Entangling Topological Invariants

Kazuki Ikeda; Yaron Oz

Entangling Topological Invariants

T0 review · 0 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that a rank-pq occupied bundle over S4 is a tensor product of factors exactly when its second Chern number vanishes modulo gcd(p,q); for 2x2 factorization this means C2 must be even.

desk verdict A sound and useful criterion for global tensor factorization of gapped bundles, with a solid surface analogue and honest numerics; minor cosmetic issues only. read the letter →

arxiv 2608.03634 v1 pith:MZK4I52U submitted 2026-08-04 quant-ph cond-mat.mes-hallhep-thmath-phmath.MP

Kazuki Ikeda , Yaron Oz This is my paper · ORCID

classification quant-phcond-mat.mes-hallhep-thmath-phmath.MP MSC 55R1057R22

keywords topologicalinvariantstensor-productfactorizationclutchingmapmixedChernclasscrossedThoulesspumpsecondnumberoccupiedprojectortomographysubsystemlabels

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

An isolated set of occupied bands can look locally like a product of two independent subsystems without being a global product: the local tensor-product descriptions may fail to glue consistently around the parameter space. The paper develops a general obstruction, the subsystem-gluing quotient, and applies it in two settings. On a closed surface with a specified array of rank-one sectors, the quotient removes all Chern contributions attributable to a single label and leaves mixed Chern classes, measurable through a crossed Thouless pump. On the four-sphere, where only the factor dimensions p and q are specified, the quotient acts on the clutching map of the rank-pq occupied bundle, and the paper proves that global factorization is possible exactly when C2 is divisible by gcd(p,q). For p=q=2, any odd C2 is a fatal obstruction.

What carries the argument

The engine is the subsystem-gluing quotient: independent changes of basis in the two candidate factors map through the tensor-product homomorphism and generate a subgroup of the occupied frame transformations; the obstruction is the cokernel of the induced map on topological data. For a specified line resolution on a closed surface this quotient is coker Φ ≅ $Z^{{(N_A-1)(N_B-1)}}$ on sector Chern tables, with the mixed classes χ_ab = C_ab - C_{a,b+1} - C_{a+1,b} + C_{a+1,b+1}. For a rank-pq bundle over S4, clutching identifies the relevant data with third homotopy, and the tensor-product map is Ψ(m,n)=qm+pn, whose cokernel is Z_{gcd(p,q)}; the residue η_{p,q}=C2 mod gcd(p,q) is the complete obstruction. The auxiliary machinery includes the Wilczek-Zee curvature of the occupied projector, the Gauss-Codazzi term for moving sector lines, and a transition-function reconstruction protocol that extracts the clutching winding from projector tomography.

What would settle it

Construct or compile a rank-four occupied bundle over S4 whose transition map g:S3→SU(4) has winding 1 but whose reconstructed equatorial transition matrix is gauge-equivalent to g_A⊗g_B with g_A,g_B in SU(2); the resulting even winding would contradict the claimed odd-C2 obstruction. Equivalently, run the paper's three-transmon tomography on the odd-C2 bundle and look for a winding parity of 0 instead of 1.

Watch

Extended reading notes

Core claim

The paper's central result is Theorem 1: for p,q>=2, a rank-pq complex vector bundle E over S4 admits a tensor-product decomposition E = E_A ⊗ E_B (with the factors carrying trivial determinant) if and only if C2(E) is divisible by gcd(p,q). Equivalently, eta_{p,q}(E) = C2(E) mod gcd(p,q) is the complete obstruction. The proof goes through clutching: E is classified by an equatorial transition map g:S3→SU(pq) whose winding is C2, while product frames g_A⊗g_B have windings q m + p n, forming the subgroup qZ + pZ = gcd(p,q)Z. Because the reduction from the product subgroup SU(p)xSU(q) to the two factor bundles lifts trivially (the relevant $H^{2}$(S4;Z_d) vanishes), divisibility is not only necessary but sufficient. The smallest nontrivial case is p=q=2, where odd C2 forbids a global 2x2 factorization. On surfaces, with a specified line resolution, the same quotient removes row- and column-additive Chern numbers and leaves the mixed Chern class, realized in Hamiltonians as a crossed pump.

Load-bearing premise

The whole argument assumes the occupied multiplet is an isolated, uniformly gapped bundle over the entire parameter space, so the spectral projector is a smooth vector bundle; without that gap the clutching and Chern-number language does not apply.

Editorial extensions

If this is right

A rank-pq gapped multiplet over S4 with C2 not divisible by gcd(p,q) cannot be realized as two independent subsystem bundles, no matter how the local frame is chosen.
For two qubits, the parity of C2 is the factorization obstruction: even C2 may, after suitable framing, factorize, while odd C2 never does.
In label-conserving two-dimensional systems, the mixed Chern number χ is a measurable transport response: inserting a 2π flux in one label transfers χ units of the other label's charge.
The obstruction is read out directly from the occupied spectral projector: reconstructing the equatorial transition matrix and computing its winding modulo d gives the residue η_{p,q}.

Reading between the lines

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

The gcd divisibility criterion is not conceptually tied to S4's clutching; the same cokernel argument should yield analogous factorization obstructions once the image of the tensor-product map on the relevant homotopy groups of another base space is known, so one can pose similar questions for rank-pq bundles over other spheres or for higher Chern classes.
The surface mixed Chern class and the S4 residue are two resolutions of one idea, and one could seek a dimensional-reduction or suspension bridge between them, although the paper does not construct such a map.
The crossed-pump protocol makes a sharp quantitative prediction that could be tested in cold-atom or photonic simulators: in an eight-level QWZ-type model with vanishing one-label responses, flux insertion in label A moves exactly χ=4 units of label B.
The transition-function tomography protocol could serve as a practical diagnostic in quantum devices: it would tell whether a candidate qubit 'subsystem' is a genuine global tensor factor or only a local description, as demonstrated on a three-transmon register.

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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

Short version: this paper earns its central claim. Theorem 1 — a rank-pq bundle over S4 admits a global tensor factorization iff C2 is divisible by gcd(p,q) — is proved properly in Appendix C, including the converse and the central-quotient lift. The proof is elementary homotopy theory, but that does not make it trivial: the statement as a physical factorization obstruction, with the parity-of-C2 consequence for 2x2 factorization, is the kind of crisp criterion that people will want to cite.

The surface material is the more original part. The cokernel of the row/column additive Chern map, the Gauss–Codazzi correction for moving sector lines, and the crossed-pump readout form a coherent package. The eight-level examples are well chosen: they show that mixed Chern numbers can be nonzero while the total and one-label responses vanish, and the numerics converge cleanly. The finite-time pump includes finite-size and disorder checks, and the three-transmon tomography protocol is worked out at the circuit level with honest finite-shot analysis.

Soft spots are mostly cosmetic. The data availability statement points to a GitHub repository without a direct link or commit hash; that is a reproducibility annoyance, not a scientific flaw. The hardware protocol is simulation-only — the authors say so explicitly — so the readout contribution is a proposal, not a measurement. The S4 criterion is short enough that its novelty sits in the physical framing rather than in the mathematics; readers should not expect a deep new theorem. The surface quotient requires a specified line resolution; without that, the surface case does not give a complete obstruction, and the paper could have said a bit more about what is lost in that situation. None of these touch the central argument, which holds up. Citation pattern is clean: the relevant spin-Chern, valley-Chern, and 4D quantum Hall references are cited, and the new claims are distinguished from those earlier constructions.

I agree with the second-pass stress-test: I did not find a load-bearing flaw. The gapped-multiplet assumption is explicit and is verified in the examples with the analytic bound of Eq. (37). If I were refereeing, I would recommend acceptance with minor revisions — mostly the code link and a small amount of expanded discussion.

Referee Report

0 major / 6 minor

Summary. The manuscript introduces a framework for deciding when an isolated occupied multiplet of a gapped Hamiltonian admits a globally consistent tensor-product (subsystem) structure, and for extracting the topological obstruction when it does not. In the line-resolved setting on a closed surface, the authors define a quotient of the sector Chern table by row- and column-additive contributions, obtaining mixed Chern classes; they show that when sector lines move inside a non-Abelian occupied space, the curvature acquires a Gauss–Codazzi term, and that in the label-conserving limit the same integer appears as a crossed Thouless pump. In the second setting, where only the factor dimensions p and q are specified on S^4, they prove (Theorem 1, Appendix C) that a rank-pq bundle with trivial determinant factors as a tensor product of determinant-trivial bundles if and only if C2(E) is divisible by gcd(p,q); equivalently, eta_{p,q}(E)=C2(E) mod gcd(p,q) is the complete obstruction, so in particular odd C2 forbids a 2x2 factorization. The paper supports these results with finite eight-level Hamiltonians, analytic gap bounds, numerical convergence checks, and a three-transmon occupied-projector tomography protocol for reconstructing the clutching winding parity.

Significance. If correct, the result is a clean and useful addition to the topological band literature: it converts a bundle-theoretic factorization question into an integer divisibility test and supplies two experimentally oriented readouts, a crossed pump for the mixed Chern class and transition-function tomography for the clutching parity. The main theorem is fully proven in Appendix C, including the converse construction via Bezout windings and the lift through the central mu_d quotient, and the numerical checks converge cleanly. The obstruction is parameter-free and falsifiable, and the finite-shot analysis is explicit about its modeling assumptions. The paper should be of interest to researchers working on multipartite topological responses, subsystem Chern numbers, and quantum simulation of four-dimensional topology.

minor comments (6)

[Table I] Table I as rendered appears internally inconsistent: with the listed values of integral f_proj close to 3.84, the entries 0.175 x 10^{-3}, 0.167 x 10^{-6}, and 0.161 x 10^{-13} for integral Xi_AB cannot sum to the stated Chern number 4; presumably the Gauss-Codazzi entries should be 0.175, 0.167, 0.161 and the powers of ten belong to the residual column, but the table should be reformatted to remove the ambiguity.
[Eqs. (5) and (28)] In Eqs. (5) and (28), the quotient is written as 'Z gcd(p,q)' without a subscript; it should be \mathbb{Z}_{\gcd(p,q)}.
[Data Availability] The Data Availability statement says the numerical data and source code are in a GitHub repository but gives no URL or identifier; a permanent link or DOI is needed for reproducibility.
[Appendix D4] In Appendix D4, the text should state explicitly at the beginning that n_s is the total number of shots pooled over the four occupied preparations in one Pauli basis, since the first paragraph describes two different implementations of the ensemble and the meaning of n_s is used throughout the shot-count estimates.
[Eq. (D3)] In the discussion of Eq. (D3), the statement that '-tau_z supplies two occupied and two unoccupied spectator states' could be made explicit by noting that in the kappa_z=-1 block the occupied states are the tau_z=+1 eigenstates, which have energy -1 in the flattened Hamiltonian.
[Section IV.C] In Section IV.C, the phrase 'transition-function tomography' is slightly stronger than what the protocol measures; the protocol reconstructs the homotopy class (winding number) of the clutching map, not the pointwise gauge-dependent transition function, and the text should say so to avoid overstating the result.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the core theorem reduces to standard external algebraic topology, and the surface responses are consistent realizations of defined invariants.

full rationale

The derivation chain is not circular. The surface mixed invariant chi is defined as the quotient of sector Chern data by row- and column-additive tables (Eqs. (3), (8), (9)); the crossed pump is a transport realization of the same combination through the standard sector-pumping relation (Eq. (21)), i.e. a consistency equivalence between responses, not a fitted parameter renamed as a prediction. The clutching criterion in Theorem 1 is proved in Appendix C from standard external facts: pi3(SU(n)) = Z, H^2(S4;Z) = 0, and H^2(S4;Z_d) = 0. The image qZ+pZ of the tensor-product winding map is computed directly (Eq. (27)), and the converse uses Bezout integers, so the residue modulo gcd(p,q) is a genuine nontrivial obstruction rather than a restatement of the quotient definition. Example parameters (v*, lambda, theta0) configure the numerical paths but do not enter the invariants. The only self-citations [12,13] are contextual remarks, not load-bearing; no uniqueness claim is imported from them.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central theorems are parameter-free; the hand-picked amplitudes, coupling strengths, and flux offsets configure the illustrative models only. The axioms are standard bundle topology and the physical gap assumption, which the examples verify numerically.

free parameters (4)

mixing amplitudes v* = (11,10,7,4)/20
Hand-chosen amplitudes of the four momentum-dependent mixing channels in Eq. (14); selected to keep energy and label-resolution gaps open along the path, not fitted to a target observable.
mixing endpoint epsilon = 6/5
Endpoint of the path in Eq. (15) where the label-conserving limit is maximally deformed; chosen for demonstration.
interblock coupling lambda = 5/4
Coupling of V_cpl in Eq. (35); chosen so the analytic gap lower bound Delta >= 5/16 remains positive, preserving C2=1.
initial flux offset theta0 = 0.37
Initial A-flux in the pump simulation in Appendix B; chosen away from the finite-size edge anticrossing.

assumptions (5)

standard math Clutching classification: over S4, isomorphism classes of determinant-trivial rank-n complex vector bundles are classified by the winding W3 of an equatorial clutching map S3 -> SU(n), with pi3(SU(n)) = Z and W3 = C2.
Used in Theorem 1 and Appendix C, Eq. (C1), to identify bundles by the clutching integer.
domain assumption The spectral projector P of a gapped occupied multiplet defines a smooth Hermitian vector bundle over the parameter space, with Berry connection A = P dP.
The general framework assumes this; the examples verify a uniform gap, e.g., the analytic lower bound in Eq. (37).
standard math The Gauss-Codazzi identity for a subbundle: the integral of (1/2pi i)[Tr(Q F_P) + Tr(Q(P dQ P)^2)] equals c1(L_s).
Eqs. (17)-(18) and Appendix A3; standard subbundle geometry.
standard math Vanishing cohomology: H^2(S4; Z_d)=0 for d=gcd(p,q), so a reduction of an SU(pq) bundle to G_{p,q} lifts to SU(p) times SU(q).
Appendix C, proof of (i) implies (ii); follows from H^2(S4;Z)=0 and universal coefficients.
domain assumption Pauli-basis tomography of the three-transmon register reconstructs the rank-four occupied projector, and the winding of the reconstructed transition function classifies C2 parity.
Appendix D4; simulated with multinomial shot noise and unmitigated symmetric readout errors, not a hardware demonstration.

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Pith. "Pith review of Entangling Topological Invariants." pith.science (2026). https://pith.science/paper/MZK4I52U

@misc{pith2026260803634,
  author       = {Pith},
  title        = {Pith review of: Entangling Topological Invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZK4I52U}},
  note         = {Machine review of arXiv:2608.03634}
}

abstract

An isolated occupied multiplet may admit local tensor-product descriptions without a globally consistent subsystem structure. We characterize the obstruction by comparing the transition functions of the occupied multiplet with those generated by independent basis changes in the two candidate subsystems. When a decomposition into rank-one sectors over a closed surface is specified, the resulting quotient removes row- and column-additive Chern data and yields mixed Chern classes. Momentum-dependent mixing of the sector labels adds the Gauss--Codazzi curvature of the moving lines, while in the label-conserving limit the mixed class is measured by a crossed Thouless pump. When only the factor dimensions $p$ and $q$ are specified, the comparison is made at the level of the clutching map of a rank-$pq$ bundle over $S^4$. Product frames generate winding numbers in $q\mathbb Z+p\mathbb Z$, so global factorization is possible exactly when $C_2$ is divisible by $\gcd(p,q)$; in particular, odd $C_2$ obstructs a $2\times2$ factorization. We illustrate the two settings with finite eight-level Hamiltonians and give pumping and occupied-projector tomography protocols for their readout.

Figures

Figures reproduced from arXiv: 2608.03634 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Boundary spectrum and cross pump. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. Clutching quotient and subsystem-factorization test. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: (a) displays the resulting nearest-neighbor circuit for 𝑀2 = 𝑌 𝑍 𝑋; changing only the local basis rotations gives the other Pauli words. Panel (b) shows the state-preparation and local-basis readout circuit used to reconstruct the occupied projector. Appendix D gives t…

**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_p012_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

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Works this paper leans on

39 extracted references · 35 canonical work pages

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The quotient identifies Chern tables thatdifferbyadditiveone-labelcontributions

Mixed Chern quotient Each sector integer𝐶𝑎𝑏 =𝑐 1(𝐿𝑎𝑏)[𝑋] is fixed by the specified decomposition. The quotient identifies Chern tables thatdifferbyadditiveone-labelcontributions. Let Λ=Z 𝑁𝐴×𝑁𝐵 and define Φ:Z 𝑁𝐴⊕Z 𝑁𝐵−→Λ, Φ(𝑢,𝑣) 𝑎𝑏 =𝑢 𝑎+𝑣 𝑏.(A1) The sublatticeΛone=imΦ contains the additive Chern tables attributable to either label separately. The mixed Cher...

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This gives the binary form of the projected-feature diagnostic [4, 5]

Label resolution and entanglement spectrum For a binary microscopic label with𝑄+=(1+𝑍)/2, 𝑋=𝑃𝑍𝑃=2𝑃𝑄 +𝑃−𝑃.(A9) If𝜉𝑗 are the nonzero eigenvalues of𝑄+𝑃𝑄+ and𝜇𝑗 the corre- sponding eigenvalues of𝑃𝑍𝑃inRan𝑃, then 𝜇𝑗 =2𝜉 𝑗−1, 𝜀 ent 𝑗 =log 1−𝜉 𝑗 𝜉𝑗 =−2 artanh𝜇 𝑗.(A10) At each binary stage, a compressed-label gap closes exactly when the corresponding internal-part...

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(14) in the order (𝑣𝐴,𝑣𝐵,𝑣(𝜎) 𝐴𝐵,𝑣(0) 𝐴𝐵)

Gauss–Codazzi formula and numerical convergence In Eq.(15), the entries ofv★ correspond to the coefficients in Eq. (14) in the order (𝑣𝐴,𝑣𝐵,𝑣(𝜎) 𝐴𝐵,𝑣(0) 𝐴𝐵). They give comparable weight to the two single-label rotations and smaller amplitudes to the joint channels. The parameter𝜖 controls the common scale. The energy and label-resolution gaps remain open ...

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Spectral flattening makes the occupied quartet exactly degenerate

Finite-shot tomography At each parameter point, tomography reconstructs the rank- four occupied projector𝑃. Spectral flattening makes the occupied quartet exactly degenerate. For a three-qubit proof of principle, the8×8Hamiltonian is diagonalized classically and the four occupied eigenvectors are compiled offline into exact state-preparation circuits [25]...

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Boundary anomaly and ribbon model The edge data are 𝐾=diag(1,−1,−1,1), 𝑡𝐴=(1,1,−1,−1) 𝑇, 𝑡 𝐵=(1,−1,1,−1) 𝑇.(B1) The mixed anomaly coefficient is𝑡𝑇 𝐴𝐾−1𝑡𝐵 =4 . A complete set of symmetric Haldane null vectors would have to satisfy ℓ𝑇 𝑖 𝐾−1ℓ𝑗 =0, ℓ 𝑇 𝑖 𝐾−1𝑡𝐴=ℓ 𝑇 𝑖 𝐾−1𝑡𝐵=0.(B2) Such a complete isotropic set would force the anomaly pairing of𝑡𝐴 and𝑡𝐵 to vanis...

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Theboundarytwistandmomentum offsetsaredefinedinEq

Pump protocol and robustness The thermodynamic spectral-flow calculation on the open-𝑥, periodic-𝑦cylindertransfersfourunitsof 𝐵chargewhileleaving the total and𝐴 charges unchanged, in agreement with the finite-timeprotocolbelow. Theboundarytwistandmomentum offsetsaredefinedinEq. (B4)andtheprecedingdiscussion;the transported𝐵 density is localized on the tw...

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The two- dimensionalbasisis C2 𝐴⊗C 2 𝐵⊗C 2 orb

Eight-level model and second Chern number The surface and clutching calculations use eight-state Hilbert spaces with rank-four occupied multiplets. The two- dimensionalbasisis C2 𝐴⊗C 2 𝐵⊗C 2 orb. Forthefour-dimensional model we choose a concrete superconducting encoding, Hamb=C 2 𝜅⊗C 2 𝜏⊗C 2 𝜎≃H𝑞𝜅⊗H𝑞𝜏⊗H𝑞𝜎,(D1) withcomputationalstates |𝜅𝜏𝜎⟩↔|𝑞 𝜅𝑞𝜏𝑞𝜎⟩. Alin...

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LetP= ∯ 𝑗𝑃𝑗 be a Pauli word with nontrivial support 𝑆 and weight𝑤=|𝑆|

Native-gate compilation on a three-transmon chain Every term in Eqs.(D3) and (D6) is a Pauli word of weight at most three. LetP= ∯ 𝑗𝑃𝑗 be a Pauli word with nontrivial support 𝑆 and weight𝑤=|𝑆| . Choose single-qubit basis changes𝐵𝑋 =𝐻 , 𝐵𝑌 =𝐻𝑆 †, and𝐵𝑍 =𝐼 , and set𝐵( P)=∯ 𝑗𝐵𝑃𝑗 on the support, so that𝐵( P)P𝐵( P)† = ˛ 𝑗∈𝑆 𝑍𝑗. If 𝑡∈𝑆is a target and 𝐶𝑆 = Ö 𝑗∈𝑆...

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

The quotient identifies Chern tables thatdifferbyadditiveone-labelcontributions

Mixed Chern quotient Each sector integer𝐶𝑎𝑏 =𝑐 1(𝐿𝑎𝑏)[𝑋] is fixed by the specified decomposition. The quotient identifies Chern tables thatdifferbyadditiveone-labelcontributions. Let Λ=Z 𝑁𝐴×𝑁𝐵 and define Φ:Z 𝑁𝐴⊕Z 𝑁𝐵−→Λ, Φ(𝑢,𝑣) 𝑎𝑏 =𝑢 𝑎+𝑣 𝑏.(A1) The sublatticeΛone=imΦ contains the additive Chern tables attributable to either label separately. The mixed Cher...

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[2] [2]

This gives the binary form of the projected-feature diagnostic [4, 5]

Label resolution and entanglement spectrum For a binary microscopic label with𝑄+=(1+𝑍)/2, 𝑋=𝑃𝑍𝑃=2𝑃𝑄 +𝑃−𝑃.(A9) If𝜉𝑗 are the nonzero eigenvalues of𝑄+𝑃𝑄+ and𝜇𝑗 the corre- sponding eigenvalues of𝑃𝑍𝑃inRan𝑃, then 𝜇𝑗 =2𝜉 𝑗−1, 𝜀 ent 𝑗 =log 1−𝜉 𝑗 𝜉𝑗 =−2 artanh𝜇 𝑗.(A10) At each binary stage, a compressed-label gap closes exactly when the corresponding internal-part...

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[3] [3]

(14) in the order (𝑣𝐴,𝑣𝐵,𝑣(𝜎) 𝐴𝐵,𝑣(0) 𝐴𝐵)

Gauss–Codazzi formula and numerical convergence In Eq.(15), the entries ofv★ correspond to the coefficients in Eq. (14) in the order (𝑣𝐴,𝑣𝐵,𝑣(𝜎) 𝐴𝐵,𝑣(0) 𝐴𝐵). They give comparable weight to the two single-label rotations and smaller amplitudes to the joint channels. The parameter𝜖 controls the common scale. The energy and label-resolution gaps remain open ...

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[4] [4]

Spectral flattening makes the occupied quartet exactly degenerate

Finite-shot tomography At each parameter point, tomography reconstructs the rank- four occupied projector𝑃. Spectral flattening makes the occupied quartet exactly degenerate. For a three-qubit proof of principle, the8×8Hamiltonian is diagonalized classically and the four occupied eigenvectors are compiled offline into exact state-preparation circuits [25]...

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[5] [5]

Boundary anomaly and ribbon model The edge data are 𝐾=diag(1,−1,−1,1), 𝑡𝐴=(1,1,−1,−1) 𝑇, 𝑡 𝐵=(1,−1,1,−1) 𝑇.(B1) The mixed anomaly coefficient is𝑡𝑇 𝐴𝐾−1𝑡𝐵 =4 . A complete set of symmetric Haldane null vectors would have to satisfy ℓ𝑇 𝑖 𝐾−1ℓ𝑗 =0, ℓ 𝑇 𝑖 𝐾−1𝑡𝐴=ℓ 𝑇 𝑖 𝐾−1𝑡𝐵=0.(B2) Such a complete isotropic set would force the anomaly pairing of𝑡𝐴 and𝑡𝐵 to vanis...

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[6] [6]

Theboundarytwistandmomentum offsetsaredefinedinEq

Pump protocol and robustness The thermodynamic spectral-flow calculation on the open-𝑥, periodic-𝑦cylindertransfersfourunitsof 𝐵chargewhileleaving the total and𝐴 charges unchanged, in agreement with the finite-timeprotocolbelow. Theboundarytwistandmomentum offsetsaredefinedinEq. (B4)andtheprecedingdiscussion;the transported𝐵 density is localized on the tw...

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[7] [7]

The two- dimensionalbasisis C2 𝐴⊗C 2 𝐵⊗C 2 orb

Eight-level model and second Chern number The surface and clutching calculations use eight-state Hilbert spaces with rank-four occupied multiplets. The two- dimensionalbasisis C2 𝐴⊗C 2 𝐵⊗C 2 orb. Forthefour-dimensional model we choose a concrete superconducting encoding, Hamb=C 2 𝜅⊗C 2 𝜏⊗C 2 𝜎≃H𝑞𝜅⊗H𝑞𝜏⊗H𝑞𝜎,(D1) withcomputationalstates |𝜅𝜏𝜎⟩↔|𝑞 𝜅𝑞𝜏𝑞𝜎⟩. Alin...

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[8] [8]

LetP= ∯ 𝑗𝑃𝑗 be a Pauli word with nontrivial support 𝑆 and weight𝑤=|𝑆|

Native-gate compilation on a three-transmon chain Every term in Eqs.(D3) and (D6) is a Pauli word of weight at most three. LetP= ∯ 𝑗𝑃𝑗 be a Pauli word with nontrivial support 𝑆 and weight𝑤=|𝑆| . Choose single-qubit basis changes𝐵𝑋 =𝐻 , 𝐵𝑌 =𝐻𝑆 †, and𝐵𝑍 =𝐼 , and set𝐵( P)=∯ 𝑗𝐵𝑃𝑗 on the support, so that𝐵( P)P𝐵( P)† = ˛ 𝑗∈𝑆 𝑍𝑗. If 𝑡∈𝑆is a target and 𝐶𝑆 = Ö 𝑗∈𝑆...

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