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Geometric-Symmetry Logical Gate and Local-Probe Selectivity in the Three-Leg AKLT Ladder

T0 review · 1 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read In the three-leg AKLT ladder, a local probe's reach into the edge code is fixed by its spin rank: rank-ℓ operators decay with the ℓ-th transfer-matrix sector's correlation length, and leg exchange yields a six-qubit permutation gate.

desk verdict A careful, exact-solvable analysis of the three-leg AKLT ladder that delivers a concrete geometric gate and a selection-rule decay law; the central claims hold up, with only the usual unproven injectivity caveat. read the letter →

arxiv 2608.03861 v1 pith:6HBEA5P5 submitted 2026-08-04 quant-ph

Jingnuo Han , Youning Li This is my paper · ORCID
classification quant-ph
keywords AKLTladdersymmetry-protectedtopologicalordermatrixproductstatelogicalgatelocalprobeWigner–EckartselectionruletransferHaldanephase
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 studies the three-leg AKLT ladder, an exactly solvable spin ladder whose open-boundary ground space encodes six logical qubits protected by SO(3)×Z2 symmetry. It establishes two linked facts. First, the leg-exchange symmetry acts on the encoded space as a logical permutation, SWAP13⊗SWAP13, that cannot be factored into single-qubit gates—something no single-chain AKLT system offers. Second, a single-site local probe with spherical-tensor rank ℓ can reach the edge code only through the L=ℓ angular-momentum sector of the transfer matrix, so its distinguishability decays as A e^{-dist/ξ(L=ℓ)} with the sector correlation length. This gives a quantitative, symmetry-resolved law for how much local operators can see of boundary-encoded quantum information.

What carries the argument

The transfer matrix of the MPS, acting on the doubled virtual space (C²)⊗3⊗(C²)⊗3, together with its SO(3)×Z2 sector decomposition. Because the transfer matrix commutes with the adjoint Casimir, it block-diagonalizes into angular-momentum sectors L=0,1,2,3, refined by leg-exchange parity. The Wigner–Eckart selection rule then forces a rank-ℓ probe to propagate only through the L=ℓ sector, so its decay length is ξ(L=ℓ) = -1/ln|λ_L/λ_0|.

What would settle it

Compute δ_edge for the quadrupole probe (S^z)^2 in the spin-1 AKLT chain at N=20: the paper predicts the position-independent value (2√6/9)3^{-(N-1)} ≈ 4.68×10^{-10}. If an independent MPS calculation gives a different value or a position-dependent profile, the trace-subtraction and channel decomposition behind Eq. (80) are wrong. A second check: in the three-leg ladder, a P_tb-odd rank-1 probe must decay with ξ(L=1,−)=0.7907 rather than the even-sector length 1.3630.

Watch

Extended reading notes

Core claim

The central claim is that local accessibility of the SPT edge code is governed by a Wigner–Eckart selection rule: in the exact MPS representation, a rank-ℓ single-site operator couples the L=0 sector of the transfer matrix only to the L=ℓ sector, so the distinguishability measure δ_edge decays as δ_edge(dist)≃A e^{-dist/ξ(L=ℓ)} for 1≪dist≪N. For the three-leg ladder the relevant sectors L=1,2,3 coexist with correlation lengths 1.3630, 0.5418, and 0.2960, making the hierarchy observable; in the single chain no L=2 sector exists, so a quadrupole probe is exactly blind in the thermodynamic limit, with only a size-dependent residue (2√6/9)3^{-(N-1)}. Independently, the leg-exchange symmetry is s

Load-bearing premise

The argument assumes the ground-state tensor is injective and the two dominant boundary matrices derived from it are invertible; if either failed, the normalization step that converts raw boundary states into an orthonormal code basis would break, and the exponential decay law would not follow in its stated form.

Editorial extensions

If this is right

  • Local probes with the same SO(3) tensor rank produce identical distinguishability profiles; the operator's strength and microscopic form do not enter the decay rate.
  • In the spin-1 AKLT chain, rank-2 operators are exactly blind to the edge code in the thermodynamic limit, leaving only the position-independent finite-size residue (2√6/9)3^{-(N-1)}.
  • In the three-leg ladder, rank-2 noise decays with ξ=0.5418 before leveling off onto a size-dependent finite-size plateau, while rank-1 probes decay with ξ=1.3630.
  • The leg-exchange symmetry yields the logical permutation gate O(P_tb)=SWAP13⊗SWAP13, which is not a product of single-qubit unitaries; the gate set remains non-universal and no finite code distance is defined because boundary probes act with full strength.
  • The same selection-rule machinery predicts that multi-site error operators couple through Clebsch–Gordan composition, |ℓ1−ℓ2|≤L≤ℓ1+ℓ2, so two separated local errors should show exponential decay of distinguishability with their separation.

Reading between the lines

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

  • If the selection rule holds for arbitrary local error sets, passive protection can be tuned by symmetry labels: noise composed solely of rank-2 or higher tensors is exponentially suppressed in the bulk with short decay lengths, while rank-1 dephasing remains the dominant accessible channel.
  • The leg-exchange gate, although not universal, points to lattice geometry as a resource for implementing multi-qubit permutations in SPT codes without addressing individual qubits; combining such geometric gates with measurements could enlarge the accessible logical gate set.
  • Extending the same tensor construction to five- and seven-leg ladders or to cyclic tubes would test whether the hierarchy ξ(L=1)>ξ(L=2)>⋯ persists and whether cyclic leg permutations generate further discrete logical gates—a direction the paper explicitly leaves open.
  • A practical consequence of the boundary-probe behavior is that any useful passive code must treat the first unit cell as fully exposed; the protection is inherently about bulk operators, so realistic error models should separate boundary and bulk noise channels.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper constructs an exact matrix product state (MPS) representation for the three-leg AKLT ladder with on-site symmetry group SO(3)×Z2, and uses it to study boundary-encoded quantum information. It derives the parent Hamiltonian, verifies the nontrivial Haldane phase via cohomology (H^2 = Z2), string order, and entanglement spectrum. It shows that global SO(3) rotations induce a continuous family of logical rotations on the K = 64 edge code space, while the geometric top–bottom leg exchange induces the logical gate O(P_tb) = SWAP13⊗SWAP13, a genuinely multi-qubit permutation not decomposable into single-qubit unitaries. The main quantitative result is a symmetry-resolved decay law for the distinguishability measure δ_edge of local probes: a rank-ℓ single-site operator couples to the edge subspace only through the L = ℓ sector of the transfer matrix, giving δ_edge(dist) ≈ A e^{-dist/ξ(L=ℓ)} (Eq. 80). The derivation uses an exact channel decomposition of the transfer matrix (Appendix G4), and the predicted decay lengths are confirmed numerically for rank-1 and rank-2 operators. A finite-size, position-independent channel is also identified and analyzed.

Significance. If correct, the paper provides a clean, exactly solvable demonstration that lattice geometry (leg exchange) expands the set of symmetry-protected logical gates beyond what single-chain SPT systems offer, and that the SO(3) tensor rank of a local probe controls its exponential reach into the bulk. The analytic channel decomposition in Appendix G is a valuable technique, and the numerical verification is thorough: sector completeness at 10^-14, wrong-sector matrix elements at 10^-16, and fitted decay lengths agreeing with the sector correlation lengths to within a few percent. The paper is honest about the scope (gate set not universal, no finite code distance, passive protection). These are strong positives.

major comments (1)
  1. [Appendix G4a, Eqs. (G13), (G21)–(G22), Eq. (80)] The central decay law (80) is derived under Assumptions (A1)–(A3), stated but not proved for the three-leg ladder. In particular, (A3) requires the reshaped dominant fixed points r̂0 and l̂0 to be invertible; this is used to normalize P̃ in (G13) and to factor the traces in (G21), whose vanishing in (G22) eliminates all channels except L = ℓ. If r̂0 or l̂0 were singular, the Gram matrix would have rank < 64 for large N and the selection-rule decay law would not follow in the stated form. The numerical evidence (rank K = 64, sector completeness 1.03×10^-14) is suggestive but is not a direct check of the thermodynamic fixed-point matrices. Please either prove these assumptions for this AKLT construction—e.g., injectivity and positivity of the fixed point—or provide direct numerical verification (e.g., determinants/condition numbers of r̂0 and l̂0, and semisimplicity of T). This is the load
minor comments (5)
  1. [Appendix G4 (after Eq. G29 and Table IV)] The paragraph beginning 'The scalar channel (a) is proportional to I_K...' appears twice, verbatim. Please remove the duplicate.
  2. [Appendix G4i, Eq. (G29)] The notation '1/√4 2' in the displayed derivation is ambiguous; it should read (1/√K) × 2 with K = 4, or an explicit multiplication sign should be inserted.
  3. [Sec. V A] In the sentence 'According to the Knill–Laflamme conditions [3], When considering...', the word 'When' should be lowercase.
  4. [Appendix G4a, Eq. (G8)] The spectral decomposition should state explicitly that the eigenbasis is chosen to be symmetry-adapted with respect to J²_adj, so that the L labels are well defined even when an eigenvalue is shared across sectors (e.g., λ/λ0 = −0.0341 appears in three sub-sectors in Table II).
  5. [Sec. V B, Table III] The fitted decay lengths differ from the theoretical sector lengths by 1–6% (e.g., 1.3459 vs 1.3630 for S^z_rung; 0.5090 vs 0.5418 for ∑(S^z_leg)^2). The text explains this as a fitting-window effect but does not report the restricted-fit values that 'can bring every probe to the sector prediction.' Including those values would quantitatively substantiate the claim of agreement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (80) is derived from the transfer-matrix spectrum, and the fitted decay lengths are independent confirmations rather than inputs.

full rationale

The paper's central result, δ_edge(dist) ≃ A e^{-dist/ξ(L=ℓ)} (Eq. 80), is obtained in Appendix G from an exact channel decomposition (Eqs. G9–G15) of the transfer matrix, with the Wigner–Eckart selection rule (Eq. G24) applied to the tensor operator T_F. The sector correlation lengths ξ(L=ℓ) in Eq. (78) are computed directly from the transfer-matrix eigenvalues (Table III), before and independently of the numerical fits in Table II; no fitted parameter is fed back into the derivation. The leg-exchange gate O(P_tb)=SWAP13⊗SWAP13 (Eq. 71) is derived from the virtual representation V(P_tb) obtained by solving the intertwining null-space system (Appendix B) and verified to machine precision; non-factorizability is checked against all tensor products of single-qubit unitaries (Appendix F). Citations to [4–7], [18,19], and [35–37] are external standard results, not self-citations, and no 'uniqueness theorem' from the authors is invoked to forbid alternatives. The only caveat is Appendix G4a, which states Assumptions (A1)–(A3): injectivity and diagonalizability of T, and invertibility of the reshaped fixed-point matrices r̂0 and l̂0. These are stated ('We assume the following') but not proved for the three-leg ladder. That is a rigor/completeness gap, not a circularity: none of the assumptions contains Eq. (80) or the fitted values, and the numerical sector completeness (1.03×10^{-14}) makes them plausible. The derivation chain is therefore self-contained against the model; the numerical checks are confirmatory, not constitutive.

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

The central derivation introduces no free physical parameters or invented entities. The only fitted numbers appear in numerical verification and do not shape the theory. The analytical claims rest on standard MPS, group cohomology, and representation theory, plus the non-proven but reasonable injectivity/invertibility assumptions of Appendix G4.

free parameters (3)
  • Exponential prefactor A in δ_edge fits = not reported per probe; fit-dependent
    Used only in ordinary least-squares verification of the decay law δ_edge = A e^{-dist/ξ}; not an input to the analytical derivation.
  • Plateau offset C in rank-2 ladder fits = ≈1.4e-6 at N=20; ≈6.5e-15 at N=60
    Nonlinear fit parameter in δ_edge = A e^{-dist/ξ} + C for the summed quadratic ladder probes; absorbs the finite-size channel, not used in the derivation.
  • Fitted decay length ξ_fit = 0.9105 (chain rank-1), 1.3459 (ladder rank-1), 0.5090 (ladder rank-2)
    Fitted per operator to compare with the theoretical sector correlation lengths ξ(L=ℓ); verification only, not load-bearing.
assumptions (5)
  • standard math Standard MPS and transfer-matrix formalism: ground state given by uniform MPS, overlaps given by transfer-matrix powers, entanglement spectrum from fixed points.
    Used in Sec. II and Appendix E; standard results from Refs. [17-20, 37].
  • domain assumption Injective MPS, diagonalizable transfer matrix, and invertible reshaped fixed-point matrices r̂0, l̂0 (Appendix G4, assumptions A1-A3).
    Needed for the channel decomposition Eq. (G12)-(G15) and the decay law Eq. (G28); stated but not proven for the three-leg ladder.
  • standard math Group cohomology facts: Künneth formula, H^2(SO(3),U(1))=Z2, H^1(SO(3),U(1))=0, H^2(Z2,U(1))=0.
    Sec. III A and Appendix C; establishes the SPT classification used to identify the Haldane phase.
  • standard math Wigner-Eckart theorem and Schur's lemma apply to the transfer-matrix operator T_F as an SO(3) tensor operator.
    Sec. V B and Appendix G4; yields the rank-ℓ selection rule and sector block structure.
  • domain assumption The parent Hamiltonian is frustration-free, the constructed MPS is its ground state, and the bulk is gapped.
    Sec. II A; gappedness is stated as expected rather than proved, but the central MPS claims do not require an explicit gap bound.

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Pith. "Pith review of Geometric-Symmetry Logical Gate and Local-Probe Selectivity in the Three-Leg AKLT Ladder." pith.science (2026). https://pith.science/paper/6HBEA5P5

@misc{pith2026260803861,
  author       = {Pith},
  title        = {Pith review of: Geometric-Symmetry Logical Gate and Local-Probe Selectivity in the Three-Leg AKLT Ladder},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6HBEA5P5}},
  note         = {Machine review of arXiv:2608.03861}
}
abstract

Symmetry-protected topological (SPT) phases provide a platform for encoding quantum information in protected boundary degrees of freedom. Here we study the three-leg Affleck-Kennedy-Lieb-Tasaki (AKLT) ladder as an exactly solvable SPT system with on-site symmetry $SO(3)\times \mathbb{Z}_2$. Using an exact matrix product state construction, we characterize the symmetry action on the edge encoding space and the accessibility of this space by local operators. We find that the continuous $SO(3)$ symmetry induces boundary rotations, while the leg-exchange symmetry generates a geometry-dependent logical permutation of edge qubits. Furthermore, by introducing a distinguishability measure motivated by the Knill--Laflamme condition, we derive a symmetry-resolved decay law for local accessibility. The decay is controlled by a selection rule raised from the Wigner--Eckart theorem, whereby a rank-$\ell$ local operator couples only to the $\mathcal L=\ell$ transfer-matrix sector, with a decay length determined by the corresponding correlation length. We further identify a finite-size channel that is independent of the probe operator position. These results establish a quantitative connection between SPT symmetry, lattice geometry, and the protection of boundary-encoded quantum information.

Figures

Figures reproduced from arXiv: 2608.03861 by the authors.

Figure 1
Figure 1. FIG. 1: Thermodynamic-limit string order [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Entanglement spectra of the chain, the two-leg ladder, and the three-leg ladder. [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (a) Edge-distinguishability [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗

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Reference graph

Works this paper leans on

54 extracted references · 44 canonical work pages

  1. [1]

    , d phys,(B1) 35 whereα g =±1 is a phase factor

    Linear constraint system The intertwining relation for a symmetry operationgreads ug :A s →A s g = X s′ [ug]ss′As′ =α g V(g)A sV(g) −1,∀s= 1, . . . , d phys,(B1) 35 whereα g =±1 is a phase factor. Multiplying both sides on the right byVgives As g V=α g V As.(B2) This is a linear equation in theD 2 unknown entries ofV. To make the linear structure explicit...

    work page

  2. [2]

    The results are summarized in Table IV

    V erification After extraction, we verify the intertwining relation by computing the maximum residual ϵ= max s ∥As g −α g V AsV −1∥F /max s ∥As∥F,(B5) where∥ · ∥F denotes the Frobenius norm. The results are summarized in Table IV. All errors are at machine precision, confirming the correctness of the extraction

    work page

  3. [3]

    A posteriori identification with Pauli tensor products To identify the extracted matrices with known operators, we compare eachV(g) with all 4 3 = 64 three-qubit Pauli tensor products{σ µ1 ⊗σ µ2 ⊗σ µ3}(whereσ 0 =I,σ 1 =σ x, 36 TABLE IV: Verification of the intertwining relation for each symmetry generator. Generator Null-space dim.α g ϵ gz 1 +1 4.83×10 −1...

    work page

  4. [4]

    The many-body state can be written as |Ψ⟩= DX α=1 |Lα⟩ ⊗ |Rα⟩,(E1) where|L α⟩and|R α⟩are the half-chain states associated with virtual boundary indexα

    Schmidt decomposition in the thermodynamic limit Consider an infinite uniform MPS cut into left and right half-infinite parts at a single virtual bond of dimensionD. The many-body state can be written as |Ψ⟩= DX α=1 |Lα⟩ ⊗ |Rα⟩,(E1) where|L α⟩and|R α⟩are the half-chain states associated with virtual boundary indexα. These states are generally not orthonor...

    work page

  5. [5]

    The transfer matrix is defined as (Eq

    Commutation of the edge state with the virtual symmetry We prove that the productG LGR of the left and right transfer-matrix fixed points com- mutes with the virtual SO(3) representationV(g) =u 1/2(g)⊗M . The transfer matrix is defined as (Eq. 13) T= dphys−1X s=0 As ⊗ As,(E8) acting on the doubled virtual space of dimensionD 2. The left and right fixed po...

    work page

  6. [6]

    Eigenvalue spectrum ofSW AP 13 andO(P tb) The operator SW AP13 acts on the three-qubit space (C 2)⊗3 by exchanging the first and third qubits. Its +1 eigenspace consists of states symmetric under this exchange: V+ = span{|000⟩,|010⟩,|111⟩,|101⟩, 1√ 2 (|001⟩+|100⟩), 1√ 2 (|011⟩+|110⟩)},(F1) of dimension 6, while the−1 eigenspace, V− = span{ 1√ 2 (|001⟩ − |...

    work page

  7. [7]

    The 64×64 matrix is reshaped into a rank-12 tensor with six input and six output indices, each of dimension 2

    Non-factorizability into single-qubit gates We verify thatO(P tb) cannot be written asU 1 ⊗ · · · ⊗U6 by the following procedure. The 64×64 matrix is reshaped into a rank-12 tensor with six input and six output indices, each of dimension 2. For each logical qubitq, an effective single-qubit matrix is extracted by fixing all other indices in the|0⟩state: [...

    work page

  8. [8]

    16) ug :A s →A s g = X s′ [ug]ss′As′ =α g V(g)A sV(g) −1, α g =±1,(G1) 44 whereu(g) is the physical representation andV(g) =u 1/2(g)⊗3 acts on the virtual space V= (C 2)⊗3

    Symmetry sectors of the transfer matrix The MPS tensorA s satisfies the intertwining relation (Eq. 16) ug :A s →A s g = X s′ [ug]ss′As′ =α g V(g)A sV(g) −1, α g =±1,(G1) 44 whereu(g) is the physical representation andV(g) =u 1/2(g)⊗3 acts on the virtual space V= (C 2)⊗3. Using the unitarity ofu(g), one shows [V(g)⊗V(g)]T[V(g)⊗V(g)] −1 =T∀g∈SO(3).(G2) Unde...

    work page

Show all 54 references
  1. [9]

    Spin-1 chain sectors The virtual space is spin- 1 2 (D= 2). The doubled space hasD 2 = 4 dimensions and decomposes as 1 2 ⊗ 1 2 = (L=0)⊕(L=1).(G6) This gives a one-dimensionalL= 0 sector and a three-dimensionalL= 1 sector, and no L ≥2 sector. The transfer-matrix eigenvalues no...

  2. [10]

    Three-leg ladder sectors The virtual space is ( 1 2 )⊗3 ∼= 3 2 ⊕ 1 2 ⊕ 1 2 (D= 8). The doubled space hasD 2 = 64 dimensions and decomposes as 8⊗8 = 5·(L=0)⊕9·(L=1)⊕5·(L=2)⊕1·(L=3),(G7) with total dimension 5 + 27 + 25 + 7 = 64, in agreement with the numerical diagonalization. ...

  3. [11]

    (80) This appendix derives Eq

    Channel decomposition and derivation of Eq. (80) This appendix derives Eq. (80) from the spectrum of the transfer matrix. Throughout, ηL ≡ |λL/λ0|, so thatξ(L) =−1/lnη L by Eq. (78). We writeη 1 for the largest such ratio over all non-leading eigenvalues. a. Spectral decomposi...

  4. [12]

    P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical review A52, R2493 (1995)

    1995

  5. [13]

    B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics 87, 307 (2015)

    2015

  6. [14]

    Knill and R

    E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Physical Review A55, 900 (1997)

    1997

  7. [15]

    Pollmann, E

    F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems, Physical Review B—Condensed Matter and Materials Physics85, 075125 (2012)

    2012

  8. [16]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, and X.-G. Wen, Classification of gapped symmetric phases in one- dimensional spin systems, Physical Review B—Condensed Matter and Materials Physics83, 035107 (2011)

    2011

  9. [17]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B87, 155114 (2013)

    2013

  10. [18]

    Schuch, D

    N. Schuch, D. P´ erez-Garc ´ ıa, and I. Cirac, Classifying quantum phases using matrix product states and projected entangled pair states, Phys. Rev. B84, 165139 (2011)

    2011

  11. [19]

    Senthil, Symmetry-protected topological phases of quantum matter, Annu

    T. Senthil, Symmetry-protected topological phases of quantum matter, Annu. Rev. Condens. Matter Phys.6, 299 (2015)

    2015

  12. [20]

    Kennedy and H

    T. Kennedy and H. Tasaki, Hidden symmetry breaking and the haldane phase in s= 1 quantum spin chains, Communications in mathematical physics147, 431 (1992)

    1992

  13. [21]

    P´ erez-Garc ´ ıa, M

    D. P´ erez-Garc ´ ıa, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac, String order and symmetries in quantum spin lattices, Physical review letters100, 167202 (2008)

    2008

  14. [22]

    Li and F

    H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification¡? format?¿ of topological order in non-abelian fractional quantum hall effect states, Physical review letters101, 010504 (2008). 54

    2008

  15. [23]

    Pollmann, A

    F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topolog- ical phase in one dimension, Physical Review B81, 064439 (2010)

    2010

  16. [24]

    Affleck and T

    I. Affleck and T. Kennedy, E. h. lieb, and h. tasaki, Phys. Rev. Lett59, 799802 (1987)

    1987

  17. [25]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Valence bond ground states in isotropic quantum antiferromagnets, Communications in Mathematical Physics115, 477 (1988)

    1988

  18. [26]

    Kennedy, Exact diagonalisations of open spin-1 chains, Journal of Physics: Condensed Matter2, 5737 (1990)

    T. Kennedy, Exact diagonalisations of open spin-1 chains, Journal of Physics: Condensed Matter2, 5737 (1990)

    1990

  19. [27]

    F. D. M. Haldane, Nonlinear field theory of large-spin heisenberg antiferromagnets: semiclas- sically quantized solitons of the one-dimensional easy-axis n´ eel state, Physical review letters 50, 1153 (1983)

    1983

  20. [28]

    Fannes, B

    M. Fannes, B. Nachtergaele, and R. F. Werner, Finitely correlated states on quantum spin chains, Communications in mathematical physics144, 443 (1992)

    1992

  21. [29]

    Perez-Garc ´ ıa, F

    D. Perez-Garc ´ ıa, F. Verstraete, M. M. Wolf, and J. I. Cirac, Matrix product state representa- tions, Quantum Information and Computation7, 401 (2007)

    2007

  22. [30]

    Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics326, 96 (2011)

    U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics326, 96 (2011)

    2011

  23. [31]

    J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Reviews of Modern Physics93, 045003 (2021)

    2021

  24. [32]

    Raussendorf and H

    R. Raussendorf and H. J. Briegel, A one-way quantum computer, Physical review letters86, 5188 (2001)

    2001

  25. [33]

    G. K. Brennen and A. Miyake, Measurement-based quantum computer in the gapped ground state¡? format?¿ of a two-body hamiltonian, Physical review letters101, 010502 (2008)

    2008

  26. [34]

    S. D. Bartlett, G. K. Brennen, A. Miyake, and J. M. Renes, Quantum computational renor- malization in the haldane phase, Physical review letters105, 110502 (2010)

    2010

  27. [35]

    T.-C. Wei, I. Affleck, and R. Raussendorf, Affleck-kennedy-lieb-tasaki state on a honeycomb lattice is a universal quantum computational resource, Physical review letters106, 070501 (2011)

    2011

  28. [36]

    Wei, Quantum spin models for measurement-based quantum computation, Advances in Physics: X3, 1461026 (2018)

    T.-C. Wei, Quantum spin models for measurement-based quantum computation, Advances in Physics: X3, 1461026 (2018)

    2018

  29. [37]

    D. V. Else, I. Schwarz, S. D. Bartlett, and A. C. Doherty, Symmetry-protected phases for 55 measurement-based quantum computation, Physical review letters108, 240505 (2012)

    2012

  30. [38]

    D. T. Stephen, D.-S. Wang, A. Prakash, T.-C. Wei, and R. Raussendorf, Computational power of symmetry-protected topological phases, Physical review letters119, 010504 (2017)

    2017

  31. [39]

    D. T. Stephen, W. W. Ho, T.-C. Wei, R. Raussendorf, and R. Verresen, Universal measurement-based quantum computation in a one-dimensional architecture enabled by dual- unitary circuits, Physical Review Letters132, 250601 (2024)

    2024

  32. [40]

    K. C. Smith, E. Crane, N. Wiebe, and S. Girvin, Deterministic constant-depth preparation of the aklt state on a quantum processor using fusion measurements, PRX Quantum4, 020315 (2023)

    2023

  33. [41]

    Miyake, Quantum computation on the edge of a symmetry-protected topological order, Physical review letters105, 040501 (2010)

    A. Miyake, Quantum computation on the edge of a symmetry-protected topological order, Physical review letters105, 040501 (2010)

    2010

  34. [42]

    Bacon, S

    D. Bacon, S. T. Flammia, and G. M. Crosswhite, Adiabatic quantum transistors, Physical Review X3, 021015 (2013)

    2013

  35. [43]

    D. J. Williamson and S. D. Bartlett, Symmetry-protected adiabatic quantum transistors, New Journal of Physics17, 053019 (2015)

    2015

  36. [44]

    de Groot, A

    C. de Groot, A. Turzillo, and N. Schuch, Symmetry protected topological order in open quan- tum systems, Quantum6, 856 (2022)

    2022

  37. [45]

    Dagotto and T

    E. Dagotto and T. Rice, Surprises on the way from one-to two-dimensional quantum magnets: The ladder materials, Science271, 618 (1996)

    1996

  38. [46]

    A. R. Edmonds,Angular momentum in quantum mechanics, Vol. 4 (Princeton university press, 1996)

    1996

  39. [47]

    Tung, Group theory in physics: An introduction to symmetry principles, group repre- sentations, and special functions in classical and quantum physics (2013)

    W.-K. Tung, Group theory in physics: An introduction to symmetry principles, group repre- sentations, and special functions in classical and quantum physics (2013)

    2013

  40. [48]

    Zauner, D

    V. Zauner, D. Draxler, L. Vanderstraeten, M. Degroote, J. Haegeman, M. M. Rams, V. Sto- jevic, N. Schuch, and F. Verstraete, Transfer matrices and excitations with matrix product states, New Journal of Physics17, 053002 (2015)

    2015

  41. [49]

    Nash and S

    C. Nash and S. Sen,Topology and geometry for physicists(Elsevier, 1988)

    1988

  42. [50]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry-protected topological orders in interacting bosonic systems, Science338, 1604 (2012)

    2012

  43. [51]

    M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information(Cambridge university press, 2010). 56

    2010

  44. [52]

    Eastin and E

    B. Eastin and E. Knill, Restrictions on transversal encoded quantum gate sets, Physical review letters102, 110502 (2009)

    2009

  45. [53]

    K. S. Brown,Cohomology of groups(Springer Science & Business Media, 2012)

    2012

  46. [54]

    C. C. Moore, Extensions and low dimensional cohomology theory of locally compact groups groups. i, Transactions of the American Mathematical Society113, 40 (1964). 57

    1964

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