From b5897d55d925da9ce291f200c449224cf1145c6b Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Sun, 15 Dec 2024 23:16:32 -0800 Subject: [PATCH] refs --- .../properties/block/distributed_storage/lrc/lcc.yml | 2 ++ codes/quantum/groups/topological/quantum_double.yml | 2 +- .../oscillators/fock_state/rotation/chebyshev.yml | 4 +++- .../oscillators/fock_state/rotation/number_phase.yml | 4 +++- codes/quantum/properties/asymmetric_qecc.yml | 2 ++ codes/quantum/properties/block/block_quantum.yml | 2 +- .../block/topological/topological_abelian.yml | 11 +---------- .../subsystem/translationally_invariant_subsystem.yml | 7 +++++++ .../qubits/subsystem/topological/kitaev_honeycomb.yml | 2 ++ 9 files changed, 22 insertions(+), 14 deletions(-) diff --git a/codes/classical/properties/block/distributed_storage/lrc/lcc.yml b/codes/classical/properties/block/distributed_storage/lrc/lcc.yml index 17d7c55a1..0d00bc8a3 100644 --- a/codes/classical/properties/block/distributed_storage/lrc/lcc.yml +++ b/codes/classical/properties/block/distributed_storage/lrc/lcc.yml @@ -42,6 +42,8 @@ relations: detail: 'There are relations between LDCs and LTCs \cite{doi:10.1007/978-3-642-15369-3_50}.' - code_id: quantum_locally_recoverable detail: 'There is not a natural quantum version of LCCs \cite[Thm. 9]{arxiv:2311.08653}.' + - code_id: analog + detail: 'LCCs can also be defined over the real or complex numbers, and there are no complex 2-query LCCs \cite{arxiv:1009.4375}.' # Begin Entry Meta Information diff --git a/codes/quantum/groups/topological/quantum_double.yml b/codes/quantum/groups/topological/quantum_double.yml index ed34d203d..10a8279c8 100644 --- a/codes/quantum/groups/topological/quantum_double.yml +++ b/codes/quantum/groups/topological/quantum_double.yml @@ -17,7 +17,7 @@ description: | 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, while local excitations are characterized by anyons. - Different types of anyons are labeled by irreducible representations of the group's quantum double algebra, \(D(G)\) (a.k.a. Drinfield center) \cite{arxiv:1006.5479}. + Different types of anyons are labeled by irreducible representations of the group's quantum double algebra, \(D(G)\) (a.k.a. Drinfield center) \cite{arxiv:1006.5479,arxiv:2310.19661}. Not all isomorphic non-Abelian groups give rise to different quantum doubles \cite{arxiv:math/0605530}. 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}. diff --git a/codes/quantum/oscillators/fock_state/rotation/chebyshev.yml b/codes/quantum/oscillators/fock_state/rotation/chebyshev.yml index 6ef398dee..f1b85fca3 100644 --- a/codes/quantum/oscillators/fock_state/rotation/chebyshev.yml +++ b/codes/quantum/oscillators/fock_state/rotation/chebyshev.yml @@ -9,7 +9,9 @@ name: 'Chebyshev code' introduced: '\cite{arxiv:1811.01450}' description: | - Single-mode bosonic Fock-state code that can be used for error-corrected sensing of a signal Hamiltonian \({\hat n}^s\), where \({\hat n}\) is the occupation number operator. Codewords for the \(s\)th-order Chebyshev code are + Single-mode bosonic Fock-state code that can be used for error-corrected sensing of a signal Hamiltonian \({\hat n}^s\), where \({\hat n}\) is the occupation number operator. + + Codewords for the \(s\)th-order Chebyshev code are \begin{align} \begin{split} \ket{\overline 0} &=\sum_{k \text{~even}}^{[0,s]} \tilde{c}_k \Ket{\left\lfloor M\sin^2\left( k\pi/{2s}\right) \right\rfloor},\\ diff --git a/codes/quantum/oscillators/fock_state/rotation/number_phase.yml b/codes/quantum/oscillators/fock_state/rotation/number_phase.yml index d266e5229..4d79fea5c 100644 --- a/codes/quantum/oscillators/fock_state/rotation/number_phase.yml +++ b/codes/quantum/oscillators/fock_state/rotation/number_phase.yml @@ -15,7 +15,9 @@ alternative_names: # Ouyang description: | - Bosonic rotation code consisting of superpositions of Pegg-Barnett phase states \cite{doi:10.1088/0305-4470/19/18/030}, + Bosonic rotation code consisting of superpositions of Pegg-Barnett phase states \cite{doi:10.1088/0305-4470/19/18/030}. + + Pegg-Barnett phase states are expressed in terms of Fock states as \begin{align} |\phi\rangle \equiv \frac{1}{\sqrt{2\pi}}\sum_{n=0}^{\infty} \mathrm{e}^{\mathrm{i} n \phi} \ket{n}. \end{align} diff --git a/codes/quantum/properties/asymmetric_qecc.yml b/codes/quantum/properties/asymmetric_qecc.yml index 0cab1910e..6882afc46 100644 --- a/codes/quantum/properties/asymmetric_qecc.yml +++ b/codes/quantum/properties/asymmetric_qecc.yml @@ -53,6 +53,8 @@ relations: detail: 'Random Clifford deformation can improve performance of surface codes against biased noise \cite{arxiv:2201.07802,arxiv:2211.02116}.' - code_id: xysurface detail: 'XY surface codes perform well against biased noise \cite{arxiv:1708.08474}.' + - code_id: xyz_product + detail: 'XYZ product codes can be used to protect against biased noise \cite{arxiv:2408.03123}.' - code_id: xyz_color detail: 'XYZ color codes perform well against biased noise \cite{arxiv:2203.16534}.' - code_id: twisted_xzzx diff --git a/codes/quantum/properties/block/block_quantum.yml b/codes/quantum/properties/block/block_quantum.yml index 6616ea307..c59a718ad 100644 --- a/codes/quantum/properties/block/block_quantum.yml +++ b/codes/quantum/properties/block/block_quantum.yml @@ -56,7 +56,7 @@ protection: | \begin{defterm}{Quantum GV bound} \label{topic:quantum-gv-bound} - The \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{doi:10.1109/TIT.2004.838088} (see also Refs. \cite{arxiv:quant-ph/9602022,arxiv:quant-ph/9906131,doi:10.1109/18.959288,doi:10.1016/j.jmaa.2007.08.023}) for Galois qudits states that a \hyperref[topic:quantum-weight-enumerator]{pure} \([[n,k,d]]_q\) Galois-qudit stabilizer code exists if + The \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{doi:10.1109/TIT.2004.838088} (see also Refs. \cite{arxiv:quant-ph/9602022,arxiv:quant-ph/9906131,doi:10.7907/m0xg-zs21,doi:10.1109/18.959288,doi:10.1016/j.jmaa.2007.08.023}) for Galois qudits states that a \hyperref[topic:quantum-weight-enumerator]{pure} \([[n,k,d]]_q\) Galois-qudit stabilizer code exists if \begin{align} \frac{q^{n-k+2}-1}{q^{2}-1}>\sum_{j=1}^{d-1}(q^{2}-1)^{j-1}\binom{n}{j}~. \end{align} diff --git a/codes/quantum/properties/block/topological/topological_abelian.yml b/codes/quantum/properties/block/topological/topological_abelian.yml index 8a096be36..e37452059 100644 --- a/codes/quantum/properties/block/topological/topological_abelian.yml +++ b/codes/quantum/properties/block/topological/topological_abelian.yml @@ -57,20 +57,11 @@ relations: parents: - code_id: topological 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}. - 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 _meta: diff --git a/codes/quantum/properties/subsystem/translationally_invariant_subsystem.yml b/codes/quantum/properties/subsystem/translationally_invariant_subsystem.yml index 4a1481646..190414256 100644 --- a/codes/quantum/properties/subsystem/translationally_invariant_subsystem.yml +++ b/codes/quantum/properties/subsystem/translationally_invariant_subsystem.yml @@ -19,6 +19,7 @@ description: | 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. + Lattice 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. features: rate: | @@ -43,6 +44,12 @@ relations: 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.' + - code_id: topological_abelian + 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. + Lattice 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. + 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/qubits/subsystem/topological/kitaev_honeycomb.yml b/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml index 2427fedb0..3bb4c1640 100644 --- a/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml +++ b/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml @@ -48,6 +48,8 @@ relations: During this process, the square lattice is effectively expanded to a hexagonal lattice \cite[Fig. 12]{arxiv:2211.03798}.' - code_id: hexagonal detail: 'The Kitaev honeycomb model is defined on the honeycomb lattice.' + - code_id: topological + detail: 'The Kitaev honeycomb 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}. This includes the (non-Abelian) Ising-anyon topological order \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.' # Begin Entry Meta Information