weight of encoded fermionic rotations proportional to the block radius rather than to the system size. A sympathetic reader would care because these resource estimates bear on whether near-term quantum hardware can prepare useful starting states for simulating correlated quantum materials.

What carries the argument

The load-bearing mechanism is the Wannierized block-distillation cycle. In each of three staggered hexagonal-block layers, the restricted correlation matrix is diagonalized, the least-entangled 'frozen' modes are distilled out, and the remaining courier modes are re-localized by solving an orthogonal Procrustes problem that aligns them with seed orbitals in the triangular regions where three blocks meet. The optimal unitary is $w_c^{(B)} = u_c^{(B)}(t_c^{(B)})^\dagger$ obtained from the singular value decomposition $\sigma_c^{(B)} = v_c^{(B)\dagger} s_c^{(B)} = t_c^{(B)}\Sigma_c^{(B)} u_c^{(B)\dagger}$, and this step keeps the courier modes inside their hexagon and centered on the next coarse-grained sites. The second mechanism is the expanding $\mathbb{Z}_2$ topological-order fermion-to-qubit encoding, which refines the encoding lattice by constant-depth circuits at each RG step so that fermionic rotations compile to Pauli strings of weight $O(r)$ rather than $O(L)$.

What would settle it

Compute the smallest singular value of $\sigma_c^{(B)} = v_c^{(B)\dagger} s_c^{(B)}$ across all blocks and RG steps for the Haldane model at representative trivial, Chern, and Dirac points; any value at numerical machine precision, or any per-site error that fails to decay exponentially with block radius at fixed system size, would falsify the compression claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the obstruction to extending 1D correlation-matrix compression to 2D—the uncontrolled spread of the 'courier' modes that carry entanglement between blocks—can be removed by a Wannierization step that localizes the courier modes onto seed orbitals in the triangular regions where three staggered block layers meet. Once these modes are localized, the recursive block-distillation procedure on the honeycomb lattice becomes geometrically self-similar, and the fitted error per site behaves as $\epsilon \sim e^{-\alpha r}$, with $\alpha$ finite even at the Dirac-semimetal critical point. From this scaling the paper derives a fermionic circuit depth $D \sim \log L\, \log^4(1/\epsilon)$ and, with the expanding $\mathbb{Z}_2$ topological-order encoding, a qubit circuit depth $D \sim \log L\, \log^5(1/\epsilon)$ with gate count $G(L,\epsilon) \sim O(L^2 \log L\, \log^5(1/\epsilon))$ for an $L\times L$ system. The thresholded version of the algorithm shows a qualitative distinction between phases: the trivial insulator terminates after a constant number of RG steps, while the Chern insulator and Dirac semimetal continue to improve at every available step, indicating that these long-range-entangled phases resist representation by shallow circuits and require the full GMERA recursion.

Load-bearing premise

The load-bearing premise is that the overlap matrix aligning each block's leftover entanglement-carrying modes with its localized target orbitals never develops an exact zero mode; the paper assumes this follows from the distilled modes being topologically trivial, but it is not proved.

Editorial extensions

If this is right

For an $L\times L$ Haldane-model ground state, the GMERA approximation error per site decays as $\epsilon\sim e^{-\alpha r}$, with $\alpha$ independent of system size and finite at the Dirac-semimetal critical point.
Preparing such a state with error $\epsilon$ costs fermionic circuit depth $O(\log L\, \log^4(1/\epsilon))$; with the expanding $\mathbb{Z}_2$ encoding the qubit circuit depth becomes $O(\log L\, \log^5(1/\epsilon))$ with gate count $O(L^2 \log L\, \log^5(1/\epsilon))$.
Trivial insulators can be truncated to a Gaussian PEPS whose RG terminates after a constant number of steps, whereas Chern insulators and Dirac semimetals show non-terminating RG, confirming that they require the full logarithmic-depth circuit.
Using qubit reuse, the number of physical qubits needed drops to $O(L\log L\, \log^2(1/\epsilon))$, and to $O(L\,\log^2(1/\epsilon))$ for truncated trivial-insulator states.
Augmenting GMERA with constant-depth adiabatic dressing yields empirical upper bounds on the qubit and gate resources needed for interacting gapped states adiabatically connected to free-fermion states, including correlated Chern insulators but not fractionalized ones.

Reading between the lines

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

A testable extension the paper leaves implicit is whether the exponential-in-$r$ scaling survives weak quenched disorder or smooth inhomogeneity; if it does, GMERA would provide a local-preparation route for disordered topological insulators, where translation invariance is not available.
The expanding $\mathbb{Z}_2$ encoding is argued in the appendix to be independent of Gaussianity; if that holds, it can serve as the fermion-to-qubit backend for non-Gaussian interacting MERA circuits, making constant-Pauli-weight encoding a general resource rather than a free-fermion convenience.
Because the Dirac semimetal is captured with a finite $\alpha$, the 2D GMERA wavefunction is a natural variational ansatz for 2D critical fermionic states; a concrete check would be comparing its entanglement entropy profile with the expected logarithmic enhancement over area-law scaling.
The resource estimates implicitly treat two-qubit gates as the dominant cost; on a fixed-connectivity architecture, long-range rotations in early RG layers would add SWAP overhead that the logarithmic-depth promise does not count, so the quoted depth is best interpreted for all-to-all-connected hardware.

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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

Two things to know. First, this is a genuinely new numerical methods paper: the Wannierization procedure for localizing courier modes is a real fix for the known failure of naive 2d GMPS, and the expanding Z2 topological-order F2QE is a clean way to keep Pauli weight tied to block radius rather than system size. Second, read the Dirac-semimetal scaling claim with care. The exponential error decay that supports the logarithmic-depth complexity statement rests on four block radii and three system sizes, and that is exactly the regime where a power-law tail can look exponential in the fitted window.

What is good. The paper correctly diagnoses why direct 1d-to-2d extension fails and then does something specific about it. The Wannierization step (Procrustes alignment of courier modes to seeds in triangular overlap regions) is explained well and is, as far as I can tell, new. The thresholded-distillation results are a nice discriminator: trivial insulators terminate the RG, Chern and Dirac states do not. The data collapse across L=24, 48, 96 for gapped phases is consistent, and the near-linear relation between fitted decay rate and inverse correlation length is a compelling sanity check. The self-cited precursors (ZER, holographic simulation) are directly relevant, so the citation pattern is not a problem.

Where it is soft. The stress-test note lands. For the DSM point, Fig. 5 covers only r = 2, 4, 6, 8. Generic truncation of a scale-invariant state at scale r produces power-law errors, and the finite-size gap ~1/L can flatten that into an apparent exponential over such a short window. If the true scaling is a power law, the depth estimate in Sec. VI.A becomes polynomial in 1/epsilon, not log^5, and the advertised advantage over polynomial-depth Gaussian preparation weakens. The paper itself admits the fits are empirical and that slow drifts with L cannot be ruled out; I would want larger r, a convergence check at fixed physical correlation length, and error bars on the slopes.

Second, the Wannierization well-posedness is asserted, not shown: the Procrustes solution requires sigma_c^(B) to have strictly positive singular values, and the paper says this is "expected" from topological triviality. That is plausible but unproven, and a single degenerate case would break the recursion. A numerical sweep or a short analytic argument would close this. Also, no code or data is provided and no error bars appear; for a methods paper with mostly numerical evidence, that is a reasonable revision request, not a fatal flaw.

Overall: the central idea is sound and the gapped-phase evidence is credible. The weak point is overclaiming the critical-point scaling. This paper deserves a serious referee, and I would send it out with a request for code/data and the additional checks above. If those come back clean, it is a solid and useful contribution.

Referee Report

3 major / 5 minor

Summary. The paper presents a two-dimensional Gaussian MERA (2d GMERA) algorithm that compresses the correlation matrix of a Gaussian fermion state into a logarithmic-depth quantum circuit. The algorithm adapts the one-dimensional Fishman-White compression idea to two dimensions by adding a Wannierization step intended to keep courier modes localized at the junctions of successive block layers. The authors test the method on the Haldane model on the honeycomb lattice for trivial insulator, Chern insulator, and Dirac semimetal phases, reporting exponential decay of the per-site approximation error with block radius r and system-size-independent decay rates. From this numerical scaling they infer circuit depths D ~ log L log^5(1/epsilon) and gate counts O(L^2 log L log^5(1/epsilon)), and they introduce an expanding Kitaev-honeycomb fermion-to-qubit encoding that turns the F2QE overhead from O(L) into O(r). The paper also discusses thresholded distillation, which automatically terminates for trivial insulators but not for Chern insulators or Dirac semimetals, and proposes a qubit-efficient sequential implementation using mid-circuit measurement and reset.

Significance. If the central empirical claim holds, the paper would establish a significant resource bound: area-law-entangled Gaussian fermion states, including topological and critical examples, could be prepared with logarithmic depth and near-linear gate overhead, which is an exponential improvement over generic Gaussian-state preparation in two dimensions. The inclusion of Chern insulators and Dirac semimetals is particularly notable, since these states are expected to resist shallow-circuit and PEPS representations, and the thresholded-distillation results provide a clean numerical contrast between short-range and long-range entanglement structures. The proposed expanding topological-order fermion-to-qubit encoding is a concrete and potentially reusable contribution that addresses a practical overhead in implementing local fermionic circuits on qubit architectures. The paper is appropriately cautious that Gaussian states are classically simulable, and it frames the contribution as an empirical upper bound useful for state-preparation subroutines.

major comments (3)

[III.A.b, Eq. (8)] The Wannierization step is well-defined only when the overlap matrix sigma_c^(B) = v_c^(B)^dagger s_c^(B) has strictly positive singular values. The paper states that this 'is expected to be satisfied' because the frozen modes are topologically trivial, but no proof or numerical verification is provided. If any singular value vanishes or becomes very small, the Procrustes solution w_c^(B) = u_c^(B) t_c^(B)^dagger degenerates and the courier modes cannot be localized to the triangular regions, so the entire GMERA recursion is not defined. Since every disentangling layer depends on this step, the authors should either prove the positivity from the spectral and locality properties of the state, or report the smallest singular values obtained in the numerical simulations for all blocks, layers, radii, and phases studied.
[IV.B, Fig. 5(ci,cii)] The central claim that the approximation error decays exponentially in block radius is supported by only four values, r = 2, 4, 6, 8, for each phase. This is especially concerning for the Dirac semimetal point, where the state is scale-invariant and has algebraically decaying correlations; generic truncation at a length scale r would instead produce power-law errors. A log-linear fit over a factor of four in r cannot reliably distinguish e^{-alpha r} from a slow power law such as r^{-4}, and the finite-size gap ~1/L can make algebraic tails look exponential in a narrow window. This point is load-bearing because the logarithmic-depth resource estimate in Section VI.A depends directly on exponential decay in r. The authors should extend the numerical range to larger r where feasible, perform an explicit comparison of exponential versus power-law fits with residuals, and provide system-size control at the critical point, or supply an analytic argument for exponential convergence based on the structure of the GMERA recursion.
[VI.A] The conversion from the numerical error scaling to the resource estimate D ~ log L log^5(1/epsilon) uses the relation r ~ log(1/epsilon) without precisely defining whether epsilon is the per-site RMS error of Eq. (11) or a total error for the L x L system. The gate count G ~ (L/r)^2 log L (B^2 r) also assumes that every RG level has the same block size and that the exponential decay rate alpha remains bounded away from zero as L increases. These assumptions should be stated explicitly, and the finite-size scaling of alpha should be quantified, since the numerical data show collapse for L = 24, 48, 96 but cannot rule out slow drifts in L. Without this clarification, the advertised complexity statement is stronger than what the current evidence supports.

minor comments (5)

[IV.B, Eq. (11)] Equation (11) appears with corrupted mathematical symbols in the typeset text; the definition of the RMS error-per-site should be presented cleanly and should state explicitly whether the square root is included in the definition.
[Fig. 5] The fitted lines in Fig. 5 are reported without error bars on the data points, without details of the fitting procedure (e.g., whether the intercept is free, how many points are used, and what the residuals are), and without uncertainty estimates on the fitted slopes alpha shown in Fig. 5(d). These details are needed to assess the robustness of the exponential-decay claim.
[Appendix A] There is a typo in 'Bugoliubov-de-Gennes'; it should be 'Bogoliubov-de-Gennes'. Similar typographical issues appear in the introduction ('cirtical', 'simulated correlated' should be 'simulating correlated') and in Fig. 1 ('filed' should be 'filled').
[I and III] The relationship to the Zipper entanglement renormalization (ZER) method of Ref. [5] is described as 'loosely' and 'a strictly local circuitized version,' but no precise comparison is given. A short statement of which parts of ZER are kept, which are modified, and why the modification avoids the known obstructions would help readers position the contribution.
[IV] No code or data repository is mentioned. Since the central claims are empirical, releasing the code used to generate Figs. 5 and 6 would substantially improve reproducibility and would allow others to check the Wannierization positivity issue directly.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: error scaling is externally benchmarked against the exact correlation matrix; self-cited precursors are motivational.

full rationale

The central claim—exponential decay of the per-site approximation error in block radius r (Section IV.B, Fig. 5; Section VI.A)—is an empirical benchmark, not a construction. The 2d GMERA consumes the exact correlation matrix C (Eq. 1), builds disentanglers from its block restriction (Eq. 4) and from the Wannierization/Procrustes steps (Eqs. 5-9), and the error (Eq. 11) compares the resulting C^approx against the exact C; no fitted parameter enters the algorithm, and the fitted slopes alpha are post hoc characterizations of measured errors, explicitly labeled as such ('the fitted slope alpha'). The complexity estimates (r >= log(1/epsilon), G ~ (L^2 log L) log^5(1/epsilon)) are algebraic consequences of that measured scaling, not predictions forced by input. Self-citations to ZER (Ref. [5]) and to holographic/MCMR works (Refs. [2,9,14,16]) motivate the Wannierization and resource-reuse optimizations, but the Wannierization is fully specified by the paper's own equations, and no uniqueness or no-go result from the authors' prior work is invoked to select the ansatz; the no-go constraints cited ([7,8]) are external. The acknowledged caveats—the positive-singularity condition on Eq. (8) that is 'expected to be satisfied,' and the impossibility of ruling out slow L-drifts—are unproven premises and statistical limits (correctness risk), not circular reductions. Score 2 reflects only the presence of minor non-load-bearing self-citations.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central resource claims rest on the standard Gaussian correlation-matrix formalism, plus several unproven structural assumptions: Wannierization is unobstructed, coarse-graining preserves lattice geometry, and lattice refinement preserves gauge constraints. The only tunable inputs are block radius r and threshold zeta; alpha is a post hoc fit. No invented physical entities are introduced; the expanding F2QE is a circuit construction rather than a new particle or force.

free parameters (3)

block radius r
Tunable truncation parameter controlling accuracy; error decays as exp(-alpha r). Chosen by hand for each run; not fitted to data.
threshold zeta (thresholded distillation) = 10^-3, 10^-4, 10^-5
Chosen by hand to define frozen versus courier modes in thresholded distillation; affects termination and accuracy.
fitted slope alpha = varies by phase
Slope of log-error versus r, extracted from numerical data to characterize scaling; not an input to the algorithm.

assumptions (5)

standard math Gaussian pure states are fully characterized by their two-point correlation matrix via Wick's theorem
Used throughout Section II and Appendix A; standard result in free-fermion physics.
ad hoc to paper Wannierization is unobstructed: all singular values in Eq. (8) are strictly positive because frozen modes are topologically trivial
Stated in Section III.A.b without proof; if false the Procrustes solution is degenerate and the algorithm fails.
domain assumption The GMERA coarse-graining preserves the honeycomb geometry so the procedure can be iterated
Assumed in Section III; verified by construction for one lattice, not proven general.
domain assumption Gapped interacting states adiabatically connected to free-fermion states can be prepared by constant-depth adiabatic dressing on top of the GMERA circuit
Invoked in Section VI.A relying on Hastings-Wen quasiadiabatic continuation; not part of the GMERA compression itself.
domain assumption The lattice refinement circuit preserves flux and charge constraints (B_P = 1) during the expanding F2QE
Claimed in Appendix B.2 with a stabilizer construction; no rigorous proof of robustness under composition.

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Cite this review

Pith. "Pith review of Entanglement renormalization circuits for $2d$ Gaussian Fermion States." pith.science (2026). https://pith.science/paper/5WVDSRQH

@misc{pith2026250604200,
  author       = {Pith},
  title        = {Pith review of: Entanglement renormalization circuits for $2d$ Gaussian Fermion States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5WVDSRQH}},
  note         = {Machine review of arXiv:2506.04200}
}

abstract

The simulation of entangled ground-states of quantum materials remains challenging for classical computational methods in more than one spatial dimension, and is a prime target for quantum computational advantage. To this end, an important goal is to identify efficient quantum state preparation protocols that minimize the physical qubit number and circuit depth resources required to capture higher-dimensional quantum correlations. This work introduces a quantum circuit compression algorithm for Gaussian fermion states based on the multi-scale entanglement renormalization ansatz (MERA), which provides an exponential reduction in the circuit depth required to approximate highly-entangled ground-states relevant for quantum materials simulations. The algorithm, termed two-dimensional Gaussian MERA ($2d$ GMERA), extends MERA techniques to compress higher-dimensional Gaussian states. Through numerical simulations of the Haldane model on a honeycomb lattice, the method is shown to accurately capture area-law entangled states including topologically trivial insulators, Chern insulators, and critical Dirac semimetals. While Gaussian states alone are classically simulable, this approach establishes empirical upper bounds on quantum resources needed to prepare free fermion states that are adiabatically connected to correlated ground states, providing guidance for implementing these protocols on near-term quantum devices and offering a foundation for simulating more complex quantum materials. Finally, we develop a novel fermion-to-qubit encoding scheme, based on an expanding $2d$ topological order, that enables implementing fermionic rotations via qubit Pauli rotations with constant Pauli weight independent of system size.

Figures

Figures reproduced from arXiv: 2506.04200 by the authors.

**Figure 2.** FIG. 2. Dual interpretation of the 1 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Comparison of GMPS implementation in quasi-1 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4. 2d GMERA procedure on the honeycomb lattice for [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

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Reference graph

Works this paper leans on

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For each hexagonal block B, form the projector onto its courier modes: P (B) c = v(B) c v(B)† c (5)

work page
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, λ(B,O) n ) contains eigenvalues arranged in descending order

Restrict this projector to each triangular overlap- ping region, O, and diagonalize it: P (B,O) c = s(B,O)Γ(B,O)s(B,O)† (6) where Γ (B,O) = diag(λ(B,O) 1 , . . . , λ(B,O) n ) contains eigenvalues arranged in descending order

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Consider a system of fermions with creation operators {ˆc† i}, where the index i can represent the spatial coordinate, spin, or orbital

Gaussian fermion states This appendix briefly reviews some notation and for- malism for Gaussian fermion states. Consider a system of fermions with creation operators {ˆc† i}, where the index i can represent the spatial coordinate, spin, or orbital. For simplicity, we consider...
[44]

F ermionic strings in Kitaev honeycomb model The Kitaev honeycomb model [23] provides an effi- cient fermion-to-qubit encoding (F2QE) for implement- ing fermionic operations in 2D systems (see [25] for a re- cent experimental implementation of this encoding). Un- like the Jord...
[45]

Kitaev honeycomb model: Lattice Refinement Circuit Attempting to directly implement a fermionic MERA circuit using a 2 d F2QE encoding would result in large circuit overheads since early (IR) steps of the MERA correspond to entangling distant (with the UV metric) fermionic mod...
[46]

Qubit-efficient implementation with sequential circuits and MCMR If our primary goal is to measure physical observables rather than preparing the entire ground state of a 2D sys- tem, we can significantly reduce qubit resources by imple- menting the 2D GMERA procedure as a seq...
[47]

Initialize the bond qubits in a state representing the configuration at the IR (infrared) level
[48]

Prepare the first row of blocks according to the GMERA procedure
[49]

Measure any desired observables on this row 18 ﬁne grain : Initial qubits equivalent 12 8 10 1 2 4 3 56 7 11 91 2 4 3 56 : Injected qubits Topologically 2 12 7 4 6 1 3 5 8 10 9 11 a a 2 (a) 6 1 2 3 4 5 P 6 PA PB 5 1 2 4 3 7 8 :2 3 y z y z |i⟩8 |+⟩7 x z(b) FIG. 8. Lattice refin...
[50]

Reset the physical qubits while preserving the bond qubits
[51]

Prepare the second row using the bond qubits to mediate correlations with the first row
[52]

For the injected qubits (5, 6) shown in Fig

Continue this process for all rows in the system Additionally, when implementing the expanding Ki- taev honeycomb model encoding, we need to ensure that the flux constraints are properly maintained during the sequential preparation. For the injected qubits (5, 6) shown in Fig....
[53]

σz 5σy 6 = 1 (flux constraint)
[54]

σx 5 σx 6 = 1 (charge conservation) When the σx 5 σx 6 = 1 constraint passes through the sta- bilizer circuit, it transforms into: σx 1 σy 6 σz 5σx 4 σy 3 σz 2 = 1 (B9) This ensures that the flux constraint BP = 1 is pre- served, maintaining the topological properties of the Z...

Pith tools

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

[1] [1]

For each hexagonal block B, form the projector onto its courier modes: P (B) c = v(B) c v(B)† c (5)

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[2] [2]

, λ(B,O) n ) contains eigenvalues arranged in descending order

Restrict this projector to each triangular overlap- ping region, O, and diagonalize it: P (B,O) c = s(B,O)Γ(B,O)s(B,O)† (6) where Γ (B,O) = diag(λ(B,O) 1 , . . . , λ(B,O) n ) contains eigenvalues arranged in descending order

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[3] [3]

4(b), we have 18r courier modes to distribute among 6 triangular regions O

For the first disentangling step in Fig. 4(b), we have 18r courier modes to distribute among 6 triangular regions O. Thus, for each triangular region, we select the first 6 r eigenmodes s(O) c correspond to Γ(B,O) as seeds for Wannierization

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[4] [4]

We formulate the Wannierization as an orthogo- nal Procrustes problem [19], seeking the optimal unitary transformation w(B) c that best aligns the 7 courier modes with the seeds identified in (c): w(B) c = arg min w ∥v(B) c w− s(B) c ∥F (7) where s(B) c = ⊕O′s s(O) c combines all the seed vec- tors from the triangular regions associated with block B, and ...

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[5] [5]

These courier modes will be centered at the triangular regions, forming the sites of the coarse-grained lattice for the next step

The resulting Wannierized courier modes ˜ v(B) c = v(B) c w(B) c are orthonormal and maximally localized within the triangular overlapping regions, and at the same time ensuring strict locality in the hexag- onal region B. These courier modes will be centered at the triangular regions, forming the sites of the coarse-grained lattice for the next step. c. ...

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[6] [6]

The number and arrangement of these blocks will depend on the lattice geometry

Block Structure: Define appropriate blocks with radius r that cover the entire lattice. The number and arrangement of these blocks will depend on the lattice geometry

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[7] [7]

For the honeycomb lattice, three layers are sufficient, but other lattices may require different numbers

Layer Organization: Determine the minimum number of disentangling layers needed to ensure en- tanglement within the same length scale are consid- ered. For the honeycomb lattice, three layers are sufficient, but other lattices may require different numbers

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[8] [8]

These regions will serve as the cen- ters for Wannierization of courier modes

Overlapping Regions : Identify the junction points where blocks from different disentangling layers meet. These regions will serve as the cen- ters for Wannierization of courier modes

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