We investigate topological phase transitions in a family of deformed $\mathbb Z_3$ toric-code wavefunctions prepared from a cluster state by local deformations and projective measurements. Their norms map to the $Q=3$ Potts model for single-parameter deformations and to a three-state Ashkin--Teller-like (AT $_3$ ) construction with two independent four-spin couplings in the general case. Projected entangled-pair-state (PEPS) and variational uniform matrix-product-state (VUMPS) calculations identify the toric-code (TC) phase and phases in which electric ( $e$ ) anyons are confined or condensed. These phases are separated by critical structures with central charges $c=4/5$ , $8/5$ , and isolated $c=1$ antiferromagnetic (AFM) endpoints. A normalized finite-distance $e$ -anyon pair-state norm provides a Fredenhagen--Marcu-type check of the confinement boundary, while the topological data of the quantum double $D(\mathbb Z_3)$ imply a topological entanglement entropy $γ=\log3$ throughout the gapped toric-code phase. Relative to the $\mathbb Z_2$ case, the absence of sign-change folding leaves the AFM endpoints unfolded, and the extreme deformation reaches square ice with an emergent $U(1)$ one-form symmetry, Hilbert-space fragmentation, and exact scar configurations.We investigate topological phase transitions in a family of deformed $\mathbb Z_3$ toric-code wavefunctions prepared from a cluster state by local deformations and projective measurements. Their norms map to the $Q=3$ Potts model for single-parameter deformations and to a three-state Ashkin--Teller-like (AT$_3$) construction with two independent four-spin couplings in the general case. Projected entangled-pair-state (PEPS) and variational uniform matrix-product-state (VUMPS) calculations identify the toric-code (TC) phase and phases in which electric ($e$) anyons are confined or condensed. These phases are separated by critical structures with central charges $c=4/5$, $8/5$, and isolated $c=1$ antiferromagnetic (AFM) endpoints. A normalized finite-distance $e$-anyon pair-state norm provides a Fredenhagen--Marcu-type check of the confinement boundary, while the topological data of the quantum double $D(\mathbb Z_3)$ imply a topological entanglement entropy $γ=\log3$ throughout the gapped toric-code phase. Relative to the $\mathbb Z_2$ case, the absence of sign-change folding leaves the AFM endpoints unfolded, and the extreme deformation reaches square ice with an emergent $U(1)$ one-form symmetry, Hilbert-space fragmentation, and exact scar configurations.

quant-pharXiv:2603.09107Mar 11, 20260 citations

Quantum error correction beyond the toric code: dynamical systems meet encoding

Garima Rajpoot, Komal Kumari, Sudhir R. Jain

arXiv:2307.04418The European Physical Journal Special TopicsAug 2, 20231 citation

Self-Dual Higgs Transitions: Toric Code and Beyond

Wenjie Ji, Ryan A. Lanzetta, Zheng Zhou +1 more

The toric code, when deformed in a way that preserves the self-duality Z_{2} symmetry exchanging the electric and magnetic excitations, admits a transition to a topologically trivial state that spontaneously breaks the Z_{2} symmetry. Numerically, this transition was found to be continuous, which makes it particularly enigmatic given the longstanding absence of a continuum field-theoretic description. In this Letter we propose such a continuum field theory for the transition dubbed the SO(4)_{2,-2} Chern-Simons-Higgs (CSH) theory. We show that our field theory provides a natural "mean-field" understanding of the phase diagram. Moreover, it can be generalized to an entire series of theories, namely the SO(4)_{k,-k} CSH theories, labeled by an integer k. For each k>2, the theory describes an analogous transition involving different non-Abelian topological orders, such as the double Fibonacci order (k=3) and the S_{3} quantum double (k=4). For k=1, we conjecture that the corresponding CSH transition is in fact infrared dual to the 3d Ising transition, in close analogy with the particle-vortex duality of a complex scalar.The toric code, when deformed in a way that preserves the self-duality Z_{2} symmetry exchanging the electric and magnetic excitations, admits a transition to a topologically trivial state that spontaneously breaks the Z_{2} symmetry. Numerically, this transition was found to be continuous, which makes it particularly enigmatic given the longstanding absence of a continuum field-theoretic description. In this Letter we propose such a continuum field theory for the transition dubbed the SO(4)_{2,-2} Chern-Simons-Higgs (CSH) theory. We show that our field theory provides a natural "mean-field" understanding of the phase diagram. Moreover, it can be generalized to an entire series of theories, namely the SO(4)_{k,-k} CSH theories, labeled by an integer k. For each k>2, the theory describes an analogous transition involving different non-Abelian topological orders, such as the double Fibonacci order (k=3) and the S_{3} quantum double (k=4). For k=1, we conjecture that the corresponding CSH transition is in fact infrared dual to the 3d Ising transition, in close analogy with the particle-vortex duality of a complex scalar.

arXiv:2601.20945Physical Review LettersJun 22, 20260 citations

Clifford-hierarchy stabilizer formalism with applications to twisted quantum doubles

Christopher Fechisin, Kieran Cooney, Mathi Raja +4 more

Recent work has shown that quantum double models can be used to circumvent the Bravyi-Koenig bound. In particular, this is achieved through code-switching protocols in which non-Pauli stabilizer phases appear as intermediate codes. We introduce the $X\mathcal C^{(k)}_d$ stabilizer formalism, which generalizes the Pauli stabilizer formalism by combining Pauli- $X$ operators with diagonal operators drawn from a fixed level $k$ of the Clifford hierarchy on prime-dimensional qudits. This yields a natural class of beyond-Pauli stabilizer models, including many twisted quantum double phases. We classify a large class of these $X\mathcal C^{(k)}_d$ stabilizer states, showing that they are uniform superpositions over affine-linear subspaces with diagonal phases. In particular, when the stabilizers of a state require operators from level $k$ of the Clifford hierarchy, we show that the diagonal gates required to produce the phases in the wavefunction sit one level higher. We further show that every 2D twisted quantum double ${\cal D}^ω(\mathbb{Z}_p^n)$ with $ω$ a product of type-III cocycles and $p$ an odd prime admits a realization using only Clifford ( $k=2$ ) stabilizers, and we conjecture that phases where $ω$ contains a type-I or type-II cocycle require level- $p$ Clifford-hierarchy stabilizers for $p$ -dimensional qudits. Finally, we demonstrate that a pair of qudit toric codes may be deterministically prepared in a logical magic state by measuring $X\mathcal C^{(k)}_d$ stabilizers. Together, these results provide a concrete stabilizer framework for topological codes beyond Pauli stabilizer models and their logical gates.Recent work has shown that quantum double models can be used to circumvent the Bravyi-Koenig bound. In particular, this is achieved through code-switching protocols in which non-Pauli stabilizer phases appear as intermediate codes. We introduce the $X\mathcal C^{(k)}_d$ stabilizer formalism, which generalizes the Pauli stabilizer formalism by combining Pauli-$X$ operators with diagonal operators drawn from a fixed level $k$ of the Clifford hierarchy on prime-dimensional qudits. This yields a natural class of beyond-Pauli stabilizer models, including many twisted quantum double phases. We classify a large class of these $X\mathcal C^{(k)}_d$ stabilizer states, showing that they are uniform superpositions over affine-linear subspaces with diagonal phases. In particular, when the stabilizers of a state require operators from level $k$ of the Clifford hierarchy, we show that the diagonal gates required to produce the phases in the wavefunction sit one level higher. We further show that every 2D twisted quantum double ${\cal D}^ω(\mathbb{Z}_p^n)$ with $ω$ a product of type-III cocycles and $p$ an odd prime admits a realization using only Clifford ($k=2$) stabilizers, and we conjecture that phases where $ω$ contains a type-I or type-II cocycle require level-$p$ Clifford-hierarchy stabilizers for $p$-dimensional qudits. Finally, we demonstrate that a pair of qudit toric codes may be deterministically prepared in a logical magic state by measuring $X\mathcal C^{(k)}_d$ stabilizers. Together, these results provide a concrete stabilizer framework for topological codes beyond Pauli stabilizer models and their logical gates.

quant-pharXiv:2610.00464Oct 2, 20260 citations

The β-Ladder of Exactly Solvable Quantum Models: XXZ, TFIM, String Nets, and Quantum Doubles as a Single ISA Family

Ian Richard C Buckley

We show that every exactly solvable quantum spin model with a phase transition is a different β-schedule of the Origami instruction-set architecture (ISA). The unifying equation is q = exp(iπβ): the quantum group deformation parameter q underlying the XXZ chain, the transverse-field Ising model (TFIM), Levin–Wen string-net models, and Kitaev's quantum doubles is identically the Gibbs ISA inverse temperature. This gives a single β-axis on which every topological phase of matter has a precise address. The main results are: (i) The XXZ Bethe-ansatz two-magnon R-matrix is the FLIP eigenvalue at q = exp(iπβ), with the BKT transition as a TWIST failure at β = 1/2 (exact, by cos(π/2) = 0). (ii) The 1D XY model and the 2D Kitaev toric code share β = 1/4 and quantum group SU(2)₂ — the same ISA programme in different geometries. (iii) For non-Abelian string-net phases in the SU(2)k family, the quantum dimension d{1/2}(β) = 2 cos(πβ) is an exact phase label, verified for k = 1 through 15. (iv) The cohomological chain H⁰/H¹/H² = Islands/Flatland/The Deep corresponds to classical/toric-code/Walker–Wang codes. Fracton phases require twisted cohomology H*(M; L) beyond standard H². The β-ladder (Table I) places every major model on a single axis: β = 0 (Heisenberg SU(2)), β = 1/5 (Fibonacci anyons, SU(2)₃), β = 1/4 (Ising anyons / toric code, SU(2)₂), β = 1/3 (semion / XY model, SU(2)₁), β = 1/2 (BKT transition, TWIST failure). All results are verified numerically in experiments x446a–x446d. This paper belongs to the Origami ISA series. Paper 443 (Planck's Constant in Disguise, doi:10.5281/zenodo.20752384) establishes the semiring deformation; Paper 445 (Kitaev Menagerie, doi:10.5281/zenodo.20752352) treats the Kitaev models as a special case; Paper 447 (Opcode Rosetta Stone, doi:10.5281/zenodo.20761260) extends the ISA dictionary to twelve models. Keywords exactly solvable models, quantum groups, XXZ model, BKT transition, topological phases, string-net models, toric code, Walker-Wang model, fracton phases, instruction set architecture, beta deformation, Yang-Baxter equation, Bethe ansatz, quantum dimension, Luttinger liquid, cohomology, tensor networks, quantum computingWe show that every exactly solvable quantum spin model with a phase transition is a different β-schedule of the Origami instruction-set architecture (ISA). The unifying equation is q = exp(iπβ): the quantum group deformation parameter q underlying the XXZ chain, the transverse-field Ising model (TFIM), Levin–Wen string-net models, and Kitaev's quantum doubles is identically the Gibbs ISA inverse temperature. This gives a single β-axis on which every topological phase of matter has a precise address. The main results are: (i) The XXZ Bethe-ansatz two-magnon R-matrix is the FLIP eigenvalue at q = exp(iπβ), with the BKT transition as a TWIST failure at β = 1/2 (exact, by cos(π/2) = 0). (ii) The 1D XY model and the 2D Kitaev toric code share β = 1/4 and quantum group SU(2)₂ — the same ISA programme in different geometries. (iii) For non-Abelian string-net phases in the SU(2)k family, the quantum dimension d{1/2}(β) = 2 cos(πβ) is an exact phase label, verified for k = 1 through 15. (iv) The cohomological chain H⁰/H¹/H² = Islands/Flatland/The Deep corresponds to classical/toric-code/Walker–Wang codes. Fracton phases require twisted cohomology H*(M; L) beyond standard H². The β-ladder (Table I) places every major model on a single axis: β = 0 (Heisenberg SU(2)), β = 1/5 (Fibonacci anyons, SU(2)₃), β = 1/4 (Ising anyons / toric code, SU(2)₂), β = 1/3 (semion / XY model, SU(2)₁), β = 1/2 (BKT transition, TWIST failure). All results are verified numerically in experiments x446a–x446d. This paper belongs to the Origami ISA series. Paper 443 (Planck's Constant in Disguise, doi:10.5281/zenodo.20752384) establishes the semiring deformation; Paper 445 (Kitaev Menagerie, doi:10.5281/zenodo.20752352) treats the Kitaev models as a special case; Paper 447 (Opcode Rosetta Stone, doi:10.5281/zenodo.20761260) extends the ISA dictionary to twelve models. Keywords exactly solvable models, quantum groups, XXZ model, BKT transition, topological phases, string-net models, toric code, Walker-Wang model, fracton phases, instruction set architecture, beta deformation, Yang-Baxter equation, Bethe ansatz, quantum dimension, Luttinger liquid, cohomology, tensor networks, quantum computing

Jun 19, 20260 citations

Topological phase transition of deformed

{\mathbb Z}_3

toric code

We investigate topological phase transitions in a family of deformed

\mathbb Z_3

toric-code wavefunctions prepared from a cluster state by local deformations and projective measurements. Their norms map to the

Q=3

Potts model for single-parameter deformations and to a three-state Ashkin--Teller-like (AT

_3

) construction with two independent four-spin couplings in the general case. Projected entangled-pair-state (PEPS) and variational uniform matrix-product-state (VUMPS) calculations identify the toric-code (TC) phase and phases in which electric (

e

) anyons are confined or condensed. These phases are separated by critical structures with central charges

c=4/5

8/5

, and isolated

c=1

antiferromagnetic (AFM) endpoints. A normalized finite-distance

e

-anyon pair-state norm provides a Fredenhagen--Marcu-type check of the confinement boundary, while the topological data of the quantum double

D(\mathbb Z_3)

imply a topological entanglement entropy

γ=\log3

throughout the gapped toric-code phase. Relative to the

\mathbb Z_2

case, the absence of sign-change folding leaves the AFM endpoints unfolded, and the extreme deformation reaches square ice with an emergent

U(1)

one-form symmetry, Hilbert-space fragmentation, and exact scar configurations.

X\mathcal C^{(k)}_d

stabilizer formalism, which generalizes the Pauli stabilizer formalism by combining Pauli-

X

operators with diagonal operators drawn from a fixed level

k

of the Clifford hierarchy on prime-dimensional qudits. This yields a natural class of beyond-Pauli stabilizer models, including many twisted quantum double phases. We classify a large class of these

X\mathcal C^{(k)}_d

stabilizer states, showing that they are uniform superpositions over affine-linear subspaces with diagonal phases. In particular, when the stabilizers of a state require operators from level

k

of the Clifford hierarchy, we show that the diagonal gates required to produce the phases in the wavefunction sit one level higher. We further show that every 2D twisted quantum double

{\cal D}^ω(\mathbb{Z}_p^n)

with

ω

a product of type-III cocycles and

p

an odd prime admits a realization using only Clifford (

k=2

) stabilizers, and we conjecture that phases where

ω

contains a type-I or type-II cocycle require level-

p

Clifford-hierarchy stabilizers for

p

-dimensional qudits. Finally, we demonstrate that a pair of qudit toric codes may be deterministically prepared in a logical magic state by measuring

X\mathcal C^{(k)}_d

stabilizers. Together, these results provide a concrete stabilizer framework for topological codes beyond Pauli stabilizer models and their logical gates.