From 8c55e2baf0b5c58758f6e7ca7e629bea2e081eb7 Mon Sep 17 00:00:00 2001 From: VVA2024 Date: Tue, 9 Jul 2024 09:43:25 -0400 Subject: [PATCH] topological refs --- codes/quantum/properties/block/topological/spt.yml | 3 +++ codes/quantum/properties/block/topological/topological.yml | 7 ++++++- .../quantum/properties/hamiltonian/commuting_projector.yml | 6 ++++-- codes/quantum/properties/hamiltonian/frustration_free.yml | 7 +++++++ codes/quantum/properties/hamiltonian/hamiltonian.yml | 5 +++-- codes/quantum/properties/qecc_finite.yml | 2 +- codes/quantum/quantum_into_quantum.yml | 5 ++--- .../stabilizer/topological/surface/higher_d/3d_surface.yml | 3 ++- 8 files changed, 28 insertions(+), 10 deletions(-) diff --git a/codes/quantum/properties/block/topological/spt.yml b/codes/quantum/properties/block/topological/spt.yml index 88a7e4712..4b41602c2 100644 --- a/codes/quantum/properties/block/topological/spt.yml +++ b/codes/quantum/properties/block/topological/spt.yml @@ -16,6 +16,9 @@ protection: | SPT codes typically do not offer protection against generic errors, but can protect against noise that respects the underlying symmetry. +notes: + - 'Review on generalized (i.e., non-tensor-product) symmetries \cite{arxiv:2204.03045}.' + relations: parents: - code_id: topological diff --git a/codes/quantum/properties/block/topological/topological.yml b/codes/quantum/properties/block/topological/topological.yml index 6d98bf315..a6374e3c1 100644 --- a/codes/quantum/properties/block/topological/topological.yml +++ b/codes/quantum/properties/block/topological/topological.yml @@ -93,6 +93,10 @@ protection: | This condition implies that any operator supported solely on \(A\) cannot distinguish the global projector from the local one \cite{arxiv:1001.4363,arxiv:2405.19412}. \end{defterm} + A notion of topological order generalizing both the \hyperref[topic:cleaning-lemma]{cleaning lemma} and the \hyperref[topic:tqo]{TQO conditions} is \textit{homogeneous topological order} \cite{arxiv:2009.13551}. + Related topological order definitions include equivalence under course-graining (i.e., renormalization group) \cite{arxiv:1406.5090,arxiv:1407.8203}. + See \cite[Sec. 4]{arxiv:2009.13551} for a discussion. + features: rate: 'The logical dimension \(K\) of 2D topological codes described by unitary modular fusion categories depends on the type of manifold \(\Sigma^2\) that is tesselated to form the many-body system. @@ -109,7 +113,7 @@ features: notes: - - 'Ref. \cite[Appx. F]{arxiv:cond-mat/0506438}\cite{doi:10.7907/5NDZ-W890,arxiv:0707.1889,doi:10.1017/9781009212717,arxiv:1508.02595,arxiv:1610.03911,arxiv:2205.05565} for introductions to topological phases.' + - 'Ref. \cite[Appx. F]{arxiv:cond-mat/0506438}\cite{doi:10.7907/5NDZ-W890,arxiv:0707.1889,arxiv:1508.02595,arxiv:1610.03911,doi:10.1017/9781316226308,arxiv:2205.05565} for introductions to topological phases.' - 'See \href{https://anyonwiki.github.io/}{AnyonWiki} for lists of categories relevant to anyons.' @@ -125,6 +129,7 @@ relations: detail: 'There exist necessary and sufficient conditions for a family of cluster states to exhibit the TQO-1 property \cite{arxiv:2112.02502}.' + # Begin Entry Meta Information _meta: # Change log - most recent first diff --git a/codes/quantum/properties/hamiltonian/commuting_projector.yml b/codes/quantum/properties/hamiltonian/commuting_projector.yml index 98f260a19..9b0c2f1b5 100644 --- a/codes/quantum/properties/hamiltonian/commuting_projector.yml +++ b/codes/quantum/properties/hamiltonian/commuting_projector.yml @@ -12,7 +12,7 @@ description: | Hamiltonian-based code whose Hamiltonian terms can be expressed as orthogonal projectors (i.e., Hermitian operators with eigenvalues 0 or 1) that commute with each other. protection: | - Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the TO conditions, meaning that a notion of a phase can be defined \cite{arxiv:1001.4363,arxiv:1001.0344,arxiv:1109.1588,arxiv:1810.02428,arxiv:2010.15337}. + Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the \hyperref[topic:tqo]{TQO conditions}, meaning that a notion of a phase can be defined \cite{arxiv:1001.4363,arxiv:1001.0344,arxiv:1109.1588,arxiv:1810.02428,arxiv:2010.15337}. This notion can be extended to semi-hyperbolic manifolds \cite{arxiv:2405.19412}. 2D topological order on qubit manifolds requires weight-four Hamiltonian terms, i.e., it cannot be stabilized via weight-two or weight-three terms on nearly Euclidean geometries of qubits or qutrits \cite{arxiv:quant-ph/0308021,arxiv:1102.0770,arxiv:1803.02213}. @@ -22,9 +22,11 @@ protection: | relations: parents: - code_id: hamiltonian + detail: 'Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the \hyperref[topic:tqo]{TQO conditions}, meaning that a notion of a phase can be defined \cite{arxiv:1001.4363,arxiv:1001.0344,arxiv:1109.1588,arxiv:1810.02428,arxiv:2010.15337}. + This notion can be extended to semi-hyperbolic manifolds \cite{arxiv:2405.19412}.' cousins: - code_id: topological - detail: 'Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the TO conditions, meaning that a notion of a phase can be defined \cite{arxiv:1001.4363,arxiv:1001.0344,arxiv:1109.1588,arxiv:1810.02428,arxiv:2010.15337}. + detail: 'Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the \hyperref[topic:tqo]{TQO conditions}, meaning that a notion of a phase can be defined \cite{arxiv:1001.4363,arxiv:1001.0344,arxiv:1109.1588,arxiv:1810.02428,arxiv:2010.15337}. This notion can be extended to semi-hyperbolic manifolds \cite{arxiv:2405.19412}.' diff --git a/codes/quantum/properties/hamiltonian/frustration_free.yml b/codes/quantum/properties/hamiltonian/frustration_free.yml index 5a612d9cb..71e074c5c 100644 --- a/codes/quantum/properties/hamiltonian/frustration_free.yml +++ b/codes/quantum/properties/hamiltonian/frustration_free.yml @@ -11,6 +11,9 @@ name: 'Frustration-free Hamiltonian code' description: | Hamiltonian-based code whose Hamiltonian is frustration free, i.e., whose ground states minimize the energy of each term. +protection: | + Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the \textit{local topological quantum order} (LTQO) condition (cf. the \hyperref[topic:tqo]{TQO conditions}), meaning that a notion of a phase can be defined \cite{arxiv:1109.1588,arxiv:2110.11194}. + features: encoders: - 'Lindbladian-based dissipative encoding can be constructed for a codespace that is the ground-state subspace of a frustration-free Hamiltonian \cite{arxiv:0809.0613,arxiv:1112.4860,arxiv:0803.1447,arxiv:1802.00010}.' @@ -19,10 +22,14 @@ features: relations: parents: - code_id: hamiltonian + - code_id: topological + detail: 'Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the local topological quantum order condition (cf. the \hyperref[topic:tqo]{TQO conditions}), meaning that a notion of a phase can be defined \cite{arxiv:1109.1588,arxiv:2110.11194}.' cousins: - code_id: commuting_projector detail: 'Frustration-free Hamiltonians can contain non-commuting projectors; an example is the AKLT model \cite{doi:10.1007/978-3-662-06390-3_18}. On the other hand, commuting-projector Hamiltonians can be frustrated; an example is the 1D classical Ising model on a circle for odd \(n\).' + - code_id: topological + detail: 'Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the local topological quantum order condition (cf. the \hyperref[topic:tqo]{TQO conditions}), meaning that a notion of a phase can be defined \cite{arxiv:1109.1588,arxiv:2110.11194}.' # Begin Entry Meta Information diff --git a/codes/quantum/properties/hamiltonian/hamiltonian.yml b/codes/quantum/properties/hamiltonian/hamiltonian.yml index 3670b8684..74333744f 100644 --- a/codes/quantum/properties/hamiltonian/hamiltonian.yml +++ b/codes/quantum/properties/hamiltonian/hamiltonian.yml @@ -31,7 +31,7 @@ description: | Hamiltonians realizing different phases cannot be adiabatically deformed into one another without a closing of the energy gap between the ground and excited states. Such adiabatic deformations naively would be generated by non-local Hamiltonians. - However, Hastings and Wen \cite{arXiv:cond-mat/0503554} (see also \cite{arXiv:quant-ph/0601019}) showed that adiabatic evolution can in fact be generated by a quasi-local operator; such evolution is often called \textit{quasi-adiabatic evolution}, \textit{quasi-adiabatic continuation}, or \textit{spectral flow}. + However, Hastings and Wen \cite{arXiv:cond-mat/0503554} (see also \cite{arXiv:quant-ph/0601019,arxiv:2205.10460}) showed that adiabatic evolution can in fact be generated by a quasi-local operator; such evolution is often called \textit{quasi-adiabatic evolution}, \textit{quasi-adiabatic continuation}, or \textit{spectral flow}. The unitary operation generated by a quasi-local Hamiltonian can be simulated by a quantum circuit, with the time of evolution determining the depth of the circuit. States in two different phases \textit{cannot} be deformed into one another via such a circuit \cite{arxiv:1004.3835}. @@ -46,7 +46,8 @@ protection: | A no-go theorem states that open-boundary MPS that form a degenerate ground-state space of a gapped local Hamiltonian yield codes with distance that is only constant in the number of qubits \(n\), so MPS excitation ansatze have to be used to achieve a distance scaling nontrivially with \(n\) \cite{arxiv:1902.02115} (see also Ref. \cite{arxiv:1407.3413}). notes: - - 'Reviews of quantum phases of matter \cite{arxiv:1508.02595,doi:10.1017/9781009212717}' + - 'Reviews of various quantum phases of matter and many-body systems \cite{doi:10.1007/978-3-662-02520-8,arxiv:1311.2717,arxiv:1508.02595,arxiv:1203.4565,doi:10.1017/9781009212717,doi:10.1007/978-3-030-41265-4,doi:10.1017/CBO9781139020916,doi:10.1017/9781316480649}.' + - 'Book on rigorous results on stability of non-topological phases \cite{doi:10.1017/CBO9780511819681}.' relations: diff --git a/codes/quantum/properties/qecc_finite.yml b/codes/quantum/properties/qecc_finite.yml index 903d19784..526097cef 100644 --- a/codes/quantum/properties/qecc_finite.yml +++ b/codes/quantum/properties/qecc_finite.yml @@ -35,7 +35,7 @@ protection: | \end{align} where the \textit{QEC matrix} elements \(c_{ij}\) are arbitrary complex numbers. \end{defterm} - The Knill-Laflamme conditions can alternatively be expressed in terms of the \hyperref[topic:complementary-channel]{complementary channel}, or in an information-theoretic way via a data processing inequality \cite{arxiv:quant-ph/9604022}\cite[Eq. (29)]{arxiv:quant-ph/9604034}. + The Knill-Laflamme conditions can alternatively be expressed in terms of the \hyperref[topic:complementary-channel]{complementary channel}, or in an information-theoretic way via a data processing inequality \cite{arxiv:quant-ph/9604022,arxiv:quant-ph/9702031}\cite[Eq. (29)]{arxiv:quant-ph/9604034}. They have been extended to sequences of multiple errors and rounds of correction \cite{arxiv:2405.17567}. \begin{defterm}{Degeneracy} diff --git a/codes/quantum/quantum_into_quantum.yml b/codes/quantum/quantum_into_quantum.yml index 94d7600ad..796aa739b 100644 --- a/codes/quantum/quantum_into_quantum.yml +++ b/codes/quantum/quantum_into_quantum.yml @@ -13,10 +13,9 @@ description: | Code designed for transmission of quantum and, optionally, classical information through a quantum channel for the purposes of robust storage, communication, or sensing. Transmission can be performed with side information or entanglement. -# relations: -# parents: -# - code_id: eacq +notes: + - 'States of block quantum codes can be classified in terms of the complexity of their underlying encoding circuit; see \href{https://complexityzoo.net/Complexity_Zoo_Exhibit}{Complexity Zoo exhibit}.' # Begin Entry Meta Information _meta: diff --git a/codes/quantum/qubits/stabilizer/topological/surface/higher_d/3d_surface.yml b/codes/quantum/qubits/stabilizer/topological/surface/higher_d/3d_surface.yml index 61af85b04..d4e9fe01c 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/higher_d/3d_surface.yml +++ b/codes/quantum/qubits/stabilizer/topological/surface/higher_d/3d_surface.yml @@ -67,7 +67,8 @@ relations: - code_id: higher_dimensional_surface - code_id: 3d_stabilizer - code_id: topological_abelian - detail: 'The 3D Kitaev surface code realizes 3D \(\mathbb{Z}_2\) gauge theory with bosonic charge and loop excitations (BcBl).' + detail: 'The 3D Kitaev surface code realizes 3D \(\mathbb{Z}_2\) gauge theory with bosonic charge and loop excitations (BcBl). + The welded surface code does not satisfy homogeneous topological order \cite{arxiv:2009.13551}.' - code_id: xyz_product detail: 'The 3D planar (3D toric) code can be obtained from a hypergraph product of three repetition (cyclic) codes \cite[Exam. A.1]{arxiv:2311.01328}, but done in a different way than the Chamon code; see \cite[Sec. 3.4]{arxiv:2011.09746}.' cousins: