From b7f326e42c2a9dfde57d21186a0f27846cdacdcb Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Tue, 30 Jan 2024 17:18:48 +0100 Subject: [PATCH] trans inv subsystem --- .../block/topological/topological_abelian.yml | 10 +++-- .../topological_stabilizer/2d_stabilizer.yml | 2 +- .../topological_stabilizer/3d_stabilizer.yml | 13 ++----- .../topological_stabilizer/fracton.yml | 20 ++++++++-- .../translationally_invariant_stabilizer.yml | 9 +++-- .../translationally_invariant_subsystem.yml | 37 +++++++++++++++++++ .../convolutional/quantum_convolutional.yml | 0 .../stab_5_1_2_convolutional.yml | 0 .../stabilizer/fracton/checkerboard.yml | 37 +++++++++++++++++++ .../{topological => }/fracton/haah_cubic.yml | 0 .../{topological => }/fracton/xcube.yml | 6 ++- .../topological/color/doubled_color.yml | 3 +- .../topological/color/subsystem_color.yml | 1 + .../subsystem/topological/five_squares.yml | 1 + .../subsystem/topological/heavy_hex.yml | 1 + .../topological/subsystem_three_fermion.yml | 1 + .../surface/3d_subsystem_surface.yml | 1 + .../surface/subsystem_rotated_surface.yml | 1 + .../topological/surface/subsystem_surface.yml | 1 + .../subsystem/topological/qudit_znone.yml | 1 + .../qudits/subsystem/topological/semion.yml | 1 + .../subsystem/topological/zthree_znine.yml | 1 + .../qudits/topological/tqd_abelian.yml | 4 +- 23 files changed, 123 insertions(+), 28 deletions(-) create mode 100644 codes/quantum/properties/stabilizer/topological_stabilizer/translationally_invariant_subsystem.yml rename codes/quantum/qubits/stabilizer/{topological => }/convolutional/quantum_convolutional.yml (100%) rename codes/quantum/qubits/stabilizer/{topological => }/convolutional/stab_5_1_2_convolutional.yml (100%) create mode 100644 codes/quantum/qubits/stabilizer/fracton/checkerboard.yml rename codes/quantum/qubits/stabilizer/{topological => }/fracton/haah_cubic.yml (100%) rename codes/quantum/qubits/stabilizer/{topological => }/fracton/xcube.yml (74%) diff --git a/codes/quantum/properties/block/topological/topological_abelian.yml b/codes/quantum/properties/block/topological/topological_abelian.yml index b4e4c0d3b..f80e7b099 100644 --- a/codes/quantum/properties/block/topological/topological_abelian.yml +++ b/codes/quantum/properties/block/topological/topological_abelian.yml @@ -45,15 +45,17 @@ relations: parents: - code_id: topological cousins: - - code_id: qudit_subsystem_stabilizer - detail: 'All Abelian bosonic topological orders can be realized as modular-qudit subsystem stabilizer codes by starting with an Abelian quantum double model (slightly different from that of Ref. \cite{arxiv:2112.11394}) along with a family of Abelian TQDs that generalize the double semion anyon theory and \hyperref[topic:gauging-out]{gauging out} certain bosonic anyons \cite{arxiv:2211.03798}. - The stabilizer generators of the new subsystem code may no longer be geometrically local. - Non-Abelian topological orders are purported not to be realizable with Pauli stabilizer codes \cite{arxiv:1605.03601}.' - code_id: hamiltonian detail: 'Subsystem stabilizer code Hamiltonians described by an Abelian anyon theory do not always realize the corresponding anyonic topological order in their ground-state subspace and may exhibit a rich phase diagram. For example, the Kitaev honeycomb Hamiltonian admits the anyon theories of the 16-fold way, i.e., all minimal modular extensions of the \(\mathbb{Z}_2^{(1)}\) Abelian non-chiral non-modular anyon theory \cite{arxiv:cond-mat/0506438}\cite[Footnote 25]{arXiv:2211.03798}.' - code_id: walker_wang detail: 'Any Abelian anyon theory \(A\) can be realized at one of the surfaces of a 3D Walker-Wang model whose underlying theory is an Abelian TQD containing \(A\) as a subtheory \cite{arxiv:1907.02075,arxiv:2202.05442}\cite[Appx. H]{arxiv:2211.03798}.' + - code_id: 3d_stabilizer + detail: 'Qubit 3D stabilizer codes are conjectured to admit either fracton phases or abelian topological phases that are equivalent to multiple copies of the 3D surface code and/or the 3D fermionic surface code \cite{arXiv:1908.08049}.' + - code_id: translationally_invariant_subsystem + detail: 'All 2D Abelian bosonic topological orders can be realized as modular-qudit lattice subsystem codes by starting with an Abelian quantum double model (slightly different from that of Ref. \cite{arxiv:2112.11394}) along with a family of Abelian TQDs that generalize the double semion anyon theory and \hyperref[topic:gauging-out]{gauging out} certain bosonic anyons \cite{arxiv:2211.03798}. + The stabilizer generators of the new subsystem code may no longer be geometrically local. + Non-Abelian topological orders are purported not to be realizable with Pauli stabilizer codes \cite{arxiv:1605.03601}.' # Begin Entry Meta Information diff --git a/codes/quantum/properties/stabilizer/topological_stabilizer/2d_stabilizer.yml b/codes/quantum/properties/stabilizer/topological_stabilizer/2d_stabilizer.yml index 5e8d9271c..5b14e27b8 100644 --- a/codes/quantum/properties/stabilizer/topological_stabilizer/2d_stabilizer.yml +++ b/codes/quantum/properties/stabilizer/topological_stabilizer/2d_stabilizer.yml @@ -9,7 +9,7 @@ code_id: 2d_stabilizer name: '2D topological stabilizer code' description: | - Translationally invariant stabilizer code in two spatial dimensions. + Lattice stabilizer code in two spatial dimensions. Any prime-qudit code can be converted to several copies of the prime-qudit 2D surface code along with some trivial codes \cite{arXiv:1812.11193}. diff --git a/codes/quantum/properties/stabilizer/topological_stabilizer/3d_stabilizer.yml b/codes/quantum/properties/stabilizer/topological_stabilizer/3d_stabilizer.yml index c7c0a43c0..5ce799458 100644 --- a/codes/quantum/properties/stabilizer/topological_stabilizer/3d_stabilizer.yml +++ b/codes/quantum/properties/stabilizer/topological_stabilizer/3d_stabilizer.yml @@ -9,18 +9,11 @@ code_id: 3d_stabilizer name: '3D topological stabilizer code' description: | - Translationally invariant stabilizer code in three spatial dimensions. + Lattice stabilizer code in three spatial dimensions. + Qubit codes are conjectured to admit either fracton phases or abelian topological phases that are equivalent to multiple copies of the 3D surface code and/or the 3D fermionic surface code \cite{arXiv:1908.08049}. - Three-dimensional qubit codes can be characterized by four - coarse classes \cite{arXiv:1908.08049}: +# 1. \textit{Abelian topological phase}: Excitations are mobile in all 3 dimensions, as is typical in a topological code. Such codes are conjectured to be equivalent to a \(\mathbb{Z}_2\) gauge theory, i.e., multiple copies of the 3D surface code or the 3D fermionic surface code. - 1. \textit{Abelian topological phase}: Excitations are mobile in all 3 dimensions, as is typical in a topological code. Such codes are conjectured to be equivalent to a \(\mathbb{Z}_2\) gauge theory, i.e., multiple copies of the 3D surface code or its variant where the charge excitation is a fermion. - - 2. \textit{Foliated type-I fracton phase}: Excitations are mobile in less than 3 dimensions, but codes can be grown by \textit{foliation}, i.e., stacking copies of the 2D surface code. - - 3. \textit{Fractal type-I fracton phase}: Excitations are mobile in less than 3 dimensions, and codes are not foliated. - - 4. \textit{Type-II fracton phase}: Excitations are not mobile in any dimension and there are no string operators. relations: diff --git a/codes/quantum/properties/stabilizer/topological_stabilizer/fracton.yml b/codes/quantum/properties/stabilizer/topological_stabilizer/fracton.yml index d5dfca005..b505f7cbf 100644 --- a/codes/quantum/properties/stabilizer/topological_stabilizer/fracton.yml +++ b/codes/quantum/properties/stabilizer/topological_stabilizer/fracton.yml @@ -7,18 +7,30 @@ code_id: fracton # includes Galois and modular name: 'Fracton stabilizer code' -#introduced: '\cite{arXiv:quant-ph/9705052}' +introduced: '\cite{arXiv:1101.1962}' -description: 'A 3D translationally invariant stabilizer code whose codewords make up the ground-state space of a Hamiltonian in a fracton phase.' +description: | + A 3D translationally invariant stabilizer code whose codewords make up the ground-state space of a Hamiltonian in a fracton phase. + Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted. + Fracton codes are further classified into three sub-types: + \begin{enumerate}[(1)] + \item \textit{Foliated type-I fracton phase}: Excitations are mobile in less than 3 dimensions, but codes can be grown by \textit{foliation}, i.e., stacking copies of the 2D surface code and applying a constant-depth circuit. + \item \textit{Fractal type-I fracton phase}: Excitations are mobile in less than 3 dimensions, and codes are not foliated. + \item \textit{Type-II fracton phase}: Excitations are not mobile in any dimension and there are no string operators. + \end{enumerate} relations: parents: - code_id: 3d_stabilizer - detail: '3D stabilizer codes can realize fracton orders. Conversely, fracton codes need not be translationally invariant, and can realize multiple phases on one lattice.' cousins: - code_id: topological - detail: 'Fracton phases can be understood as topological defect networks, meaning that they can be described in the language of topological quantum field theory with defects \cite{arXiv:2002.05166,arxiv:2112.14717}.' + detail: 'Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted. + Fracton phases can be understood as topological defect networks, meaning that they can be described in the language of topological quantum field theory with defects \cite{arXiv:2002.05166,arxiv:2112.14717}.' + - code_id: surface + detail: 'Foliated type-I fracton phase codes can be grown by \textit{foliation}, i.e., stacking copies of the 2D surface code.' + + # Conversely, fracton codes need not be translationally invariant, and can realize multiple phases on one lattice. # Begin Entry Meta Information diff --git a/codes/quantum/properties/stabilizer/topological_stabilizer/translationally_invariant_stabilizer.yml b/codes/quantum/properties/stabilizer/topological_stabilizer/translationally_invariant_stabilizer.yml index a5c7ef1f7..43155b362 100644 --- a/codes/quantum/properties/stabilizer/topological_stabilizer/translationally_invariant_stabilizer.yml +++ b/codes/quantum/properties/stabilizer/topological_stabilizer/translationally_invariant_stabilizer.yml @@ -10,12 +10,15 @@ name: 'Lattice stabilizer code' introduced: '\cite{arXiv:1101.1962,arXiv:1204.1063,doi:10.7907/GCYW-ZE58}' # geometrically local would also cover hyperbolic, Euclidean overlaps with CSS +alternative_names: + - 'Topological stabilizer code' +# 1D is not very topological... + description: | A geometrically local qubit, modular-qudit, or Galois-qudit stabilizer code with qudits organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\). - If the stabilizer group is generated by site-local Pauli operators and their translations, then the code is called \textit{translationally invariant stabilizer code}. - Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional codes. + On an infinite lattice, its stabilizer group is generated by few-site Pauli operators and their translations, in which case the code is called \textit{translationally invariant stabilizer code}. + Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional versions. Lattice defects and boundaries between different codes can also be introduced. - It is possible to formulate a thermodynamic limit for lattice codes, with the 1D lattice version reducing to quantum convolutional codes. Translationally-invariant prime-qudit (\(q=p\)) stabilizer codes with \(m\) qudits per unit cell have been classified in dimensions \(D\in\{1,2\}\) in the thermodynamic limit, up to equivalence under local constant-depth Clifford circuits. Any 1D (2D) code can be converted to several copies of the 1D repetition code (prime-qudit 2D surface code) along with some trivial codes \cite{arXiv:1607.01387} (\cite{arXiv:1812.11193}). diff --git a/codes/quantum/properties/stabilizer/topological_stabilizer/translationally_invariant_subsystem.yml b/codes/quantum/properties/stabilizer/topological_stabilizer/translationally_invariant_subsystem.yml new file mode 100644 index 000000000..bedfa3ed2 --- /dev/null +++ b/codes/quantum/properties/stabilizer/topological_stabilizer/translationally_invariant_subsystem.yml @@ -0,0 +1,37 @@ +####################################################### +## This is a code entry in the error correction zoo. ## +## https://github.com/errorcorrectionzoo ## +####################################################### + +code_id: translationally_invariant_subsystem +# includes both Galois and modular + +name: 'Lattice subsystem code' +introduced: '\cite{arxiv:0908.4246}' +# geometrically local would also cover hyperbolic, Euclidean overlaps with CSS + +alternative_names: + - 'Topological subsystem code' +# 1D is not very topological... + +description: | + A geometrically local qubit, modular-qudit, or Galois-qudit subsystem stabilizer code with qudits organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\). + On an infinite lattice, its gauge and stabilizer groups are generated by few-site Pauli operators and their translations, in which case the code is called \textit{translationally invariant subsystem code}. + Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional versions, in which case the stabilizer group may no longer be generated by few-site Pauli operators. + Lattice defects and boundaries between different codes can also be introduced. + + +relations: + parents: + - code_id: oecc + cousins: + - code_id: translationally_invariant_stabilizer + detail: 'Lattice subsystem codes reduce to lattice stabilizer codes when there are no gauge qudits. The former (latter) is required to admit few-site gauge-group (stabilizer-group) generators on a lattice with boundary conditions.' + + +# Begin Entry Meta Information +_meta: + # Change log - most recent first + changelog: + - user_id: VictorVAlbert + date: '2024-01-30' diff --git a/codes/quantum/qubits/stabilizer/topological/convolutional/quantum_convolutional.yml b/codes/quantum/qubits/stabilizer/convolutional/quantum_convolutional.yml similarity index 100% rename from codes/quantum/qubits/stabilizer/topological/convolutional/quantum_convolutional.yml rename to codes/quantum/qubits/stabilizer/convolutional/quantum_convolutional.yml diff --git a/codes/quantum/qubits/stabilizer/topological/convolutional/stab_5_1_2_convolutional.yml b/codes/quantum/qubits/stabilizer/convolutional/stab_5_1_2_convolutional.yml similarity index 100% rename from codes/quantum/qubits/stabilizer/topological/convolutional/stab_5_1_2_convolutional.yml rename to codes/quantum/qubits/stabilizer/convolutional/stab_5_1_2_convolutional.yml diff --git a/codes/quantum/qubits/stabilizer/fracton/checkerboard.yml b/codes/quantum/qubits/stabilizer/fracton/checkerboard.yml new file mode 100644 index 000000000..abd701b22 --- /dev/null +++ b/codes/quantum/qubits/stabilizer/fracton/checkerboard.yml @@ -0,0 +1,37 @@ +####################################################### +## This is a code entry in the error correction zoo. ## +## https://github.com/errorcorrectionzoo ## +####################################################### + +code_id: checkerboard +physical: qubits +logical: qubits + +name: 'Checkerboard model code' +introduced: '\cite{arXiv:1505.02576}' + +description: | + A foliated type-I fracton code defined on a cubic lattice that admits weight-eight \(X\)- and \(Z\)-type stabilizer generators on cubes of the lattice. + Variants include the twisted checkerboard model \cite{arxiv:1805.06899}. + +features: + decoders: + - 'Parallelized matching decoder \cite{arxiv:1901.08061}.' + code_capacity_threshold: + - 'Independent \(X,Z\) noise: \(\sim 7.5\%\), higher than 3D surface code and color code \cite{arXiv:2112.05122}.' + + +relations: + parents: + - code_id: qubit_css + - code_id: fracton + detail: 'The checkerboard model is equivalent to two copies of the X-cube model via a local unitary \cite{arxiv:1806.08633}. Hence, it is a foliated type-I fracton code.' + cousins: + - code_id: xcube + detail: 'The checkerboard model is equivalent to two copies of the X-cube model via a local unitary \cite{arxiv:1806.08633}.' + + +_meta: + changelog: + - user_id: VictorVAlbert + date: '2024-01-30' diff --git a/codes/quantum/qubits/stabilizer/topological/fracton/haah_cubic.yml b/codes/quantum/qubits/stabilizer/fracton/haah_cubic.yml similarity index 100% rename from codes/quantum/qubits/stabilizer/topological/fracton/haah_cubic.yml rename to codes/quantum/qubits/stabilizer/fracton/haah_cubic.yml diff --git a/codes/quantum/qubits/stabilizer/topological/fracton/xcube.yml b/codes/quantum/qubits/stabilizer/fracton/xcube.yml similarity index 74% rename from codes/quantum/qubits/stabilizer/topological/fracton/xcube.yml rename to codes/quantum/qubits/stabilizer/fracton/xcube.yml index 510ad1ab5..c0f8f9e12 100644 --- a/codes/quantum/qubits/stabilizer/topological/fracton/xcube.yml +++ b/codes/quantum/qubits/stabilizer/fracton/xcube.yml @@ -11,7 +11,8 @@ name: 'X-cube model code' introduced: '\cite{arxiv:1603.04442}' description: | - A Type-I fracton code supporting a subextensive number of logical qubits. + A foliated type-I fracton code supporting a subextensive number of logical qubits. + Variants include the membrane-coupled \cite{arxiv:1701.00747} and twice-foliated \cite{arxiv:1806.08625} X-cube models. features: decoders: @@ -22,8 +23,9 @@ features: relations: parents: - - code_id: qubit_stabilizer + - code_id: qubit_css - code_id: fracton + detail: 'The X-cube model is a foliated type-I fracton code \cite{arxiv:1803.10426,arxiv:1908.08049}.' cousins: - code_id: quantum_inspired detail: 'According to Ref. \cite{arxiv:2002.11738}, a classical analogue of the X-cube model is the eight-vertex model \cite{doi:10.1063/1.1665111,doi:10.1103/PhysRevLett.26.832,doi:10.1016/0003-4916(72)90335-1}.' diff --git a/codes/quantum/qubits/subsystem/topological/color/doubled_color.yml b/codes/quantum/qubits/subsystem/topological/color/doubled_color.yml index 78c1a5b42..e6526dd9b 100644 --- a/codes/quantum/qubits/subsystem/topological/color/doubled_color.yml +++ b/codes/quantum/qubits/subsystem/topological/color/doubled_color.yml @@ -21,7 +21,8 @@ features: relations: parents: - - code_id: subsystem_color + - code_id: subsystem_css + - code_id: translationally_invariant_subsystem cousins: - code_id: divisible detail: 'Doubled color codes are constructed using a generalization of the doubling transformation \cite{arxiv:1012.4134} that combine doubly-even codes to make triply-even codes.' diff --git a/codes/quantum/qubits/subsystem/topological/color/subsystem_color.yml b/codes/quantum/qubits/subsystem/topological/color/subsystem_color.yml index 6416c4a7a..ba654936b 100644 --- a/codes/quantum/qubits/subsystem/topological/color/subsystem_color.yml +++ b/codes/quantum/qubits/subsystem/topological/color/subsystem_color.yml @@ -34,6 +34,7 @@ features: relations: parents: - code_id: subsystem_css + - code_id: translationally_invariant_subsystem - code_id: single_shot detail: 'The subsystem color code is a single-shot code \cite{arxiv:1503.08217}.' cousins: diff --git a/codes/quantum/qubits/subsystem/topological/five_squares.yml b/codes/quantum/qubits/subsystem/topological/five_squares.yml index 2525762f6..1481f150f 100644 --- a/codes/quantum/qubits/subsystem/topological/five_squares.yml +++ b/codes/quantum/qubits/subsystem/topological/five_squares.yml @@ -24,6 +24,7 @@ relations: parents: - code_id: subsystem_hypergraph detail: 'Generalized five-squares codes are special cases of subsystem hypergraph codes \cite[Sec. II.B]{arxiv:1805.12542}.' + - code_id: translationally_invariant_subsystem cousins: - code_id: surface detail: 'Decoding of five-squares codes leads to a mapping of these codes to two copies of the surface code \cite{arxiv:1012.0425,arxiv:1805.12542}.' diff --git a/codes/quantum/qubits/subsystem/topological/heavy_hex.yml b/codes/quantum/qubits/subsystem/topological/heavy_hex.yml index f5c8e9d56..f38ae2bbc 100644 --- a/codes/quantum/qubits/subsystem/topological/heavy_hex.yml +++ b/codes/quantum/qubits/subsystem/topological/heavy_hex.yml @@ -48,6 +48,7 @@ realizations: relations: parents: - code_id: subsystem_css + - code_id: translationally_invariant_subsystem cousins: - code_id: surface detail: 'Surface code stabilizers are used to measure the Z-type stabilizers of the code.' diff --git a/codes/quantum/qubits/subsystem/topological/subsystem_three_fermion.yml b/codes/quantum/qubits/subsystem/topological/subsystem_three_fermion.yml index 4b103948d..b8664a64f 100644 --- a/codes/quantum/qubits/subsystem/topological/subsystem_three_fermion.yml +++ b/codes/quantum/qubits/subsystem/topological/subsystem_three_fermion.yml @@ -20,6 +20,7 @@ description: | relations: parents: - code_id: subsystem_stabilizer + - code_id: translationally_invariant_subsystem - code_id: topological_abelian detail: 'The 3F code is a subsystme code characterized by 3F topological order \cite{arxiv:2211.03798}, which is chiral and modular.' cousins: diff --git a/codes/quantum/qubits/subsystem/topological/surface/3d_subsystem_surface.yml b/codes/quantum/qubits/subsystem/topological/surface/3d_subsystem_surface.yml index 90c431999..d2f2fedd2 100644 --- a/codes/quantum/qubits/subsystem/topological/surface/3d_subsystem_surface.yml +++ b/codes/quantum/qubits/subsystem/topological/surface/3d_subsystem_surface.yml @@ -20,6 +20,7 @@ description: | relations: parents: - code_id: subsystem_css + - code_id: translationally_invariant_subsystem - code_id: single_shot detail: 'The 3D subsystem surface code is a single-shot code \cite{arxiv:2106.02621,arxiv:2305.06365}; see Ref. \cite{arxiv:2307.08118} for an alternative formulation.' cousins: diff --git a/codes/quantum/qubits/subsystem/topological/surface/subsystem_rotated_surface.yml b/codes/quantum/qubits/subsystem/topological/surface/subsystem_rotated_surface.yml index 89829b14a..a0b5a8350 100644 --- a/codes/quantum/qubits/subsystem/topological/surface/subsystem_rotated_surface.yml +++ b/codes/quantum/qubits/subsystem/topological/surface/subsystem_rotated_surface.yml @@ -19,6 +19,7 @@ description: | relations: parents: - code_id: subsystem_css + - code_id: translationally_invariant_subsystem cousins: - code_id: subsystem_surface - code_id: rotated_surface diff --git a/codes/quantum/qubits/subsystem/topological/surface/subsystem_surface.yml b/codes/quantum/qubits/subsystem/topological/surface/subsystem_surface.yml index 8dec36d65..06d429d0b 100644 --- a/codes/quantum/qubits/subsystem/topological/surface/subsystem_surface.yml +++ b/codes/quantum/qubits/subsystem/topological/surface/subsystem_surface.yml @@ -33,6 +33,7 @@ notes: relations: parents: - code_id: subsystem_css + - code_id: translationally_invariant_subsystem cousins: - code_id: surface diff --git a/codes/quantum/qudits/subsystem/topological/qudit_znone.yml b/codes/quantum/qudits/subsystem/topological/qudit_znone.yml index 22bc9a580..ba3d39585 100644 --- a/codes/quantum/qudits/subsystem/topological/qudit_znone.yml +++ b/codes/quantum/qudits/subsystem/topological/qudit_znone.yml @@ -19,6 +19,7 @@ description: relations: parents: - code_id: qudit_subsystem_stabilizer + - code_id: translationally_invariant_subsystem - code_id: topological_abelian detail: 'The \(\mathbb{Z}_q^{(1)}\) subsystem code is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory \cite{doi:10.7907/5NDZ-W890}. The anyon theory has a single generator \(a \in \mathbb Z_N\) with \(\theta(a) =e^{\frac{2\pi i}{N}a^2}\). It is modular for odd prime \(q\) and non-modular otherwise.' cousins: diff --git a/codes/quantum/qudits/subsystem/topological/semion.yml b/codes/quantum/qudits/subsystem/topological/semion.yml index 3dcc9a728..4a9caf4e4 100644 --- a/codes/quantum/qudits/subsystem/topological/semion.yml +++ b/codes/quantum/qudits/subsystem/topological/semion.yml @@ -21,6 +21,7 @@ features: relations: parents: - code_id: qudit_subsystem_stabilizer + - code_id: translationally_invariant_subsystem - code_id: topological_abelian detail: 'The semion code is a subsystem code characterized by the chiral semion topological phase.' cousins: diff --git a/codes/quantum/qudits/subsystem/topological/zthree_znine.yml b/codes/quantum/qudits/subsystem/topological/zthree_znine.yml index f00a6b025..a1cde785e 100644 --- a/codes/quantum/qudits/subsystem/topological/zthree_znine.yml +++ b/codes/quantum/qudits/subsystem/topological/zthree_znine.yml @@ -18,6 +18,7 @@ description: | relations: parents: - code_id: qudit_subsystem_stabilizer + - code_id: translationally_invariant_subsystem - code_id: topological_abelian detail: 'The \(\mathbb{Z}_q^{(1)}\) subsystem code is characterized by a non-modular anyon theory with \(\mathbb{Z}_3\times\mathbb{Z}_9\) fusion rules.' cousins: diff --git a/codes/quantum/qudits/topological/tqd_abelian.yml b/codes/quantum/qudits/topological/tqd_abelian.yml index bc7c33bef..322eb3c91 100644 --- a/codes/quantum/qudits/topological/tqd_abelian.yml +++ b/codes/quantum/qudits/topological/tqd_abelian.yml @@ -17,10 +17,8 @@ description: | relations: parents: - - code_id: qudit_stabilizer - detail: 'All Abelian TQD codes can be realized as modular-qudit stabilizer codes by starting with an Abelian quantum double model along with a family of Abelian TQDs that generalize the double semion anyon theory and \hyperref[topic:code-switching]{condensing} certain bosonic anyons \cite{arxiv:2211.03798}.' - code_id: 2d_stabilizer - detail: 'All Abelian TQD codes can be realized as modular-qudit stabilizer codes by starting with an Abelian quantum double model along with a family of Abelian TQDs that generalize the double semion anyon theory and \hyperref[topic:code-switching]{condensing} certain bosonic anyons \cite{arxiv:2211.03798}. + detail: 'All Abelian TQD codes can be realized as modular-qudit lattice stabilizer codes by starting with an Abelian quantum double model along with a family of Abelian TQDs that generalize the double semion anyon theory and \hyperref[topic:code-switching]{condensing} certain bosonic anyons \cite{arxiv:2112.11394}. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice.' - code_id: topological_abelian detail: 'Abelian TQDs realize all modular gapped Abelian topological orders \cite{arxiv:2112.11394}.