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REVIEW 3 major objections 4 minor 69 references

For a 3+1d Z_2 1-form symmetry, the lattice operations of duality and SPT stacking generate the single-qubit Clifford group, with the semion QCA as the central element.

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2026-08-01 06:57 UTC pith:BRSJ24HN

load-bearing objection A serious, mostly rigorous paper whose headline identification is honestly flagged as conditional: the proven lattice relations and the corrected Witt-group computation are real contributions, but the semion QCA identification and hence the full Z_8/Clifford-group claim remain unproven. the 3 major comments →

arxiv 2607.21698 v1 pith:BRSJ24HN submitted 2026-07-23 quant-ph cond-mat.str-elhep-thmath.QA

Non-Invertible Symmetries Mixing with Witt Non-Trivial Quantum Cellular Automata

Kansei Inamura , Oskar Wojdel , Lukasz Fidkowski , Sakura Schafer-Nameki This is my paper
classification quant-ph cond-mat.str-elhep-thmath.QA
keywords non-invertible symmetriesquantum cellular automata1-form symmetryKramers-Wannier-Wegner dualitySPT entanglerWitt groupClifford groupcentral extension
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that the two basic operations on 3+1d quantum field theories with a Z_p 1-form symmetry—Kramers-Wannier-Wegner duality (gauging) and stacking a 1-form SPT—can be realized microscopically as quantum cellular automata acting on a lattice operator algebra. For p=2, the group these QCAs generate is a Z_8 central extension of PSL(2,Z_4)/Z_2, exactly the single-qubit Clifford group, with the order-8 semion QCA as the central element; for odd primes p, the extension splits abstractly as SL(2,Z_p)×Witt classes, but the lattice fusion rules still mix with the Z_p Clifford QCA. The paper computes explicit lattice relations, e.g. (α_KWW∘α_TW^2)^4 = α_t^2∘α_framing∘α_TW^2 and (α_KWW,p∘α_TW,p)^3 = α_t∘β̃_p^{(p-1)/2}∘α_TW,p, showing that continuum fusion rules hold only up to these non-trivial QCAs and lattice translations. This connects non-invertible symmetry fusion to the Witt-group classification of 3+1d QCAs, and the authors argue—under an explicitly stated assumption—that the full group of locality-preserving automorphisms modulo finite-depth circuits is captured by this structure.

Core claim

The central claim is that the group of locality-preserving automorphisms of the Z_2^(1)-symmetric operator algebra, generated by the KWW duality QCA and the Tsui-Wen SPT entangler QCA, is a Z_8 central extension of PSL(2,Z_4)/Z_2 isomorphic to the Clifford group of a single qubit, with the duality and entangler acting as Hadamard and phase gates. This equates a lattice computation with a categorical calculation of the pointed graded Witt group. For odd primes, the analogous group is a split extension of SL(2,Z_p) by the pointed Witt group, and the lattice fusion rule (α_KWW,p∘α_TW,p)^3 = α_t∘β̃_p^{(p-1)/2}∘α_TW,p exhibits mixing with the Z_p Clifford QCA. The authors prove the lattice relati

What carries the argument

The central object is the quotient algebra A_{Z_p^(1)}/I, formed by taking all local operators commuting with the Z_p 1-form symmetry and quotienting by the ideal generated by the 1-form symmetry generators around each cube. On this algebra, the Kramers-Wannier-Wegner duality α_KWW and the Tsui-Wen SPT entangler α_TW are defined as QCAs using half-translations and higher cup products on a cubic lattice. The group structure is then compared with the graded, syllepsis-twisted pointed Witt group Witt^pt(Z_p⊕Z_p^dual, s), whose central extension of PSL(2,Z_4)/Z_2 (for p=2) is computed to be the binary octahedral group 2O central-product Z_8, i.e. the single-qubit Clifford group. For odd p the ex

Load-bearing premise

The load-bearing unproven assumption is that any QCA on the Z_2^(1)-symmetric algebra that extends to the full algebra as a finite-depth circuit (with gates not necessarily symmetric) is equivalent, up to symmetric finite-depth circuits, to some power of the Tsui-Wen SPT entangler; this — together with the completeness of the 1-form SPT classification — is what identifies (α_KWW∘α_TW)^3 with the order-8 semion QCA and completes the Z_8 central extension.

What would settle it

Explicitly construct the order-8 semion QCA (or a commuting-projector parent Hamiltonian with semion boundary) and compare it with (α_KWW∘α_TW)^3 on the symmetric algebra modulo symmetric finite-depth circuits and translations; any inequivalence would falsify the central identification. Alternatively, find an extendable QCA that is a finite-depth circuit with non-symmetric gates but is not equivalent to α_TW^k up to symmetric FDC—that would directly disprove the assumption in §5.5.4.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the central claim holds, the single-qubit Clifford group is realized as a lattice symmetry group generated by duality and SPT-stacking, giving a concrete many-body setting for Clifford quantum computation.
  • The lattice fusion rules for non-invertible symmetries are refined by Witt-non-trivial QCAs, so QCA classes (not just translations) must be tracked when studying non-invertible symmetry on the lattice.
  • All QCAs in the conjectured Witt classification for 3+1d can be generated from elementary operations: KWW duality, Tsui-Wen entangler, and onsite charge conjugation.
  • For odd primes, although the group splits abstractly, the canonical assignment of Witt classes to extendable QCAs (e.g. p=5) still yields a Borel-subgroup structure incompatible with the naive splitting, so mixing persists in a refined sense.
  • The explicit lattice relation (4.84) matches the continuum (ST^2)^4 = Y^4 up to the framing QCA and translations, providing a checkable signature of the 3-fermion QCA in the symmetry algebra.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: for Z_4 1-form symmetry the same construction should produce a central extension by the U(1)_4 Walker-Wang QCA, generating the full Z_8 × Z_2 in the pointed Witt group; verifying this would confirm the pattern that QCA central elements are determined by the relevant anyon theory.
  • If the stated assumption is correct, it implies a general 'lattice refinement' principle: whenever a continuum defect fusion rule has a TQFT coefficient, the lattice implementation replaces that coefficient by the QCA class of the disentangler for that TQFT, up to translations.
  • The appearance of the qubit Clifford group as a symmetry of a constrained operator algebra could be used to map non-invertible symmetry operations to Clifford circuits, potentially connecting generalized symmetries to quantum error correction and measurement-based protocols.
  • The p=5 example suggests that the physically meaningful notion of 'mixing' between non-invertible symmetries and QCAs is not captured by abstract group splittings but by the compatibility of the splitting with the Witt-class labeling of extendable QCAs; this could be formalized as a Borel-subgroup refinement.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper constructs an operator-algebra/lattice realization of the non-invertible duality symmetry generated by Kramers-Wannier-Wegner duality and 1-form SPT stacking for a Z_2 (and odd-prime Z_p) one-form symmetry in 3+1d. The authors introduce a quotient local operator algebra A_{Z_2^{(1)}}/I, define QCAs α_KWW and α_TW (and their Z_p analogues), and prove direct lattice relations, notably (α_KWW∘α_TW^2)^4 = α_t^2 ∘ α_framing ∘ α_TW^2 (Eq. 4.84) and (α_KWW,p∘α_TW,p)^3 = α_t ∘ β̃_p^{(p-1)/2} ∘ α_TW,p (Eq. 6.82). They compare these with continuum defect fusion and with a categorical computation of the graded pointed Witt group, and claim that for p=2 the resulting group is the one-qubit Clifford group, with the central Z_8 generated by the semion QCA. For odd p they claim a split extension by the relevant Witt classes. The paper is transparent that the identification (ST)^3 = semion QCA is not rigorously proven and rests on an explicitly stated assumption about completeness of the 1-form SPT classification.

Significance. If the central identification is accepted, the paper gives the first microscopic lattice realization of the full non-invertible symmetry/SPT/ QCA mixing structure in 3+1d, and equates a lattice automorphism group with a categorical Witt-group calculation. The directly proven relations (4.84) and (6.82) are explicit and impressive: they show that continuum fusion rules are refined on the lattice by translations and by non-trivial QCAs, including the framing/three-fermion QCA and the Z_p Clifford QCA. The categorical computation of the graded pointed Witt groups in Section 3 and Appendices A-C is detailed, with several propositions proved from defining data. The paper also carefully and honorably marks its own limitations, including the unproved semion-QCA identification and the conjectural completeness assumptions. These limitations are not presentation issues; they are precisely what prevents the headline claim from being a theorem as written.

major comments (3)
  1. [§5.5.4, Eq. (5.51)] The identification (α_KWW ∘ α_TW)^3 = semion QCA Y is the load-bearing step for the Z_8 central extension and the Clifford-group claim, but it is explicitly not proven. The text states that the argument 'falls short of being a rigorous proof'. The derivation uses the Assumption in §5.5.4 — that any QCA extendable to an FDC on the full algebra with possibly non-symmetric gates is symmetric-FDC-equivalent to some α_TW^k — together with completeness of the Z_2^(1) SPT classification. The direct computations (4.84) are consistent, but they do not by themselves determine the Witt class of Q in Claim 1; the conclusion that the class is the semion, rather than the three-fermion or trivial class, is inferred from a Walker-Wang Hamiltonian argument and from the S_4 quotient relation m=1, j=0, both under the Assumption. If the Assumption fails, the semion-QCA identification and hence the full Z_8
  2. [§4.8.1, Eq. (4.95)] The order-2 and centrality statements for [α_framing ∘ α_TW^2] are derived from Eq. (4.95), which the paper itself says relies on the conjectural completeness of the Z_2^(1) SPT classification. Equations (4.96) and (4.99) are therefore conditional. Since these statements are used to place the element in the center and to give the group generated by S and T^2 its claimed structure, this is not a minor caveat: without (4.95), the relations (4.92) only define a quotient that could have additional central extensions or fail to be the dihedral-type group matching the continuum. The authors are transparent about this, but the manuscript should either isolate this assumption from the proven lattice identities or provide independent evidence for (4.95).
  3. [§6.6 and §6.8, Eq. (6.93)] For odd primes, the centrality and order of [α_Y] = [β̃_p^{(p-1)/2} ∘ α_TW,p] are established only under the Assumption (6.93), and §6.8 explicitly states that the centrality of the elements M_k is an assumption. The proven identity (6.82) is strong and is the main lattice result, but the subsequent claims that the group splits as SL(2,Z_p) × W_p, and that the physically relevant assignment of Witt classes is the one in (6.116), require this unproved assumption. Moreover, the discussion around Eq. (6.113)-(6.116) shows that the abstract group isomorphism is not canonical: the p=5 example admits two different Z_2×Z_2 identifications, and the physical one is selected by the extendability criterion. The manuscript should make clear that the split and the physical Witt-class assignment are conditional on (6.93), not consequences of the direct computation.
minor comments (4)
  1. [§2.2.1] Typo: 'parition function' should be 'partition function'.
  2. [§5.5.4] The notation S=α_KWW, T=α_TW is convenient, but it is heavily overloaded with the continuum operations S,T and the unitary operators S,T elsewhere in the paper. A table or an explicit 'in this section only' disclaimer would help the reader avoid ambiguity.
  3. [§5.5.4, Claim 5.4] The notation |(-1)^b⟩ in Eq. (5.62)-(5.63) is terse; defining it as a shorthand for a product of |±⟩ eigenstates is fine, but the half-translation convention in the subscript deserves one more sentence. (Avoid LaTeX backslash in the notation for clarity.)
  4. [§4.2.2, Definition 4.2] The support of an equivalence class in the quotient algebra is defined as an intersection over all representatives. The reader would benefit from an explicit example showing how this differs from the support of a chosen representative, since Lemma 4.3 already provides the clean characterization.

Circularity Check

0 steps flagged

No significant circularity. Lattice relations (4.84) and (6.82) are independent direct computations matched post hoc to continuum/Witt results; the semion-QCA step completing the Z8 extension is honestly flagged as an unproven, assumption-based identification — a rigor gap, not a circular reduction.

full rationale

The three structures the paper compares — continuum defect fusion (§2), the graded pointed Witt group (§3, Apps. A), and the lattice QCA group (§§4–6) — are computed independently, and their agreement is the result, not an input. The load-bearing lattice identities are direct operator computations: (4.84), (α_KWW∘α_TW^2)^4 = α_t^2∘α_framing∘α_TW^2, is derived in §4.7 by computing the action of both (ST^2S†)T^2(ST^2S†) and the independently defined framing QCA of [35] on the generators X_f, ũ_δe; §4.9 only then maps it onto the continuum (ST^2)^4 = Y^4. Likewise (6.82) is obtained in §6.5 by direct comparison of S_pT_p^{2k}S_pT_p^{1/(2k)}S_p with the Clifford QCA β̃_p^{(−k)} of [34]. The headline semion step (α_KWW∘α_TW)^3 = order-8 semion QCA is not produced by a circle: it rests on two explicitly stated physical inputs (completeness of the Z_2 1-form SPT classification by T^k; the claim that the X_f stabilizer Hamiltonian is a semion Walker–Wang parent), and the paper itself disclaims rigor: 'our argument falls short of being a rigorous proof' (§5.5.4) and 'We do not currently have a proof of either of these statements' (§5.5.4). The Assumption in §5.5.4 is a completeness statement about QCAs extendable to full-algebra FDCs, not a restatement of the target relation; the SPT-completeness assumptions in §4.8 and §6.6 are likewise inputs, not echoes of the conclusions. These are honest gaps/conjectures (also §1 'It is natural to conjecture that this is the full group'; §4.2 deferring rigorous definitions to '[52]'), and the paper corrects rather than hides a defect in [30] (footnote 8). Framework imports from [30] (exact sequence (3.11), H^5(Z_2⊕Z_2[3],C^×)=Z_2⊕Z_2) support the Witt-group formalism, but the load-bearing relations (ST)^3=Y, S^2=T^4=1, order 8, are rederived in-paper (Propositions A.3–A.8), and the odd-prime H^5=0 is proven in App. A.4. The Clifford-group identification (Lemma 3.4) is presentation matching of an independently computed group. No fitted parameter is renamed a prediction; no step reduces to a self-citation chain or to a definitional identity.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No fitted parameters: every construction is explicit and parameter-free. The sign/timing conventions (s_f, s_e in (6.9)-(6.10); the half-translation t^{1/2}; the integer-lift {·}_k in §5.1) and the normalization choices (â = κ^{-1}a in (3.79), S = u^2aba, T = ua in Theorem 3.3) are conventions chosen for compatibility, not numbers fitted to data. The load-bearing inputs are: the categorical framework of [30] with pointed-Witt-group input from [49]; the conjectural completeness of the 1-form SPT classification; the paper's own Assumption in §5.5.4/§6.6; the conjectural identification of graded Witt classes with symmetric QCA; and external QCA results [31,32,34,35,42,58]. All unproven assumptions are explicitly flagged by the authors.

axioms (6)
  • domain assumption The classification of 3+1d bosonic Z_p^(1) SPT phases is complete, generated by entanglers T^k (k∈Z_4 for p=2; k∈Z_p for odd p).
    Used in §4.8.1 to fix the order of [α_framing∘α_TW^2] and in §6.6 with eq. (6.93) to fix the order of [α_Y]; the authors call it 'physically reasonable' but conjectural.
  • ad hoc to paper Any symmetric QCA on A_Z2^(1)/I extendable to an FDC on the full algebra (with possibly non-symmetric gates) is symmetric-FDC-equivalent to α_TW^k for some k∈{0,1,2,3} (the §5.5.4 Assumption; odd-p version eq. (6.93)).
    Load-bearing for (ST)^3 = semion QCA, for centrality of [α_framing∘α_TW^2] and [α_Y], and hence for the Clifford-group identification in the abstract. The authors flag it: 'perhaps a little stronger' than the physical SPT assumption, and 'we do not currently have a proof'.
  • domain assumption The separator Hamiltonian of §5.5.2 admits a commuting-projector boundary realization with semion topological order, i.e., the constructed QCA lies in the semion Witt class (k=1) rather than the YY' class ((1,1)).
    Used in §5.5.4 to fix the Witt class of (ST)^3 after the group-theoretic argument fixes m=1, j=0; the authors call it a 'standard assumption' and note the alternative would be realized for Z_4^(1) symmetry.
  • domain assumption Witt-theoretic input from [30]: exact sequence (3.11), the s-twisted pre-metric group description and triviality criterion (Lemma 3.2), and the p=2 cohomology H^5(B^3(Z_2⊕Z_2),C^×) = Z_2⊕Z_2; plus [49]'s pointed Witt group (3.44)-(3.45).
    The framework is cited, not rederived; the paper corrects eq. (5.17) of [30] to its eq. (3.17) and derives the odd-p H^5 vanishing itself (Appendix A.4).
  • domain assumption The conjectural classification of 3+1d QCA by the (pointed) Witt group of abelian anyon theories, and its promotion to the graded Witt group for symmetric QCA.
    The paper explicitly says the Witt classification of ordinary QCA is conjectural and that 'it is natural to conjecture' the graded promotion; the order-8 semion QCA has not been explicitly written down.
  • domain assumption External QCA facts: β̃_p^{(p-1)/2} has order 2 (p≡1 mod 4) / 4 (p≡3 mod 4) modulo translations and FDCs, from [34] with ancilla-removal [58]; the framing QCA of [35] is in the 3-fermion class; the 3-fermion QCA (order 2) is the seminal example [31,32].
    Used directly in §4.8-4.9 and §6.6 for the order and Witt class of the lattice products; these are external results the paper does not rederive.

pith-pipeline@v1.3.0-alltime-deepseek · 72414 in / 23747 out tokens · 199215 ms · 2026-08-01T06:57:55.465734+00:00 · methodology

0 comments
read the original abstract

Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a $\mathbb{Z}_p$ one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality $S$, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT $T$. In the continuum, they form a central extension of $PSL(2,\mathbb{Z}_4)$ for $p=2$, and of $SL(2,\mathbb{Z}_p)$ for odd primes $p$, whose central elements are invertible theories with purely gravitational response. These central extensions are governed by a twisted, graded generalization of the Witt group of abelian anyon theories, which we determine. For $p=2$ the resulting group is the single-qubit Clifford group, with duality and entangler acting as the Hadamard and phase gates. We realize this entire structure microscopically as quantum cellular automata (QCA) acting on a certain local operator algebra associated with a spin lattice Hilbert space on a cubic lattice. Specifically, our local operator algebra is built by starting with all local operators commuting with a $\mathbb{Z}_p$ 1-form symmetry, and taking the quotient by all the (local) 1-form symmetry generators. The central elements can always be extended to the full tensor product algebra with a uniquely defined QCA class. For $p=2$ they are generated by the non-trivial semion QCA, and for odd prime $p$ they are generated by the non-trivial $\mathbb{Z}_p$ Clifford QCA. Consequently the lattice fusion rules reproduce the continuum ones only up to these QCAs and lattice translations, giving rise to fusion rules refined by QCAs.

Figures

Figures reproduced from arXiv: 2607.21698 by Kansei Inamura, Lukasz Fidkowski, Oskar Wojdel, Sakura Schafer-Nameki.

Figure 1
Figure 1. Figure 1: The half-space gauging construction of the duality [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The half-space gauging construction of the triality [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The half-space gauging construction of the quater [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The half-space gauging construction of the duality [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: The generators of the algebra of Z2 1-form sym￾metric local operators. Here, the solid lines represent edges of the dual lattice. where Σ is a closed surface on the direct lattice. We note that η(Σ) is not topological on the tensor product Hilbert space (4.1), nor as an operator on the full algebra of local operators (to be introduced in more detail later). Local operators symmetric under this non-topologi… view at source ↗
Figure 7
Figure 7. Figure 7: The symmetry operator X∂c around cube c. Here, the middle vertex represents the dual of c. for all cubes c, unless otherwise stated. Using the lan￾guage of states (on a finite lattice), this means restrict￾ing to the subspace of states for which that condition is true, which is no longer a tensor product of on-site Hilbert spaces. In the language of operator algebras, that is meant as an operator equation.… view at source ↗
Figure 8
Figure 8. Figure 8: The action of the KWW operator S on Z2 1- form symmetric operators illustrated on the dual lattice. The dashed lines represent the edges on which the operators on the left-hand side are supported. The KWW operator S 14 is a non-invertible operator that implements the gauging of the Z2 1-form symmetry generated by (4.3). Its action on Z2 1-form symmetric operators in (4.4) is given by [24] Xf S 7−→ Z δt 1 2… view at source ↗
Figure 9
Figure 9. Figure 9: The action of the Tsui-Wen entangler U on the Pauli X operator. The figure is illustrated on the dual lattice. The action of T 2 on the Pauli Z operator is trivial. Zf X X X Zf X X X Zf [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The fermion hopping operator ˜uf illustrated on the dual lattice. 4.5. Framing QCA In this subsection, we review a non-trivial QCA con￾structed in [35], which is in the same equivalence class as the 3-fermion QCAs in [31, 32]. A 3-fermion QCA was originally constructed in [32] as a disentangler of the Walker-Wang model based on the 3-fermion MTC. The original 3-fermion QCA in [32] and its simplified versi… view at source ↗
Figure 11
Figure 11. Figure 11: The fermion flux operator ˜uδe illustrated on the dual lattice. The red dot represents the dual of cube c where the operator X∂c acts. We recall our convention that Pauli X’s act before Pauli Z’s. will only compute the action on Z2 1-form symmetric lo￾cal operators. We note that such operators are generated by Xf and ˜uδe because Zδe is obtained from Xf and ˜uδe due to (4.62). Thus, it suffices to compute… view at source ↗
Figure 12
Figure 12. Figure 12: The action of Uframing on the Pauli X operator. The figure is illustrated on the dual lattice. The action of Uframing on the fermionic flux operator ˜uδe is trivial. 4.6. 3-fermion Kramers-Wannier-Wegner QCA Using the KWW operator S and the Tsui-Wen entan￾gler T 2 , we can define the following operator: D3F := ST2ST2S † = ST2ST2St −1 . (4.72) We refer to this operator as the 3-fermion Kramers￾Wannier-Wegn… view at source ↗
Figure 13
Figure 13. Figure 13: The action of the 3-fermion Kramers-Wannier [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: The flux operator Zδe for each edge e. Here, the plaquette in each diagram represents the dual of e. coboundary δe (defined by (B.9)) because Zf is not of order 2. The operator Zδe is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p039_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The Gauss law operator Gf illustrated on the direct lattice. Here, the middle plaquette in each diagram represents face f. The action (6.8) can also be expressed in terms of the cup product as Xf Sp 7−→ Y e∈E Z − R f⌣e δe , Zδe Sp 7−→ Y f∈F X R e⌣f f . (6.13) As in the Z2 case, Sp maps the symmetry operator X∂c to the identity operator. Thus, Sp is not technically a QCA on AZ (1) p , but defines a QCA on … view at source ↗
Figure 17
Figure 17. Figure 17: The symmetry operator Zδv of the gauged model. The figure is illustrated on the direct lattice, where the middle vertex is v. 1-form symmetry generated by the Wilson surface oper￾ators [36]. The symmetry operators on the lattice are given by Zb = Y e∈E Z b(e) e , (6.23) where b is an arbitrary 1-cocycle on the direct lattice. We can think of Zb as an operator supported on a closed surface on the dual latt… view at source ↗
Figure 18
Figure 18. Figure 18: The action of the powers of the Tsui-Wen entangler generator [PITH_FULL_IMAGE:figures/full_fig_p043_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The action of β˜(k) p on the Pauli X operator. The figure is drawn on the dual lattice. β˜(k) p (Zδe) = Ue(−k) δe † h β˜(k) p (Xt 1 2 (e) ) i−kse h β˜(k) p (Xt − 1 2 (e) ) ikse Y c ′∈C h β˜(k) p (X∂c′ ) ik R c ′⌣1e = Ue(−k) δe † h β˜(k) p (Xt 1 2 (e) ) i−kse h β˜(k) p (Xt − 1 2 (e) ) ikse Y c ′∈C X k R c ′⌣1e ∂c′ . (6.76) Here, the second equality follows from (6.61). By plugging (6.68) and (6.75) into th… view at source ↗
Figure 20
Figure 20. Figure 20: The action of β˜(k) p on the flux operator Zδe for an x-link e. Here, the bottom-left plaquette is the dual of e. The action on Zδe for y-links and z-links can also be illustrated similarly. See [PITH_FULL_IMAGE:figures/full_fig_p045_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The faces on the boundary of cube (•, •, •). (•, 0, 1) (•, 1, 1) (•, 1, 0) (•, 0, 0) (0, •, 0) (1, •, 0) (0, •, 1) (1, •, 1) (0, 0, •) (1, 0, •) (0, 1, •) (1, 1, •) [PITH_FULL_IMAGE:figures/full_fig_p056_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The edges on the boundary of cube (•, •, •). using the 3-fermion QCA in [31, Appendix E]. C.1. Kramers-Wannier-Wegner operator revisited As we will see later, our definition of the 3-fermion KWW operator is motivated by a specific representation of the ordinary KWW operator. As such, we start by revisiting the KWW operator in this subsection. More specifically, the goal of this subsection is to represent … view at source ↗
Figure 23
Figure 23. Figure 23: The action of the 3-fermion QCA α3F on X (A) f illustrated on the dual lattice. α3F(X (B) f ) = u˜ (A) u˜ (A) u˜ (A) u˜ (A) X (B) f X (A) ∂c X (A) ∂c , u˜ (A) u˜ (A) u˜ (A) u˜ (A) X (B) f X (A) ∂c X (A) ∂c , u˜ (A) u˜ (A) u˜ (A) u˜ (A) X (B) f X (A) ∂c X (A) ∂c [PITH_FULL_IMAGE:figures/full_fig_p060_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: The action of the 3-fermion QCA α3F on X (B) f illustrated on the dual lattice. that D3F admits an adaptive circuit representation with a single round of measurement, because U3F is not a finite￾depth circuit [31, 32]. Nevertheless, D3F does admit an adaptive circuit representation with multiple measure￾ment rounds. Indeed, as we will see later, D3F is equal to ST2ST2S † as an operator, and the latter adm… view at source ↗

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