From acb768a44bc0749257679c8f35a4f5a186f6384e Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Mon, 16 Oct 2023 14:32:05 -0400 Subject: [PATCH] capital abelian --- codes/classical/bits/reed_muller.yml | 2 +- .../tanner/regular_tanner/regular_binary_tanner.yml | 2 +- codes/classical/groups/group_classical.yml | 2 +- codes/classical/q-ary_digits/cyclic/group.yml | 8 ++++---- codes/classical/q-ary_digits/cyclic/q-ary_cyclic.yml | 2 +- codes/classical/q-ary_digits/q-ary_additive.yml | 2 +- codes/classical/rings/rings_into_rings.yml | 2 +- codes/classical/rings/rings_linear.yml | 2 +- codes/classical/spherical/numerical/tlsc.yml | 2 +- codes/quantum/categories/groupoid_surface.yml | 2 +- codes/quantum/categories/string_net.yml | 2 +- codes/quantum/groups/group_gkp.yml | 2 +- .../quantum/groups/topological/generalized_color.yml | 4 ++-- codes/quantum/groups/topological/quantum_double.yml | 8 ++++---- codes/quantum/oscillators/fock_state/matrix_qm.yml | 4 ++-- .../block/covariant/nonabelian_covariant_erasure.yml | 2 +- .../properties/block/topological/topological.yml | 4 ++-- .../qldpc/translationally_invariant_stabilizer.yml | 4 ++-- codes/quantum/qubits/dynamic_gen/da/honeycomb.yml | 2 +- codes/quantum/qubits/fermions/mbq.yml | 2 +- .../qubits/stabilizer/qldpc/algebraic/qcga.yml | 2 +- codes/quantum/qubits/stabilizer/qubit_stabilizer.yml | 2 +- .../quantum/qubits/stabilizer/topological/color.yml | 4 ++-- .../qubits/stabilizer/topological/matching.yml | 2 +- .../topological/surface/two_dim/surface/surface.yml | 2 +- .../qubits/subsystem/subsystem_quantum_parity.yml | 2 +- .../qubits/subsystem/subsystem_stabilizer.yml | 2 +- .../subsystem/topological/kitaev_honeycomb.yml | 12 ++++++------ codes/quantum/qudits/qudit_stabilizer.yml | 4 ++-- .../qudits/subsystem/qudit_subsystem_stabilizer.yml | 2 +- .../subsystem/topological/topological_abelian.yml | 10 +++++----- .../qudits/topological/quantum_double_abelian.yml | 12 ++++++------ codes/quantum/qudits/topological/tqd_abelian.yml | 12 ++++++------ .../qudits_galois/qldpc/expander_lifted_product.yml | 2 +- .../qudits_galois/qldpc/galois_topological.yml | 2 +- 35 files changed, 66 insertions(+), 66 deletions(-) diff --git a/codes/classical/bits/reed_muller.yml b/codes/classical/bits/reed_muller.yml index c8fef5105..a3bba9fe2 100644 --- a/codes/classical/bits/reed_muller.yml +++ b/codes/classical/bits/reed_muller.yml @@ -51,7 +51,7 @@ relations: - code_id: divisible detail: 'An RM\((r,m)\) code is \(2^{\left\lceil m/r\right\rceil-1}\)-divisible, according to McEliece''s theorem \cite{doi:10.1016/0097-3165(71)90066-5,doi:10.1016/0012-365X(72)90032-5}.' - code_id: group - detail: 'RM codes are group-algebra codes \cite{doi:10.1007/BF01072842,manual:{Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.}}\cite[Ex. 16.4.11]{preset:HKSalgebra}. Consider a binary vector space of dimension \( m \). Under addition, this forms a finite group with \( 2^m \) elements known as an elementary abelian 2-group -- the direct product of \( m \) two-element cyclic groups \( \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2 \). Denote this group by \( G_m \). Let \( J \) be the Jacobson radical of the \hyperref[topic:group-algebra]{group algebra} \( \mathbb{F}_2 G_m \), where \(\mathbb{F}_2=GF(2)\). RM\((r,m)\) codes correspond to the ideal \( J^{m-r} \). The length of the code is \( |G_m| = 2^m \), the distance is \( 2^{m-r} \), and the dimension is \( \sum_{i=0}^r {m \choose i} \). A similar construction exists for choices of a prime \( p\neq 2 \).' + detail: 'RM codes are group-algebra codes \cite{doi:10.1007/BF01072842,manual:{Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.}}\cite[Ex. 16.4.11]{preset:HKSalgebra}. Consider a binary vector space of dimension \( m \). Under addition, this forms a finite group with \( 2^m \) elements known as an elementary Abelian 2-group -- the direct product of \( m \) two-element cyclic groups \( \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2 \). Denote this group by \( G_m \). Let \( J \) be the Jacobson radical of the \hyperref[topic:group-algebra]{group algebra} \( \mathbb{F}_2 G_m \), where \(\mathbb{F}_2=GF(2)\). RM\((r,m)\) codes correspond to the ideal \( J^{m-r} \). The length of the code is \( |G_m| = 2^m \), the distance is \( 2^{m-r} \), and the dimension is \( \sum_{i=0}^r {m \choose i} \). A similar construction exists for choices of a prime \( p\neq 2 \).' cousins: - code_id: bch detail: 'RM\(^*(r,m)\) codes are equivalent to subcodes of BCH codes of designed distance \(2^{m-r}-1\) while RM\((r,m)\) are subcodes of extended BCH codes of the same designed distance \cite[Ch. 13]{preset:MacSlo}.' diff --git a/codes/classical/bits/tanner/regular_tanner/regular_binary_tanner.yml b/codes/classical/bits/tanner/regular_tanner/regular_binary_tanner.yml index 25223a12c..3524a2e0c 100644 --- a/codes/classical/bits/tanner/regular_tanner/regular_binary_tanner.yml +++ b/codes/classical/bits/tanner/regular_tanner/regular_binary_tanner.yml @@ -36,7 +36,7 @@ features: encoders: - 'Quadratic algorithm: This technique works for all linear block codes and encodes using matrix multiplication \cite{doi:10.1145/258533.258575}.' - - 'Using the nonabelian Fast Fourier Transform and exploiting the symmetry of the underlying graph, an encoding algorithm that requires \(O(n^{4/3})\) has been devised in \cite{doi:10.1145/258533.258575}.' + - 'Using the non-Abelian Fast Fourier Transform and exploiting the symmetry of the underlying graph, an encoding algorithm that requires \(O(n^{4/3})\) has been devised in \cite{doi:10.1145/258533.258575}.' - 'A modified construction yields codes that may be encoded in linear time yet maintain similar performance \cite{doi:10.1109/18.556668}.' decoders: diff --git a/codes/classical/groups/group_classical.yml b/codes/classical/groups/group_classical.yml index 68dd491a0..0e37ca18f 100644 --- a/codes/classical/groups/group_classical.yml +++ b/codes/classical/groups/group_classical.yml @@ -19,7 +19,7 @@ relations: # - code_id: bits_into_bits -# detail: 'Group-alphabet codes whose alphabet is based on the field \(GF(2)\), taken to be an abelian group under addition, are binary codes.' +# detail: 'Group-alphabet codes whose alphabet is based on the field \(GF(2)\), taken to be an Abelian group under addition, are binary codes.' # Begin Entry Meta Information diff --git a/codes/classical/q-ary_digits/cyclic/group.yml b/codes/classical/q-ary_digits/cyclic/group.yml index d054ebd74..11990f0ff 100644 --- a/codes/classical/q-ary_digits/cyclic/group.yml +++ b/codes/classical/q-ary_digits/cyclic/group.yml @@ -14,7 +14,7 @@ alternative_names: description: | An \( [n,k]_q \) code based on a finite group \( G \) of size \(n \). - A group-algebra code for an abelian group is called an \textit{abelian group-algebra code}. + A group-algebra code for an Abelian group is called an \textit{Abelian group-algebra code}. \subsection{Group algebra} \label{topic:group-algebra} @@ -36,17 +36,17 @@ description: | \subsection{Group-algebra code} A group-algebra code is a \( k \)-dimensional linear subspace of the \hyperref[topic:group-algebra]{group algebra} of \( G\) with coefficients in the field \(GF(q) = \mathbb{F}_q\) with \(q\) elements. To be precise, the code must be closed under permutations corresponding to the elements of the group \( G \); therefore, \( G \) must be a subgroup of the permutation automorphism group of the code, which is defined as the group of permutations of the physical bits that preserve the code space. This leads us to the formal definition of a group-algebra code: a group-algebra code is an ideal in the \hyperref[topic:group-algebra]{group algebra} \( \mathbb{F}_q G \). -#protection: 'The class of abelian group-algebra codes is very general, for example including all group-algebra codes of size \(n \leq 23 \). As such it is very difficult to say anything about the distance of abelian groups codes without specializing to a particular family' +#protection: 'The class of Abelian group-algebra codes is very general, for example including all group-algebra codes of size \(n \leq 23 \). As such it is very difficult to say anything about the distance of Abelian groups codes without specializing to a particular family' notes: - 'See \cite{preset:HKSalgebra}\cite[pg. 58]{doi:10.1007/978-94-011-3810-9} for introductions to group-algebra codes.' - - 'Not all abelian group-algebra codes are for cyclic groups (cyclic codes) or for elementary abelian \( p \) groups (e.g. Reed Muller codes \cite{doi:10.1007/BF01119999}). For example, there is a binary code with parameters \( [45,13,16] \) which is an abelian group-algebra code for the group \( G = \mathbb{Z}_3 \times \mathbb{Z}_{15} \). ' + - 'Not all Abelian group-algebra codes are for cyclic groups (cyclic codes) or for elementary Abelian \( p \) groups (e.g. Reed Muller codes \cite{doi:10.1007/BF01119999}). For example, there is a binary code with parameters \( [45,13,16] \) which is an Abelian group-algebra code for the group \( G = \mathbb{Z}_3 \times \mathbb{Z}_{15} \). ' relations: parents: - code_id: q-ary_linear detail: 'A linear code is a group-algebra code for a group \(G\) if and only if \(G\) is isomorphic to a regular subgroup of the code''s permutation automorphism group \cite{doi:10.1007/s10623-008-9261-z}\cite[Thm. 16.4.7]{preset:HKSalgebra}.' -#Note that we have an isomorphism of \( \mathbb{F} \) algebras \( \mathbb{F}\mathbb{Z}_n \cong \mathbb{F}[x]/\langle x^n-1\rangle \) by taking \( x \) to be the generator of the cyclic group. Thus we can see how cyclic codes are an example of an abelian group-algebra code.' +#Note that we have an isomorphism of \( \mathbb{F} \) algebras \( \mathbb{F}\mathbb{Z}_n \cong \mathbb{F}[x]/\langle x^n-1\rangle \) by taking \( x \) to be the generator of the cyclic group. Thus we can see how cyclic codes are an example of an Abelian group-algebra code.' # Begin Entry Meta Information diff --git a/codes/classical/q-ary_digits/cyclic/q-ary_cyclic.yml b/codes/classical/q-ary_digits/cyclic/q-ary_cyclic.yml index 05746f052..cd928c593 100644 --- a/codes/classical/q-ary_digits/cyclic/q-ary_cyclic.yml +++ b/codes/classical/q-ary_digits/cyclic/q-ary_cyclic.yml @@ -50,7 +50,7 @@ relations: - code_id: q-ary_linear - code_id: cyclic - code_id: group - detail: 'A length-\(n\) cyclic \(q\)-ary linear code is an abelian group-algebra code for the cyclic group with \(n\) elements \( \mathbb{Z}_n \).' + detail: 'A length-\(n\) cyclic \(q\)-ary linear code is an Abelian group-algebra code for the cyclic group with \(n\) elements \( \mathbb{Z}_n \).' cousins: - code_id: q-ary_ltc detail: 'Cyclic linear codes cannot be \(c^3\)-LTCs \cite{doi:10.1109/TIT.2005.851735}. Codeword symmetries are in general an obstruction to achieving such LTCs \cite{doi:10.1007/978-3-642-16367-8_12}.' diff --git a/codes/classical/q-ary_digits/q-ary_additive.yml b/codes/classical/q-ary_digits/q-ary_additive.yml index bcf12621f..b656af5fd 100644 --- a/codes/classical/q-ary_digits/q-ary_additive.yml +++ b/codes/classical/q-ary_digits/q-ary_additive.yml @@ -16,7 +16,7 @@ relations: parents: - code_id: q-ary_digits_into_q-ary_digits - code_id: group_linear - detail: 'Additive \(q\)-ary codes are linear over \(G=GF(q)\) since Galois fields are abelian groups under addition.' + detail: 'Additive \(q\)-ary codes are linear over \(G=GF(q)\) since Galois fields are Abelian groups under addition.' # Begin Entry Meta Information diff --git a/codes/classical/rings/rings_into_rings.yml b/codes/classical/rings/rings_into_rings.yml index a09501eef..1a7f04212 100644 --- a/codes/classical/rings/rings_into_rings.yml +++ b/codes/classical/rings/rings_into_rings.yml @@ -15,7 +15,7 @@ relations: - code_id: block - code_id: ecc_finite - code_id: group_classical - detail: 'A ring \(R\) is an abelian group under addition.' + detail: 'A ring \(R\) is an Abelian group under addition.' # Begin Entry Meta Information diff --git a/codes/classical/rings/rings_linear.yml b/codes/classical/rings/rings_linear.yml index f2a380b94..d54e91cb0 100644 --- a/codes/classical/rings/rings_linear.yml +++ b/codes/classical/rings/rings_linear.yml @@ -17,7 +17,7 @@ relations: parents: - code_id: rings_into_rings - code_id: group_linear - detail: '\(R\)-linear codes are linear over \(G=R\) since rings are abelian groups under addition.' + detail: '\(R\)-linear codes are linear over \(G=R\) since rings are Abelian groups under addition.' # Begin Entry Meta Information diff --git a/codes/classical/spherical/numerical/tlsc.yml b/codes/classical/spherical/numerical/tlsc.yml index c897ac194..00aac10cb 100644 --- a/codes/classical/spherical/numerical/tlsc.yml +++ b/codes/classical/spherical/numerical/tlsc.yml @@ -26,7 +26,7 @@ features: relations: parents: - code_id: slepian_group - detail: 'Polyphase codewords can be implemented by acting on the all-ones initial vector by diagonal orthogonal matrices whose entries are the codeword components \cite[Ch. 8]{preset:EricZin}. TLSC codes are generalizations of polyphase codes to other initial vectors and are examples of abelian Slepian-group codes.' + detail: 'Polyphase codewords can be implemented by acting on the all-ones initial vector by diagonal orthogonal matrices whose entries are the codeword components \cite[Ch. 8]{preset:EricZin}. TLSC codes are generalizations of polyphase codes to other initial vectors and are examples of Abelian Slepian-group codes.' # Begin Entry Meta Information diff --git a/codes/quantum/categories/groupoid_surface.yml b/codes/quantum/categories/groupoid_surface.yml index f405b71d8..7ca4dfe4d 100644 --- a/codes/quantum/categories/groupoid_surface.yml +++ b/codes/quantum/categories/groupoid_surface.yml @@ -11,7 +11,7 @@ name: 'Groupoid toric code' introduced: '\cite{arxiv:2212.01021}' description: | - Extension of the Kitaev surface code from abelian groups to groupoids, i.e., multi-fusion categories in which every morphism is an isomorphism \cite{doi:10.1112/blms/19.2.113}. + Extension of the Kitaev surface code from Abelian groups to groupoids, i.e., multi-fusion categories in which every morphism is an isomorphism \cite{doi:10.1112/blms/19.2.113}. Some models admit fracton-like features such as extensive ground-state degeneracy and excitations with restricted mobility. The robustness of these features has not yet been established. diff --git a/codes/quantum/categories/string_net.yml b/codes/quantum/categories/string_net.yml index e9526faef..4656aca9d 100644 --- a/codes/quantum/categories/string_net.yml +++ b/codes/quantum/categories/string_net.yml @@ -31,7 +31,7 @@ features: - 'Gates can be implemented through topological operations corresponding to elements of the mapping class group, which is generated by Dehn-twists along non-contractible cycles for triangulations of toroidal \cite{arXiv:1806.02358,arxiv:1806.06078} and hyperbolic \cite{arXiv:1901.11029} manifolds. Whether or not a gate set is universal depends on the choice of input category; in some cases such as the Ising category, gates can be complemented by topological charge measurements to obtain a universal gate set.' - 'Alternatively, one could encode the logical quantum information into the degenerate fusion space of a number of computational anyons. In this case, a universal logical gate set can be implemented through the braiding of the computational anyons \cite{arXiv:quant-ph/0001108,arXiv:math/0103200,arxiv:1002.2816}, e.g., for the case of the \hyperref[code:fibonacci]{Fibonacci} input category.' decoders: - - 'Fusing nonabelian anyons cannot be done in one step \cite{arxiv:hep-th/0110205}.' + - 'Fusing non-Abelian anyons cannot be done in one step \cite{arxiv:hep-th/0110205}.' - 'Syndrome measurement circuits analyzed in Ref. \cite{arXiv:1206.6048}.' - 'Clustering decoder \cite{arxiv:1607.02159}.' diff --git a/codes/quantum/groups/group_gkp.yml b/codes/quantum/groups/group_gkp.yml index 254e86cdd..902ab1a30 100644 --- a/codes/quantum/groups/group_gkp.yml +++ b/codes/quantum/groups/group_gkp.yml @@ -62,7 +62,7 @@ relations: cousins: - code_id: css detail: 'Group GKP codes are stabilized by \(X\)-type Pauli matrices representing \(H\) and all \(Z\)-type operators that are constant on \(K\). - However, the \(Z\)-type operators are not unitary for nonabelian groups.' + However, the \(Z\)-type operators are not unitary for non-Abelian groups.' - code_id: oscillator_stabilizer detail: 'The group-GKP construction encompasses all bosonic CSS codes. A single-mode qubit GKP code corresponds to the \(2\mathbb{Z}\subset\mathbb{Z}\subset\mathbb{R}\) group construction, and multimode GKP codes can be similarly described. diff --git a/codes/quantum/groups/topological/generalized_color.yml b/codes/quantum/groups/topological/generalized_color.yml index c34bddf5c..9c6ce5a5d 100644 --- a/codes/quantum/groups/topological/generalized_color.yml +++ b/codes/quantum/groups/topological/generalized_color.yml @@ -11,7 +11,7 @@ name: 'Generalized color code' introduced: '\cite{arXiv:1408.6238}' description: | - Member of a family of nonabelian topological codes, defined by a finite group \( G \), that serves as a generalization of the color code (for which \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)). + Member of a family of non-Abelian topological codes, defined by a finite group \( G \), that serves as a generalization of the color code (for which \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)). relations: @@ -22,7 +22,7 @@ relations: - code_id: group_gkp detail: 'Generalized color-code Hamiltonians should be expressable in terms of \(X\)- and \(Z\)-type operators of group-GKP codes; see \cite[Sec. 3.3]{arxiv:2111.12096}.' - code_id: quantum_double - detail: 'Generalized color code for group \(G\) on the 4.8.8 lattice is equivalent to a \(G\) quantum double model and another \(G/[G,G]\) quantum double model defined using the abelianization of \(G\).' + detail: 'Generalized color code for group \(G\) on the 4.8.8 lattice is equivalent to a \(G\) quantum double model and another \(G/[G,G]\) quantum double model defined using the Abelianization of \(G\).' # Begin Entry Meta Information diff --git a/codes/quantum/groups/topological/quantum_double.yml b/codes/quantum/groups/topological/quantum_double.yml index f5a4a1109..535b835c7 100644 --- a/codes/quantum/groups/topological/quantum_double.yml +++ b/codes/quantum/groups/topological/quantum_double.yml @@ -14,11 +14,11 @@ description: | Group-GKP stabilizer code whose codewords realize 2D modular gapped topological order defined by a finite group \(G\). The code's generators are few-body operators associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit of dimension \( |G| \) located at each edge of the tesselation). - The physical Hilbert space has dimension \( |G|^E \), where \( E \) is the number of edges in the tessellation. The dimension of the code space is the number of orbits of the conjugation action of \( G \) on \( \text{Hom}(\pi_1(\Sigma),G) \), the set of group homomorphisms from the fundamental group of the surface \( \Sigma \) into the finite group \( G \) \cite{arXiv:1908.02829}. When \( G \) is abelian, the formula for the dimension simplifies to \( |G|^{2g} \), where \( g \) is the genus of the surface \( \Sigma \). + The physical Hilbert space has dimension \( |G|^E \), where \( E \) is the number of edges in the tessellation. The dimension of the code space is the number of orbits of the conjugation action of \( G \) on \( \text{Hom}(\pi_1(\Sigma),G) \), the set of group homomorphisms from the fundamental group of the surface \( \Sigma \) into the finite group \( G \) \cite{arXiv:1908.02829}. When \( G \) is Abelian, the formula for the dimension simplifies to \( |G|^{2g} \), where \( g \) is the genus of the surface \( \Sigma \). The codespace is the ground-state subspace of the quantum double model Hamiltonian. - For nonabelian groups, alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model \cite{doi:10.1007/3-540-49208-9_31,arXiv:quant-ph/0306063,doi:10.1017/CBO9780511792908}. - The fusion space of such nonabelian anyons has dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes. + For non-Abelian groups, alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model \cite{doi:10.1007/3-540-49208-9_31,arXiv:quant-ph/0306063,doi:10.1017/CBO9780511792908}. + The fusion space of such non-Abelian anyons has dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes. protection: | Error-correcting properties established in Ref. \cite{arxiv:1908.02829}. @@ -29,7 +29,7 @@ features: - 'A depth-\(L^2\) circuit that grows the code out of a small patch on an \(L\times L\) square lattice using CMULT gates (i.e., "local moves") \cite{arxiv:0712.0348,arxiv:1101.0527}.' - 'For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit \cite{arXiv:0901.1345,arXiv:1101.0527} or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates \cite{arXiv:0712.0348,arxiv:0806.4583}.' - 'For any solvable group \(G\), ground-state preparation and anyon-pair creation can be done with an adaptive constant-depth circuit with geometrically local gates and measurements throughout \cite{arXiv:2112.01519,arXiv:2205.01933} (see Ref. \cite{arXiv:2112.03061} for specific dihedral groups). - Anyon-pair creation requires an adaptive circuit for any nonabelian \(G\) \cite{arXiv:2205.01933}.' + Anyon-pair creation requires an adaptive circuit for any non-Abelian \(G\) \cite{arXiv:2205.01933}.' general_gates: - 'Universal topological quantum computation possible for certain groups \cite{arxiv:quant-ph/0306063,arxiv:0901.1345}.' decoders: diff --git a/codes/quantum/oscillators/fock_state/matrix_qm.yml b/codes/quantum/oscillators/fock_state/matrix_qm.yml index 69efeee2e..7562b72ae 100644 --- a/codes/quantum/oscillators/fock_state/matrix_qm.yml +++ b/codes/quantum/oscillators/fock_state/matrix_qm.yml @@ -11,7 +11,7 @@ name: 'Matrix-model code' introduced: '\cite{arxiv:2211.08448}' description: | - Multimode-mode Fock-state bosonic approximate code derived from a matrix model, i.e., a nonabelian bosonic gauge theory with a large gauge group. + Multimode-mode Fock-state bosonic approximate code derived from a matrix model, i.e., a non-Abelian bosonic gauge theory with a large gauge group. The model's degrees of freedom are matrix-valued bosons \(a\), each consisting of \(N^2\) harmonic oscillator modes and subject to an \(SU(N)\) gauge symmetry. A simple matrix-model code consists of two spatially separated bosons with codewords @@ -38,7 +38,7 @@ relations: detail: 'Matrix-model codewords for simple codes are eigenstates of a matrix-model Hamiltonian.' cousins: - code_id: holographic - detail: 'Matrix-model codes are motivated by the Ads/CFT correspondence because it is manifest in continuous nonabelian gauge theories with large gauge groups \cite{arxiv:2211.08448}.' + detail: 'Matrix-model codes are motivated by the Ads/CFT correspondence because it is manifest in continuous non-Abelian gauge theories with large gauge groups \cite{arxiv:2211.08448}.' - code_id: self_correct detail: 'Matrix-model codes are similar to self-correcting memories in the sense that memory time becomes infinite in the thermodynamic limit, but with corrections being polynomial in \(N\).' diff --git a/codes/quantum/properties/block/covariant/nonabelian_covariant_erasure.yml b/codes/quantum/properties/block/covariant/nonabelian_covariant_erasure.yml index 98a5eb22a..c86156767 100644 --- a/codes/quantum/properties/block/covariant/nonabelian_covariant_erasure.yml +++ b/codes/quantum/properties/block/covariant/nonabelian_covariant_erasure.yml @@ -3,7 +3,7 @@ ## https://github.com/errorcorrectionzoo ## ####################################################### -code_id: nonabelian_covariant_erasure +code_id: non-Abelian_covariant_erasure name: '\(U(d)\)-covariant approximate erasure code' introduced: '\cite{arxiv:2007.09154}' diff --git a/codes/quantum/properties/block/topological/topological.yml b/codes/quantum/properties/block/topological/topological.yml index 5366ba660..c2ddb8265 100644 --- a/codes/quantum/properties/block/topological/topological.yml +++ b/codes/quantum/properties/block/topological/topological.yml @@ -58,8 +58,8 @@ features: \end{align} and a generalization of the formula to the non-orientable case can be found in Ref. \cite{arxiv:1612.07792}.' encoders: - - 'The unitary circuit depth required to initialize in a general topologically ordered state using geometrically local gates on an \(L\times L\) lattice is \(\Omega(L)\) \cite{arXiv:quant-ph/0603121}, irrespective of whether the ground state admits Abelian or nonabelian anyonic excitations. - However, only a finite-depth circuit and one round of measurements is required for nonabelian topological orders with a Lagrangian subgroup \cite{arxiv:2209.03964}.' + - 'The unitary circuit depth required to initialize in a general topologically ordered state using geometrically local gates on an \(L\times L\) lattice is \(\Omega(L)\) \cite{arXiv:quant-ph/0603121}, irrespective of whether the ground state admits Abelian or non-Abelian anyonic excitations. + However, only a finite-depth circuit and one round of measurements is required for non-Abelian topological orders with a Lagrangian subgroup \cite{arxiv:2209.03964}.' notes: diff --git a/codes/quantum/properties/stabilizer/qldpc/translationally_invariant_stabilizer.yml b/codes/quantum/properties/stabilizer/qldpc/translationally_invariant_stabilizer.yml index fd4a974d8..117b26aa2 100644 --- a/codes/quantum/properties/stabilizer/qldpc/translationally_invariant_stabilizer.yml +++ b/codes/quantum/properties/stabilizer/qldpc/translationally_invariant_stabilizer.yml @@ -57,8 +57,8 @@ relations: - code_id: quantum_double_abelian detail: 'Translation-invariant 2D prime-qudit topological stabilizer codes are equivalent to several copies of the prime-qudit surface code via a local constant-depth Clifford circuit \cite{arxiv:1812.11193}.' - code_id: tqd_abelian - detail: 'Translationally-invariant stabilizer codes can realize 2D modular gapped abelian topological orders \cite{arxiv:2211.03798}. - Conversely, abelian TQD codes need not be translationally invariant, and can realize multiple topological phases on one lattice.' + detail: 'Translationally-invariant stabilizer codes can realize 2D modular gapped Abelian topological orders \cite{arxiv:2211.03798}. + Conversely, Abelian TQD codes need not be translationally invariant, and can realize multiple topological phases on one lattice.' - code_id: fracton detail: 'Translationally-invariant stabilizer codes can realize fracton orders. Conversely, fracton codes need not be translationally invariant, and can realize multiple phases on one lattice.' diff --git a/codes/quantum/qubits/dynamic_gen/da/honeycomb.yml b/codes/quantum/qubits/dynamic_gen/da/honeycomb.yml index f4d45ea7e..aacd9472b 100644 --- a/codes/quantum/qubits/dynamic_gen/da/honeycomb.yml +++ b/codes/quantum/qubits/dynamic_gen/da/honeycomb.yml @@ -34,7 +34,7 @@ features: - 'The ISG has a static subgroup for all time steps \(r\geq 3\) – that is, a subgroup which remains a subgroup of the ISG for all future times – given by so-called \textit{plaquette stabilizers}. These are stabilizers consisting of products of check operators around homologically trivial paths. The syndrome bits correspond to the eigenvalues of the plaquette stabilizers. Because of the structure of the check operators, only one-third of all plaquettes are measured each round. The syndrome bits must therefore be represented by a lattice in spacetime, to reflect when and where the outcome was obtained.' fault_tolerance: - - 'One can run a fault-tolerant decoding algorithm by (1) bipartitioning the syndrome lattice into two graphs which are congruent to the Cayley graph of the free abelian group with three generators (up to boundary conditions) and (2) performing a matching algorithm to deduce errors.' + - 'One can run a fault-tolerant decoding algorithm by (1) bipartitioning the syndrome lattice into two graphs which are congruent to the Cayley graph of the free Abelian group with three generators (up to boundary conditions) and (2) performing a matching algorithm to deduce errors.' threshold: - '\(0.2\%-0.3\%\) in a controlled-not circuit model with a correlated minimum-weight perfect-matching decoder \cite{arXiv:2108.10457}.' diff --git a/codes/quantum/qubits/fermions/mbq.yml b/codes/quantum/qubits/fermions/mbq.yml index 89378e777..2716a7ecb 100644 --- a/codes/quantum/qubits/fermions/mbq.yml +++ b/codes/quantum/qubits/fermions/mbq.yml @@ -32,7 +32,7 @@ relations: - code_id: small_distance - code_id: topological_abelian detail: 'When treated as ground states of the code Hamiltonian, surface codewords realize, codewords of a single Kitaev chain realize \(\mathbb{Z}_2\) fermionic topological order. - The MZMs used to define the tetron code act as Ising anyons, which are nonabelian.' + The MZMs used to define the tetron code act as Ising anyons, which are non-Abelian.' cousins: - code_id: hamiltonian detail: 'The tetron code forms the ground-state subspace of two Kitaev Majorana chain Hamiltonians.' diff --git a/codes/quantum/qubits/stabilizer/qldpc/algebraic/qcga.yml b/codes/quantum/qubits/stabilizer/qldpc/algebraic/qcga.yml index 6e05439b7..31eeed1bd 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/algebraic/qcga.yml +++ b/codes/quantum/qubits/stabilizer/qldpc/algebraic/qcga.yml @@ -7,7 +7,7 @@ code_id: qcga physical: qubits logical: qubits -name: 'Bravyi-Cross-Gambetta-Maslov-Rall-Yoder (BCGMRY) group-algebra code' +name: 'Bravyi-Cross-Gambetta-Maslov-Rall-Yoder (BCGMRY) code' short_name: 'BCGMRY code' introduced: '\cite{arxiv:2308.07915}' diff --git a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml b/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml index 59aa0f936..23b775f69 100644 --- a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml +++ b/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml @@ -77,7 +77,7 @@ notes: relations: parents: - code_id: cws - detail: 'If the CWS set \( \mathcal{W} \) is an abelian group not containing \(-I\), then the CWS code is a stabilizer code.' + detail: 'If the CWS set \( \mathcal{W} \) is an Abelian group not containing \(-I\), then the CWS code is a stabilizer code.' - code_id: xp_stabilizer detail: 'The XP stabilizer formalism reduces to the Pauli formalism at \(N=2\).' - code_id: qudit_stabilizer diff --git a/codes/quantum/qubits/stabilizer/topological/color.yml b/codes/quantum/qubits/stabilizer/topological/color.yml index 6e108564c..53dc80726 100644 --- a/codes/quantum/qubits/stabilizer/topological/color.yml +++ b/codes/quantum/qubits/stabilizer/topological/color.yml @@ -10,7 +10,7 @@ logical: qubits name: 'Color code' introduced: '\cite{arxiv:quant-ph/0605138}' -description: 'A family of abelian topological \hyperref[code:css]{CSS stabilizer} codes defined on a \(D\)-dimensional lattice which satisfies two properties: The lattice is (1) a homogeneous simplicial \(D\)-complex obtained as a triangulation of the interior of a \(D\)-simplex and (2) is \(D+1\)-colorable. Qubits are placed on the \(D\)-simplices and generators are supported on suitable simplices \cite{doi:10.7907/059V-MG69}. For 2-dimensional color code, the lattice must be such that it is 3-valent and has 3-colorable faces, such as a honeycomb lattice. The qubits are placed on the vertices and two stabilizer generators are placed on each face \cite{arXiv:1311.0277}.' +description: 'A family of Abelian topological \hyperref[code:css]{CSS stabilizer} codes defined on a \(D\)-dimensional lattice which satisfies two properties: The lattice is (1) a homogeneous simplicial \(D\)-complex obtained as a triangulation of the interior of a \(D\)-simplex and (2) is \(D+1\)-colorable. Qubits are placed on the \(D\)-simplices and generators are supported on suitable simplices \cite{doi:10.7907/059V-MG69}. For 2-dimensional color code, the lattice must be such that it is 3-valent and has 3-colorable faces, such as a honeycomb lattice. The qubits are placed on the vertices and two stabilizer generators are placed on each face \cite{arXiv:1311.0277}.' protection: 'As with the surface code, the code distance depends on the specific kind of lattice used to define the code. More precisely, the distance depends on the homology of logical string operators \cite{arXiv:1311.0277}.' @@ -69,7 +69,7 @@ relations: detail: 'The 2D color code has been extended to Galois qudits.' - code_id: surface detail: 'The 3D color code is equivalent to multiple decoupled copies of the 2D surface code \cite{arxiv:1007.4601,arxiv:1503.02065,arXiv:1804.00866}. - Conversely, the 2D color code can \hyperref[topic:code-switching]{condense} to form the 2D surface code in nine different ways, i.e., by adding two body hopping terms along one of its three hexagonal directions to the stabilizer group and then taking the center of the resulting nonabelian group \cite{arxiv:2212.00042}.' + Conversely, the 2D color code can \hyperref[topic:code-switching]{condense} to form the 2D surface code in nine different ways, i.e., by adding two body hopping terms along one of its three hexagonal directions to the stabilizer group and then taking the center of the resulting non-Abelian group \cite{arxiv:2212.00042}.' - code_id: higher_dimensional_surface detail: 'The color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D-1\)-dimensional surface code \cite{arxiv:1007.4601,arxiv:1503.02065,arXiv:1804.00866}.' - code_id: quantum_triorthogonal diff --git a/codes/quantum/qubits/stabilizer/topological/matching.yml b/codes/quantum/qubits/stabilizer/topological/matching.yml index c04ff4d28..bc1cc7e74 100644 --- a/codes/quantum/qubits/stabilizer/topological/matching.yml +++ b/codes/quantum/qubits/stabilizer/topological/matching.yml @@ -10,7 +10,7 @@ logical: qubits name: 'Matching code' introduced: '\cite{arxiv:1501.07779}' -description: 'Member of a class of qubit stabilizer codes based on the abelian phase of the Kitaev honeycomb model.' +description: 'Member of a class of qubit stabilizer codes based on the Abelian phase of the Kitaev honeycomb model.' realizations: - 'Braiding of defects has been demonstrated for a five-qubit version of code \cite{arxiv:1609.07774}.' diff --git a/codes/quantum/qubits/stabilizer/topological/surface/two_dim/surface/surface.yml b/codes/quantum/qubits/stabilizer/topological/surface/two_dim/surface/surface.yml index b7a158720..ad2c4fc8b 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/two_dim/surface/surface.yml +++ b/codes/quantum/qubits/stabilizer/topological/surface/two_dim/surface/surface.yml @@ -15,7 +15,7 @@ alternative_names: - 'Kitaev toric code' description: | - A family of abelian topological \hyperref[code:css]{CSS stabilizer} codes + A family of Abelian topological \hyperref[code:css]{CSS stabilizer} codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a cellulation of a two-dimensional surface (with a qubit located at each edge of the diff --git a/codes/quantum/qubits/subsystem/subsystem_quantum_parity.yml b/codes/quantum/qubits/subsystem/subsystem_quantum_parity.yml index 470f426da..823ade6c9 100644 --- a/codes/quantum/qubits/subsystem/subsystem_quantum_parity.yml +++ b/codes/quantum/qubits/subsystem/subsystem_quantum_parity.yml @@ -36,7 +36,7 @@ description: | # \end{align} # Next, consider a square lattice of size \(n \times n\) with a qubit located at each vertex, where \(n = n_1 + n_2\). # Let \(\mathsf{T}_1\) (\(\mathsf{T}_2\)) be the stabilizer group consisting of \(\mathsf{S}_1\) (\(\mathsf{S}_2\)) operators acting on all columns (rows) of the lattice. -# Then, \(\mathsf{T} = \langle\mathsf{T}_1,\mathsf{T}_2\rangle\) is a nonabelian group from which we can construct an Abelian invariant subgroup \(\mathsf{S}\). +# Then, \(\mathsf{T} = \langle\mathsf{T}_1,\mathsf{T}_2\rangle\) is a non-Abelian group from which we can construct an Abelian invariant subgroup \(\mathsf{S}\). # This subgroup is generated by \(\mathsf{S}_1\) and \(\mathsf{S}_2\) for all stabilizer codeword combinations in both rows and columns. # By construction, \(\mathsf{S}\) is a stabilizer with \((n_1 - k_1)k_2\) stabilizer generators (i.e., the tensor product of \(I\) and \(Z\) operators), and \((n_2 - k_2)k_1\) stabilizer generators (i.e., the tensor product of \(I\) and \(X\) operators). diff --git a/codes/quantum/qubits/subsystem/subsystem_stabilizer.yml b/codes/quantum/qubits/subsystem/subsystem_stabilizer.yml index a5f73dace..5f87adaa9 100644 --- a/codes/quantum/qubits/subsystem/subsystem_stabilizer.yml +++ b/codes/quantum/qubits/subsystem/subsystem_stabilizer.yml @@ -40,7 +40,7 @@ features: notes: - - 'When the gauge group \( \mathsf{G} \) is abelian, the above is reduced to the standard stabilizer formalism.' + - 'When the gauge group \( \mathsf{G} \) is Abelian, the above is reduced to the standard stabilizer formalism.' relations: parents: diff --git a/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml b/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml index 19d5308be..27428c7be 100644 --- a/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml +++ b/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml @@ -11,14 +11,14 @@ name: 'Kitaev honeycomb code' introduced: '\cite{arxiv:cond-mat/0506438,arxiv:1701.05052}' description: | - Code whose logical subspace is labeled by different fusion outcomes of Ising anyons present in the nonabelian topological phase of the Kitaev honeycomb model \cite{arXiv:cond-mat/0506438}. - Each logical qubit is constructed out of four Majorana operators, which admit braiding-based gates due to their nonabelian statistics and which can be used for topological quantum computation. + Code whose logical subspace is labeled by different fusion outcomes of Ising anyons present in the non-Abelian topological phase of the Kitaev honeycomb model \cite{arXiv:cond-mat/0506438}. + Each logical qubit is constructed out of four Majorana operators, which admit braiding-based gates due to their non-Abelian statistics and which can be used for topological quantum computation. - The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code based on the \(\mathbb{Z}_2^{(1)}\) abelian anyon theory, which is non-chiral and non-modular \cite[Sec. 7.3]{arxiv:2211.03798}. + The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code based on the \(\mathbb{Z}_2^{(1)}\) Abelian anyon theory, which is non-chiral and non-modular \cite[Sec. 7.3]{arxiv:2211.03798}. The model realizes all anyon theories of the 16-fold way, i.e., all minimal modular extensions of the \(\mathbb{Z}_2^{(1)}\) anyon theory \cite{arxiv:cond-mat/0506438}\cite[Footnote 25]{arXiv:2211.03798}. - The Kitaev honeycomb code utilizes the nonabelian Ising-anyon topological order of the Kitaev honeycomb model \cite{arXiv:cond-mat/0506438} (a.k.a. \(p+ip\) superconducting phase \cite{arxiv:1104.5485}) as well as abelian \(\mathbb{Z}_2\) topological order. - More generally, the Hamiltonian realizes all 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}. + The Kitaev honeycomb code utilizes the non-Abelian Ising-anyon topological order of the Kitaev honeycomb model \cite{arXiv:cond-mat/0506438} (a.k.a. \(p+ip\) superconducting phase \cite{arxiv:1104.5485}) as well as Abelian \(\mathbb{Z}_2\) topological order. + More generally, the Hamiltonian realizes all 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}. features: encoders: @@ -32,7 +32,7 @@ relations: - code_id: majorana_stab detail: 'While the Kitaev honeycomb model is bosonic, a fermionic mapping is useful for solving and understanding the model. Logical subspace of the Kitaev honeycomb code can be formulated as a joint eigenspace of certain Majorana operators \cite[Sec. 4.1]{arxiv:1701.05052}. Logical Paulis are also be constructed out of Majorana operators.' - code_id: qudit_znone - detail: 'The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code based on the \(\mathbb{Z}_{q=2}^{(1)}\) abelian anyon theory, which is non-chiral and non-modular \cite[Sec. 7.3]{arxiv:2211.03798}.' + detail: 'The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code based on the \(\mathbb{Z}_{q=2}^{(1)}\) Abelian anyon theory, which is non-chiral and non-modular \cite[Sec. 7.3]{arxiv:2211.03798}.' cousins: - code_id: surface detail: 'The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code. diff --git a/codes/quantum/qudits/qudit_stabilizer.yml b/codes/quantum/qudits/qudit_stabilizer.yml index de897b2c4..f55df058d 100644 --- a/codes/quantum/qudits/qudit_stabilizer.yml +++ b/codes/quantum/qudits/qudit_stabilizer.yml @@ -17,7 +17,7 @@ description: | Each code can be represented by a \textit{check matrix} (a.k.a. \textit{stabilizer generator matrix}) \(H=(A|B)\), where each row \((a|b)\) is the \(q\)-ary symplectic representation of a stabilizer generator. The check matrix can be brought into standard form via Gaussian elimination \cite{arxiv:1101.1519}. - One can switch between stabilizer codes by appending another abelian subgroup of the Pauli group to the stabilizer group and taking the center of the resulting larger group. + One can switch between stabilizer codes by appending another Abelian subgroup of the Pauli group to the stabilizer group and taking the center of the resulting larger group. \begin{defterm}{Code switching} \label{topic:code-switching} Code switching is a map between stabilizer codes that is done using a stabilizer group \(\mathsf{F}\) of the \(n\)-qudit Pauli group, @@ -26,7 +26,7 @@ description: | \end{align} where \(\mathsf{Z}\) denotes taking the center of a group. Code switching may not preserve the logical information and instead implement logical measurements; conditions on \(\mathsf{S}\) and \(\mathsf{F}\) such that qubit stabilizer code switching preserves logical information are derived in \cite[Prop. II.1]{arxiv:2304.01277}. - In the context of abelian topological stabilizer codes, code switching implements \textit{anyon condensation} of any anyons represented by operators in the group \(\mathsf{F}\). + In the context of Abelian topological stabilizer codes, code switching implements \textit{anyon condensation} of any anyons represented by operators in the group \(\mathsf{F}\). \end{defterm} diff --git a/codes/quantum/qudits/subsystem/qudit_subsystem_stabilizer.yml b/codes/quantum/qudits/subsystem/qudit_subsystem_stabilizer.yml index ca03e57e3..bd4a3c48b 100644 --- a/codes/quantum/qudits/subsystem/qudit_subsystem_stabilizer.yml +++ b/codes/quantum/qudits/subsystem/qudit_subsystem_stabilizer.yml @@ -28,7 +28,7 @@ description: | One can gauge fix \cite{arxiv:quant-ph/0508131} an Abelian subgroup of the gauge group by adding it to the stabilizer group. \begin{defterm}{Gauge fixing} \label{topic:gauge-fixing} - Gauge fixing is a map between subsystem codes that is done using an abelian subgroup \(\mathsf{F}\subseteq\mathsf{G}\), + Gauge fixing is a map between subsystem codes that is done using an Abelian subgroup \(\mathsf{F}\subseteq\mathsf{G}\), \begin{align} \begin{split} \mathsf{S}&\to\left\langle \mathsf{S},\mathsf{F}\right\rangle \\ diff --git a/codes/quantum/qudits/subsystem/topological/topological_abelian.yml b/codes/quantum/qudits/subsystem/topological/topological_abelian.yml index 7c43749f9..40453648b 100644 --- a/codes/quantum/qudits/subsystem/topological/topological_abelian.yml +++ b/codes/quantum/qudits/subsystem/topological/topological_abelian.yml @@ -20,7 +20,7 @@ description: | \end{align} between all anyon pairs \(a,b\). - All 2D abelian topological orders can be understood within the subsystem stabilizer formalism \cite{arxiv:2211.03798}. + All 2D Abelian topological orders can be understood within the subsystem stabilizer formalism \cite{arxiv:2211.03798}. As such, many of the operations one can perform on such codes have both a stabilizer and a topological-phase interpretation. Stabilizer generators of 2D topological codes acting on 1D loops of qubits can be interpreted as one-form symmetries of the underlying phase realized by the code. Identification of an anyon \(a\) with the vacuum is equivalent to adding string excitation operators corresponding to \(a\) to the stabilizer group and taking the center to get another stabilizer group. @@ -45,17 +45,17 @@ features: relations: parents: - code_id: qudit_subsystem_stabilizer - detail: 'All Abelian 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}. + detail: 'All Abelian 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.' - code_id: topological detail: 'All Abelian topological orders can be realized as modular-qudit subsystem stabilizer codes \cite{arxiv:2211.03798}. - Nonabelian topological orders are purported not to be realizable with Pauli stabilizer codes \cite{arxiv:1605.03601}.' + Non-Abelian topological orders are purported not to be realizable with Pauli stabilizer codes \cite{arxiv:1605.03601}.' cousins: - 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}.' + 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}.' + 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}.' # Begin Entry Meta Information diff --git a/codes/quantum/qudits/topological/quantum_double_abelian.yml b/codes/quantum/qudits/topological/quantum_double_abelian.yml index dd035b623..b4c737a67 100644 --- a/codes/quantum/qudits/topological/quantum_double_abelian.yml +++ b/codes/quantum/qudits/topological/quantum_double_abelian.yml @@ -12,8 +12,8 @@ introduced: '\cite{arXiv:quant-ph/9707021}' description: | Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order with trivial cocycle. - The corresponding anyon theory is defined by an abelian group. - All such codes can be realized by a stack of modular-qudit surface codes because all abelian groups are Kronecker products of cyclic groups. + The corresponding anyon theory is defined by an Abelian group. + All such codes can be realized by a stack of modular-qudit surface codes because all Abelian groups are Kronecker products of cyclic groups. protection: @@ -24,11 +24,11 @@ protection: relations: parents: - code_id: tqd_abelian - detail: 'The anyon theory corresponding to abelian quantum double codes is defined by an abelian group and trivial cocycle. - 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}. - Upon gauging, Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain abelian quantum double models \cite{arxiv:hep-th/9511195,arxiv:1508.03468,arxiv:2112.12757}.' + detail: 'The anyon theory corresponding to Abelian quantum double codes is defined by an Abelian group and trivial cocycle. + 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}. + Upon gauging, Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain Abelian quantum double models \cite{arxiv:hep-th/9511195,arxiv:1508.03468,arxiv:2112.12757}.' - code_id: quantum_double - detail: 'The anyon theory corresponding to (abelian) quantum double codes is defined by an (abelian) group.' + detail: 'The anyon theory corresponding to (Abelian) quantum double codes is defined by an (Abelian) group.' # Begin Entry Meta Information diff --git a/codes/quantum/qudits/topological/tqd_abelian.yml b/codes/quantum/qudits/topological/tqd_abelian.yml index db0a84f17..b34a259b5 100644 --- a/codes/quantum/qudits/topological/tqd_abelian.yml +++ b/codes/quantum/qudits/topological/tqd_abelian.yml @@ -12,24 +12,24 @@ introduced: '\cite{arxiv:1008.0654,arxiv:1211.3695,arxiv:2107.13091}' description: | Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order. - The corresponding anyon theory is defined by an abelian group and a Type-III group cocycle that can be decomposed as a product of Type-I and Type-II group cocycles; see \cite[Sec. IV.A]{arXiv:2112.11394}. + The corresponding anyon theory is defined by an Abelian group and a Type-III group cocycle that can be decomposed as a product of Type-I and Type-II group cocycles; see \cite[Sec. IV.A]{arXiv:2112.11394}. 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}.' + 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: topological_abelian detail: 'Abelian TQDs realize all modular gapped Abelian topological orders \cite{arxiv:2112.11394}. - Conversely, every abelian anyon theory is a subtheory of some TQD \cite[Sec. 6.2]{arxiv:2211.03798}. - 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}.' + Conversely, every Abelian anyon theory is a subtheory of some TQD \cite[Sec. 6.2]{arxiv:2211.03798}. + 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: tqd - detail: 'The anyon theory corresponding to (abelian) TQD codes is defined by an (abelian) group and a Type III cocycle.' + detail: 'The anyon theory corresponding to (Abelian) TQD codes is defined by an (Abelian) group and a Type III cocycle.' cousins: - code_id: quantum_double_dihedral detail: 'Upon gauging, a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model \cite{arxiv:hep-th/9511195,arxiv:1508.03468,arxiv:2112.12757}.' - code_id: double_semion - 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 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}.' # Begin Entry Meta Information diff --git a/codes/quantum/qudits_galois/qldpc/expander_lifted_product.yml b/codes/quantum/qudits_galois/qldpc/expander_lifted_product.yml index 4f709cfc7..3d84e1a93 100644 --- a/codes/quantum/qudits_galois/qldpc/expander_lifted_product.yml +++ b/codes/quantum/qudits_galois/qldpc/expander_lifted_product.yml @@ -18,7 +18,7 @@ description: | The small classical codes used in the construction of good QLDPC codes are required to have a certain product-expansion property (Lemma 10 in Ref. \cite{arXiv:2111.03654}); it is proven that random codes satisfy said property in the thermodynamic limit. -protection: 'Code performance strongly depends on \(G\). Certain nonabelian groups yield asymptotically good QLDPC codes with parameters \([[n, k = \Theta(n), d = \Theta(n)]]\) \cite{arXiv:2111.03654}. Abelian groups like \(\mathbb{Z}_{\ell}\) for \(\ell=\Theta(n / \log n)\) yield constant-rate codes with parameters \([[n, k = \Theta(n), d = \Theta(n / \log n)]]\) \cite{arxiv:2012.04068}; this construction can be derandomized by being reformulated as a balanced product code \cite{arXiv:2012.09271}.' +protection: 'Code performance strongly depends on \(G\). Certain non-Abelian groups yield asymptotically good QLDPC codes with parameters \([[n, k = \Theta(n), d = \Theta(n)]]\) \cite{arXiv:2111.03654}. Abelian groups like \(\mathbb{Z}_{\ell}\) for \(\ell=\Theta(n / \log n)\) yield constant-rate codes with parameters \([[n, k = \Theta(n), d = \Theta(n / \log n)]]\) \cite{arxiv:2012.04068}; this construction can be derandomized by being reformulated as a balanced product code \cite{arXiv:2012.09271}.' features: rate: 'Expander lifted-product codes include the first examples \cite{arXiv:2111.03654} of (asymptotically) \textit{good QLDPC codes}, i.e., codes with asymptotically constant rate and linear distance. The existence of such codes proves the QLDPC conjecture \cite{arXiv:2103.06309}. Another notable family encodes \(k \in \Theta(n^\alpha \log n)\) logical qubits with distance \(d \in \Omega(n^{1 - \alpha} / \log n)\) for any number of physical qubits \(n\) and any real parameter \(0 \leq \alpha < 1\) \cite{arxiv:2012.04068}.' diff --git a/codes/quantum/qudits_galois/qldpc/galois_topological.yml b/codes/quantum/qudits_galois/qldpc/galois_topological.yml index d43beff46..b1c3ba544 100644 --- a/codes/quantum/qudits_galois/qldpc/galois_topological.yml +++ b/codes/quantum/qudits_galois/qldpc/galois_topological.yml @@ -21,7 +21,7 @@ relations: cousins: - code_id: quantum_double_abelian detail: 'A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits \cite{doi:10.1109/18.959288}; see Sec. 5.3 of Ref. \cite{arxiv:quant-ph/0501074}. - Galois-qudit topological surface and color codes yield abelian quantum-double codes via this decomposition.' + Galois-qudit topological surface and color codes yield Abelian quantum-double codes via this decomposition.' # Begin Entry Meta Information