We classify mobile Pauli stabilizer codes up to gapped interfaces and coarse-graining using the framework of algebraic $\mathrm{L}$ -theory. We compare this classification with that of framed TQFTs, theories that arise naturally in the continuum, highlighting a close structural relationship between the two. Our approach is formulated in the category of perfect chain complexes equipped with quadratic functor over the Laurent polynomial ring $R = \mathbb{Z}/p[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]$ , within which the collection of topological operators of Pauli stabilizer codes arise naturally as objects. In particular, we establish a bulk-boundary correspondence for lattice theories: the equivalence class of a Pauli stabilizer code up to gapped interface is described by a Clifford QCA in one dimension higher. This is done using the universal target category for stabilizer codes, which is the categorical spectrum whose existence and universal properties are introduced in this work. We conclude by highlighting subtle differences between the classification of Pauli stabilizer codes and TQFTs, leading to qualitative distinctions between lattice and continuum theories.We classify mobile Pauli stabilizer codes up to gapped interfaces and coarse-graining using the framework of algebraic $\mathrm{L}$-theory. We compare this classification with that of framed TQFTs, theories that arise naturally in the continuum, highlighting a close structural relationship between the two. Our approach is formulated in the category of perfect chain complexes equipped with quadratic functor over the Laurent polynomial ring $R = \mathbb{Z}/p[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]$, within which the collection of topological operators of Pauli stabilizer codes arise naturally as objects. In particular, we establish a bulk-boundary correspondence for lattice theories: the equivalence class of a Pauli stabilizer code up to gapped interface is described by a Clifford QCA in one dimension higher. This is done using the universal target category for stabilizer codes, which is the categorical spectrum whose existence and universal properties are introduced in this work. We conclude by highlighting subtle differences between the classification of Pauli stabilizer codes and TQFTs, leading to qualitative distinctions between lattice and continuum theories.

math-phCross-listedarXiv:2604.24847Apr 29, 20260 citations

部分次元粒子：格子模型、場の理論、背景の数学的構造

Taiichi Nakanishi

Topologically ordered phases and their continuum descriptions provide a rich arena in which quantum many-body systems, algebraic topology, and quantum field theory meet.Recently, foliated fracton phases with subdimensional particles, which are excitations constrained to move only along lower-dimensional submanifolds, have emerged as a new class of quantum matter with potential applications to robust quantum information storage.These phases are naturally tied to foliation structures of the spatial manifold and exhibit unusual ground state degeneracies and subsystem conservation laws that go beyond conventional topological order.In this thesis we investigate subdimensional particles from three complementary viewpoints: lattice stabilizer codes, mathematical structures on manifolds, and continuum gauge theories.As a warm-up, we revisit the toric code and show how its stabilizer structure, logical operators, and ground state degeneracy are encoded in cellular (co)homology.This "triangle of ideas" relating lattice models, BF-type topological quantum field theories, and topological invariants serves as a guiding paradigm.We then extend this perspective to foliated fracton phases, focusing on the plaquette Ising model and the X-cube model as paradigmatic examples.We clarify how their subextensive ground state degeneracies and rigid line and membrane operators are organized by foliation-sensitive (co)homological data of the underlying manifold.Motivated by these observations, we construct a Godbillon-Vey U(1) BF-like gauge theory coupled to a codimension-one foliation.By promoting the redundancies of the foliation data to gauge symmetries, the resulting continuum theory depends only on the foliation class and naturally realizes Wilson and higher-form operators whose mobility is restricted to the leaves.In (2+1)-and (3+1)-dimensions we show that this theory reproduces, respectively, the ground state degeneracy and operator algebra of the plaquette Ising and X-cube models, providing a foliated analogue of the BF-toric code correspondence.Finally, we introduce matter field theories coupled to the foliation background and to the BF-like gauge fields.For complex scalars, fermions, and compact scalars we construct conserved currents that obey leafwise conservation laws enforcing subdimensional kinematics.We close by outlining extensions to higher-codimension foliations, more general G-structures, and symmetry topological field theories, and by emphasizing a generalized triangle relating microscopic models, geometric invariants, and continuum field theories for subdimensional particle systems.Topologically ordered phases and their continuum descriptions provide a rich arena in which quantum many-body systems, algebraic topology, and quantum field theory meet.Recently, foliated fracton phases with subdimensional particles, which are excitations constrained to move only along lower-dimensional submanifolds, have emerged as a new class of quantum matter with potential applications to robust quantum information storage.These phases are naturally tied to foliation structures of the spatial manifold and exhibit unusual ground state degeneracies and subsystem conservation laws that go beyond conventional topological order.In this thesis we investigate subdimensional particles from three complementary viewpoints: lattice stabilizer codes, mathematical structures on manifolds, and continuum gauge theories.As a warm-up, we revisit the toric code and show how its stabilizer structure, logical operators, and ground state degeneracy are encoded in cellular (co)homology.This "triangle of ideas" relating lattice models, BF-type topological quantum field theories, and topological invariants serves as a guiding paradigm.We then extend this perspective to foliated fracton phases, focusing on the plaquette Ising model and the X-cube model as paradigmatic examples.We clarify how their subextensive ground state degeneracies and rigid line and membrane operators are organized by foliation-sensitive (co)homological data of the underlying manifold.Motivated by these observations, we construct a Godbillon-Vey U(1) BF-like gauge theory coupled to a codimension-one foliation.By promoting the redundancies of the foliation data to gauge symmetries, the resulting continuum theory depends only on the foliation class and naturally realizes Wilson and higher-form operators whose mobility is restricted to the leaves.In (2+1)-and (3+1)-dimensions we show that this theory reproduces, respectively, the ground state degeneracy and operator algebra of the plaquette Ising and X-cube models, providing a foliated analogue of the BF-toric code correspondence.Finally, we introduce matter field theories coupled to the foliation background and to the BF-like gauge fields.For complex scalars, fermions, and compact scalars we construct conserved currents that obey leafwise conservation laws enforcing subdimensional kinematics.We close by outlining extensions to higher-codimension foliations, more general G-structures, and symmetry topological field theories, and by emphasizing a generalized triangle relating microscopic models, geometric invariants, and continuum field theories for subdimensional particle systems.

Mar 23, 20260 citations

Topological subsystem bivariate bicycle codes with four-qubit check operators

Zijian Liang, Yu-An Chen

High-rate bivariate bicycle (BB) codes are promising low-overhead quantum memories, but their stabilizer checks typically have weight $6$ or higher, making syndrome extraction challenging. We introduce subsystem bivariate bicycle (SBB) codes, a translation-invariant CSS subsystem construction that realizes BB-code logical structure using local weight- $4$ gauge measurements. Their stabilizer syndromes are inferred by multiplying the corresponding gauge outcomes. We further show that nonlocal stabilizers in translation-invariant CSS subsystem codes can be detected using a determinantal-ideal criterion based on the gauge-operator commutation matrix. When this criterion excludes nonlocal stabilizers, a finite-depth Clifford circuit decouples gauge qubits and identifies the protected subsystem with a corresponding BB stabilizer code. An SBB code is topological, meaning that it has no nontrivial local logical operators, if and only if the corresponding BB code is topological. A finite search yields low-overhead examples including $[[27,6,3]]$ , $[[75,10,5]]$ , and $[[108,12,6]]$ ; the latter encodes six times more logical qubits than a subsystem surface code at the same block length and distance. These results show how gauge degrees of freedom can make high-rate BB logical structure compatible with local weight- $4$ syndrome extraction.High-rate bivariate bicycle (BB) codes are promising low-overhead quantum memories, but their stabilizer checks typically have weight $6$ or higher, making syndrome extraction challenging. We introduce subsystem bivariate bicycle (SBB) codes, a translation-invariant CSS subsystem construction that realizes BB-code logical structure using local weight-$4$ gauge measurements. Their stabilizer syndromes are inferred by multiplying the corresponding gauge outcomes. We further show that nonlocal stabilizers in translation-invariant CSS subsystem codes can be detected using a determinantal-ideal criterion based on the gauge-operator commutation matrix. When this criterion excludes nonlocal stabilizers, a finite-depth Clifford circuit decouples gauge qubits and identifies the protected subsystem with a corresponding BB stabilizer code. An SBB code is topological, meaning that it has no nontrivial local logical operators, if and only if the corresponding BB code is topological. A finite search yields low-overhead examples including $[[27,6,3]]$, $[[75,10,5]]$, and $[[108,12,6]]$; the latter encodes six times more logical qubits than a subsystem surface code at the same block length and distance. These results show how gauge degrees of freedom can make high-rate BB logical structure compatible with local weight-$4$ syndrome extraction.

quant-pharXiv:2605.04151May 7, 20260 citations

Quantum Mechanical Gauge Models

Franco Strocchi

Lecture notes in physicsNov 12, 20150 citations

Fiber Bundles and Gauge Theories

Arno Böhm, Ali Mostafazadeh, Hiroyasu Koizumi +2 more

Springer eBooksJan 1, 20030 citations

We classify mobile Pauli stabilizer codes up to gapped interfaces and coarse-graining using the framework of algebraic

\mathrm{L}

-theory. We compare this classification with that of framed TQFTs, theories that arise naturally in the continuum, highlighting a close structural relationship between the two. Our approach is formulated in the category of perfect chain complexes equipped with quadratic functor over the Laurent polynomial ring

R = \mathbb{Z}/p[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]

, within which the collection of topological operators of Pauli stabilizer codes arise naturally as objects. In particular, we establish a bulk-boundary correspondence for lattice theories: the equivalence class of a Pauli stabilizer code up to gapped interface is described by a Clifford QCA in one dimension higher. This is done using the universal target category for stabilizer codes, which is the categorical spectrum whose existence and universal properties are introduced in this work. We conclude by highlighting subtle differences between the classification of Pauli stabilizer codes and TQFTs, leading to qualitative distinctions between lattice and continuum theories.

High-rate bivariate bicycle (BB) codes are promising low-overhead quantum memories, but their stabilizer checks typically have weight

6

or higher, making syndrome extraction challenging. We introduce subsystem bivariate bicycle (SBB) codes, a translation-invariant CSS subsystem construction that realizes BB-code logical structure using local weight-

4

gauge measurements. Their stabilizer syndromes are inferred by multiplying the corresponding gauge outcomes. We further show that nonlocal stabilizers in translation-invariant CSS subsystem codes can be detected using a determinantal-ideal criterion based on the gauge-operator commutation matrix. When this criterion excludes nonlocal stabilizers, a finite-depth Clifford circuit decouples gauge qubits and identifies the protected subsystem with a corresponding BB stabilizer code. An SBB code is topological, meaning that it has no nontrivial local logical operators, if and only if the corresponding BB code is topological. A finite search yields low-overhead examples including

[[27,6,3]]

[[75,10,5]]

, and

[[108,12,6]]

; the latter encodes six times more logical qubits than a subsystem surface code at the same block length and distance. These results show how gauge degrees of freedom can make high-rate BB logical structure compatible with local weight-

4

syndrome extraction.