circuit complexity term, refs

codes/quantum/categories/gauge/dijkgraaf_witten.yml

@@ -22,10 +22,9 @@ description: |
 
 relations:
   parents:
+    - code_id: group_quantum
     - code_id: yetter_gauge_theory
-      detail: 'Replacing the two-group data in a two-gauge theory with a group and a cocycle reproduces the phase of the Dijkgraaf-Witten gauge theory, with the two theories equivalent in 2D \cite{arxiv:1309.4721}.
-      Generalizations of Ocneanu''s tube algebras \cite{manual:{Ocneanu, Adrian. "Chirality for operator algebras." Subfactors (Kyuzeso, 1993) 39 (1994).},doi:10.2969/aspm/03110235} can be used to characterize excitations in both theories \cite[Sec. 4.2]{arxiv:1909.07937}.
-      A Dijkgraaf-Witten Lagrangian can also be re-expressed as a two-group gauge theory Lagrangian by relating the electric and magnetic gauge fields via the equations of motion \cite{arxiv:1309.4721,arxiv:2112.12757,manual:{P. S. Hsin, private communication, 2024.}}.'
+      detail: 'Replacing the two-group data in a two-gauge theory with a group and a cocycle reproduces the phase of the Dijkgraaf-Witten gauge theory, with the two theories equivalent in 2D \cite{arxiv:1309.4721}. Generalizations of Ocneanu''s tube algebras \cite{manual:{Ocneanu, Adrian. "Chirality for operator algebras." Subfactors (Kyuzeso, 1993) 39 (1994).},doi:10.2969/aspm/03110235} can be used to characterize excitations in both theories \cite[Sec. 4.2]{arxiv:1909.07937}. A Dijkgraaf-Witten Lagrangian can also be re-expressed as a two-group gauge theory Lagrangian by relating the electric and magnetic gauge fields via the equations of motion \cite{arxiv:1309.4721,arxiv:2112.12757,manual:{P. S. Hsin, private communication, 2024.}}.'
 
 
 # Begin Entry Meta Information

codes/quantum/groups/topological/tqd.yml

@@ -26,19 +26,16 @@ features:
 
 relations:
   parents:
-    - code_id: group_quantum
-    - code_id: string_net
-      detail: 'String-net models reduce to TQDs for categories \(\text{Vec}^{\omega}G\), where \(G\) is a finite group, and \(\omega\) is a Type III cocycle.
-      There is a duality between a large class of string–net models and certain TQD models \cite{arxiv:1211.3695}.'
     - code_id: dijkgraaf_witten
       detail: |
         Restricting Dijkgraaf-Witten gauge theory to a 2D manifold reproduces the phase of the TQD model \cite{arxiv:0705.0665}.
         The Drinfield center of the category \(\text{Vec}^{\omega}(G)\) is used to describe bulk excitations of 3D Dijkgraaf-Witten models, and this center is equivalent to the twisted quantum double \(D^{\omega}(G)\) \cite[pg. 41]{arxiv:1905.08673}.
         TQD codewords are gauge-invariant boundary states of a 3D Dijkgraaf-Witten theory \cite[Sec. IX]{arxiv:1211.3695}.
+    - code_id: string_net
+      detail: 'String-net models reduce to TQDs for categories \(\text{Vec}^{\omega}G\), where \(G\) is a finite group, and \(\omega\) is a Type III cocycle. There is a duality between a large class of string–net models and certain TQD models \cite{arxiv:1211.3695}.'
   cousins:
     - code_id: spt
-      detail: 'A TQD code Hamiltonian can be obtained from a 2D particular SPT model by promoting the model''s background gauge field to a dynamical gauge field.
-      The same group and cocycle data classifies both 2D SPTs and TQDs \cite{arxiv:1202.3120,arxiv:1301.0861}.'
+      detail: 'A TQD code Hamiltonian can be obtained from a 2D particular SPT model by promoting the model''s background gauge field to a dynamical gauge field. The same group and cocycle data classifies both 2D SPTs and TQDs \cite{arxiv:1202.3120,arxiv:1301.0861}.'
 
 
 

codes/quantum/oscillators/hybrid_qudit_oscillator.yml

@@ -16,15 +16,15 @@ description: |
   Codewords of a simple hybrid code \cite{arxiv:1112.0825} are \(|\alpha\rangle|+\rangle\) and \(|-\alpha\rangle|V\rangle\), i.e., hyper-entangled states of the polarization \(|\pm\rangle\) and occupation-number degrees of freedom of a photon, with the latter being in a coherent state \(|\pm\alpha\rangle\).
 
 notes:
-  - 'See review \cite{arxiv:1409.3719} for an introduction to hybrid qudit-oscillator codes.'
+  - 'See reviews \cite{arxiv:1409.3719,arxiv:2407.10381} for introductions to hybrid qudit-oscillator platforms.'
 
 relations:
   parents:
     - code_id: oscillators
       detail: 'The physical Hilbert space of a hybrid qubit-oscillator code contains at least one oscillator.'
   cousins:
     - code_id: qudits_into_oscillators
-      detail: 'Hybrid code with \(n_1=0\).'
+      detail: 'A hybrid qudit-oscillator code with \(n_1=0\) is a qudit-into-oscillator code.'
 
 
 # Begin Entry Meta Information

codes/quantum/properties/approximate_qecc.yml

@@ -108,8 +108,13 @@ protection: |
 
   \subsection{Circuit complexity}
 
+  \begin{defterm}{Code space complexity}
+  \label{topic:codespace-complexity}
   One can relate robustness of an approximate quantum code to the quantum \textit{circuit complexity} \cite{arxiv:1210.1281,arxiv:1402.5674,arxiv:1301.1363,arxiv:1607.05256} of creating states in the codespace.
   For a family of block codes, scaling as order \(O(k/n)\) of a code parameter called the \textit{subsystem variance} characterizes the transition between code subspaces with low and high circuit complexity \cite{arxiv:2310.04710}.
+  \end{defterm}
+
+
 
 # Please remove calM from, first discussing the codespace case and then opening up to algebras.
 

codes/quantum/properties/block/topological/topological.yml

@@ -98,6 +98,7 @@ protection: |
   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.
 
+  Certain topological codes have nontrivial \hyperref[topic:codespace-complexity]{codespace complexity} \cite{arxiv:2310.04710}.
 
 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.

codes/quantum/properties/block/topological/topological_abelian.yml

@@ -33,11 +33,13 @@ description: |
   Gapped anyon theories admit a subgroup of bosons with trivial mutual statistics whose order squares to that of \(G\); see Ref. \cite{arxiv:2112.11394}.
   In terms of their category theoretic structure, gapped anyon theories admit a Lagrangian subgroup \cite{arxiv:1008.0654,arxiv:2107.13091}.
 
-  \subsection{3D abelian bosonic topological codes}
+# There are three types of \(\mathbb{Z}_2\) topological orders in 3D: one with bosonic charge and loop excitations (BcBl) and two with fermionic charge excitations and bosonic (FcBl) and fermionic (FcFl) loop excitations, respectively \cite{arxiv:2011.11165,arxiv:2110.14654}.
+# There exists an invariant that distinguishes these \cite{arxiv:2110.14654}.
 
-  There are three types of \(\mathbb{Z}_2\) topological orders in 3D: one with bosonic charge and loop excitations (BcBl) and two with fermionic charge excitations and bosonic (FcBl) and fermionic (FcFl) loop excitations, respectively \cite{arxiv:2011.11165,arxiv:2110.14654}.
-  There exists an invariant that distinguishes these \cite{arxiv:2110.14654}.
-  A similar pattern follows in higher dimensions.
+# In higher dimensions, one has to classify both the dimension of the stabilizer generators (and corresponding excitations) as well as their self statistics.
+# Regarding the former, there are \((p,q)\) surface codes for \(p+q=D\) realized by \(Z\)-type stabilizer generators of dimension \(p\) and \(X\)-type stabilizer generators of dimension \(q\).
+# The two corresponding types of excitations are of dimension \(p-1\) and \(q-1\), respectively.
+# Moved to higher_dimensional_surface
 
 
 features:

codes/quantum/properties/quantum_random.yml

@@ -10,6 +10,9 @@ name: 'Random quantum code'
 
 description: 'Quantum code whose construction is non-deterministic in some way, i.e., codes that utilize an elements of randomness somewhere in their construction. Members of this class range from fully non-deterministic codes (e.g., random-circuit codes), to codes whose multi-step construction is deterministic with the exception of a single step (e.g., expander lifter-product codes).'
 
+protection: |
+  Certain random codes have nontrivial \hyperref[topic:codespace-complexity]{codespace complexity} \cite{arxiv:2310.04710}.
+
 # This is more of a unifying property, not meant to be a parent to anything
 relations:
   parents:

codes/quantum/qubits/holographic/cft.yml

@@ -19,6 +19,8 @@ protection: |
   Code performance is quantified by a lower bound on the entanglement fidelity in terms of the conditional mutual information \cite[Eq. (9)]{arxiv:1611.07528}; see also \cite[Appx. A]{arxiv:1801.07271}.
   The coherent information of the combined noise and recover channel can also be perturbatively expanded \cite{arxiv:2406.09555}.
 
+  Certain CFT codes have indefinite \hyperref[topic:codespace-complexity]{codespace complexity}, and their protection depends on the minimum scaling dimension of the underlying CFT \cite{arxiv:2310.04710}.
+
 features:
   code_capacity_threshold:
     - 'Threshold under dephasing depends on the structure of the conformal field theory, with the 1D critical Ising model admitting a finite threshold against certain dephasing noise \cite{arxiv:2406.09555}.'

codes/quantum/qubits/nonstabilizer/circuit_to_hamiltonian.yml

@@ -20,6 +20,9 @@ description: |
 
   Using Markov-chain techniques, the gap of the Hamiltonian can be proven to be of order \(\Omega(D^{-2}n^{-3.09}\log^{-6} n)\) for an \(n\)-qubit input circuit of depth \(D\).
 
+protection: |
+  Circuit-to-Hamiltonian approximate codes have nontrivial \hyperref[topic:codespace-complexity]{codespace complexity} \cite{arxiv:2310.04710}.
+
 features:
   encoders:
     - 'There exists a circuit of size polynomial in \(n\) whose terms act on at most \(\log (n)+2\) qubits \cite[Thm. 3.3]{arxiv:1811.00277}.'

codes/quantum/qubits/nonstabilizer/eth.yml

@@ -28,7 +28,10 @@ description: |
   This implies that energy eigenstates around some energy \(\bar E\) are approximately locally indistinguishable from one another, as their reduced density matrices on any subsystem are both approximately thermal at energy \(\bar E\).
   In this way, global information is protected from local measurements by the environment as \(n\to\infty\).
 
-protection: 'Approximately protects against erasure errors at known locations. Translation invariance alone is sufficient for good approximate error-correcting properties in a many-body spectrum, including in integrable models \cite{arxiv:1710.04631}. The ETH code generated from the spectrum of the translation-invariant 1D Heisenberg spin chain \cite{arxiv:1710.04631} has recovery infidelity (against the erasure of a constant number of sites) scale as \(\epsilon_\text{worst}=O(1/n)\) \cite{arxiv:1902.07714}.'
+protection: |
+  Approximately protects against erasure errors at known locations. Translation invariance alone is sufficient for good approximate error-correcting properties in a many-body spectrum, including in integrable models \cite{arxiv:1710.04631}. The ETH code generated from the spectrum of the translation-invariant 1D Heisenberg spin chain \cite{arxiv:1710.04631} has recovery infidelity (against the erasure of a constant number of sites) scale as \(\epsilon_\text{worst}=O(1/n)\) \cite{arxiv:1902.07714}.'
+
+  The ETH code defined on a Heisenberg spin chain has unbouldable \hyperref[topic:codespace-complexity]{codespace complexity} \cite{arxiv:2310.04710}.
 
 features:
   decoders:

codes/quantum/qubits/stabilizer/topological/surface/higher_d/4d_13_surface.yml

@@ -28,6 +28,7 @@ relations:
     - code_id: translationally_invariant_stabilizer
     - code_id: topological_abelian
       detail: 'The \((1,3)\) 4D toric code realizes 4D \(\mathbb{Z}_2\) gauge theory with 1D \(Z\)-type and 3D \(X\)-type logical operators.'
+  cousins:
     - code_id: dfour
       detail: 'The \((1,3)\) 4D toric code on a hyper-diamond lattice admits a transversal logical \(CCCZ\) gate \cite{arxiv:2010.02238}.'
 

codes/quantum/qubits/stabilizer/topological/surface/higher_dimensional_surface.yml

@@ -15,13 +15,21 @@ alternative_names:
 
 description: |
   CSS-type extenstion of the Kitaev surface code to arbitrary manifolds.
-  The version on a Euclidean manifold of some fixed dimension is called the \(D\)\textit{-dimensional surface} or \(D\)\textit{-dimensional toric} code.
+  The version on a Euclidean manifold of some fixed dimension is called the \(D\)\textit{-dimensional "surface"} or \(D\)\textit{-dimensional toric} code.
 
   Given a cellulation of a manifold, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers
   are associated with \((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \((i+1)\)-faces.
 
-  The 4D surface code serves as a self-correcting quantum memory, while surface codes in higher dimensions can have distances not possible in lower dimensions.
+  Lattice surface codes in \(D\) spatial dimensions can be partially classified by the dimension of their stabilizer generators (and corresponding excitations).
+  There are \((p,q)\) \textit{surface codes} for \(p+q=D\) realized by \(Z\)-type stabilizer generators of dimension \(p\) and \(X\)-type stabilizer generators of dimension \(q\).
+  The two corresponding types of excitations are of dimension \(p-1\) and \(q-1\), respectively.
 
+  In addition, one has to classify the self statistics of the codes' excitations.
+  For example, there are three types of \((1,3)\) surface codes in 3D, corresponding to the three types of \(\mathbb{Z}_2\) Abelian topological orders: one with bosonic charge and loop excitations (BcBl) and two with fermionic charge excitations and bosonic (FcBl) and fermionic (FcFl) loop excitations, respectively \cite{arxiv:2011.11165,arxiv:2110.14654}.
+  There exists an invariant that distinguishes these \cite{arxiv:2110.14654}.
+
+protection: |
+  The 4D \((2,2)\) all-loop surface code serves as a self-correcting quantum memory, while surface codes in higher dimensions can have distances not possible in lower dimensions.
 
 
 features:

codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/hyperbolic_surface.yml

@@ -16,6 +16,8 @@ description: |
 protection: 'Constructions (see code children below) have yielded distances scaling favorably with the number of qubits. The use of hyperbolic surfaces allows one to circumvent bounds on code parameters (such as the \term{BPT bound}) that are valid for lattice geometries.'
 
 features:
+  general_gates:
+    - '\((1,D-1)\) surface codes on hyperbolic geometries admit a fault-tolerant implementation of \(C^D Z\) gates \cite{arxiv:2312.09111}.'
   decoders:
     - 'Hastings decoder \cite{arxiv:1312.2546}.'