Ch 8-11

Author: valbert4

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

@@ -106,15 +106,19 @@ protection: |
   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 tessellated to form the many-body system.
-  For closed orientable manifolds \cite{doi:10.1007/bf01217730,doi:10.1007/BF01238857},
-  \begin{align}
-    K=\sum_{a\in A}\left(d_{a}/D\right)^{\chi(\Sigma^{2})}~,
-  \end{align}
-  and a generalization of the formula to the non-orientable case can be found in Ref. \cite{arxiv:1612.07792}.'
+  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 tessellated to form the many-body system.
+    For closed orientable manifolds \cite{doi:10.1007/bf01217730,doi:10.1007/BF01238857},
+    \begin{align}
+      K=\sum_{a\in A}\left(d_{a}/D\right)^{\chi(\Sigma^{2})}~,
+    \end{align}
+    and a generalization of the formula to the non-orientable case can be found in Ref. \cite{arxiv:1612.07792}.
+  threshold:
+    - 'Topological-code families can be used to obtain a fault-tolerance threshold, although the numerical threshold value and overhead depend strongly on the syndrome-extraction and decoding protocol \cite[Ch. 10]{preset:GottesmanBook}.'
   encoders:
-    - 'A depth of \hyperref[topic:asymptotics]{order} \(\Omega(L)\) is necessary for a unitary circuit to initialize in a 2D topologically ordered state using geometrically local gates on an \(L\times L\) lattice \cite{arxiv:quant-ph/0603121,arxiv:quant-ph/0603114}, 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}.'
+    - |
+      A depth of \hyperref[topic:asymptotics]{order} \(\Omega(L)\) is necessary for a unitary circuit to initialize in a 2D topologically ordered state using geometrically local gates on an \(L\times L\) lattice \cite{arxiv:quant-ph/0603121,arxiv:quant-ph/0603114}, 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}.
     - 'Algorithm that takes in reduced density matrices and outputs a circuit preparing the global state in polynomial time \cite{arxiv:2410.23544}.'
   general_gates:
     - 'Ising anyon braiding and fusion were studied in a phenomenological model that was the first to study error correction with non-Abelian anyons \cite{arxiv:1311.0019}.'
@@ -132,9 +136,10 @@ relations:
     - code_id: block_quantum
       detail: 'Topological codes are block codes because an infinite family of tensor-product Hilbert spaces is required to formally define a phase of matter.'
     - code_id: hamiltonian
-      detail: 'Codespace of a topological code is typically the ground-state or low-energy subspace of a geometrically local Hamiltonian admitting a topological phase.
-      Logical qubits can also be created via lattice defects or by appropriately scheduling measurements of gauge generators (see Floquet codes).
-      Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to sufficiently weak quasi-local perturbations when they satisfy local topological quantum order together with the Local-Gap condition \cite{arxiv:1109.1588,arxiv:2110.11194}.'
+      detail: |
+        Codespace of a topological code is typically the ground-state or low-energy subspace of a geometrically local Hamiltonian admitting a topological phase.
+        Logical qubits can also be created via lattice defects or by appropriately scheduling measurements of gauge generators (see Floquet codes).
+        Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to sufficiently weak quasi-local perturbations when they satisfy local topological quantum order together with the Local-Gap condition \cite{arxiv:1109.1588,arxiv:2110.11194}.
   cousins:
     - code_id: cluster_state
       detail: 'There exist necessary and sufficient conditions for a family of cluster states to exhibit the TQO-1 property \cite{arxiv:2112.02502}.'

codes/quantum/properties/qecc.yml

@@ -55,22 +55,6 @@ protection: |
 #  - 'Links to code tables, github, GAP algebra packages, more papers \cite{arxiv:####.#####}.'
 
 features:
-  rate: |
-    The quantum channel capacity, i.e., the regularized coherent information, is the highest rate of quantum information transmission through a quantum channel with arbitrarily small error rate \cite{arxiv:quant-ph/9604015,preset:ShorMSRI,arxiv:quant-ph/0304127}. 
-    In other words, the capacity formula implies that one can achieve a transmission rate
-    \(r\) over a quantum channel \(\mathcal{E}\) iff, for sufficiently large \(n\), \(m=\lfloor r n \rfloor\),
-    and for all \(\epsilon>0\),
-    \begin{align}
-      ||\mathcal{D}\mathcal{E}\mathcal{U}-I^{\otimes m}||_1\leq \epsilon
-    \end{align}
-    for some encoding channel \(\mathcal{U}\) and some recovery channel \(\mathcal{D}\).
-    The quantum capacity \(Q\) of \(\mathcal{E}\) is defined as the supremum over \(n\) of achievable transmission rates \cite{doi:10.1017/9781316848142}.
-    See \cite[Ch. 24]{arxiv:1106.1445} for definitions and a history.     
-    
-    The fault-tolerant capacity is the capacity for the more general case where the encoding and decoding maps are also assumed to undergo noise \cite{arxiv:2009.07161}.
-
-    Doeblin coefficients \cite{preset:Doeblin37} for quantum channels have been studied \cite{arxiv:2309.08475}.
-
   decoders:
     - 'The effect of an error is a mapping of the code subspace into another, potentially overlapping, subspace. To determine, or diagnose, the effect of the error in what is known as \textit{syndrome-based decoding}, one can measure one or more operators called \textit{check operators}, which resolve code and error spaces without collapsing the quantum information inside the spaces. The eigenvalues of check operators are called \textit{error syndromes}. One \textit{round} or \textit{cycle} of quantum error correction proceeds by extracting syndromes and performing correcting operations to map the error space containing the logical information back into the codespace. For some codes, correcting operations are not necessary because one can instead track which error space contains the logical information.'
 

codes/quantum/properties/qecc_finite.yml

@@ -81,20 +81,55 @@ protection: |
   A channel \(\mathcal{E}\) is correctable if  \(\mathcal{E}^C(\rho)=\rho_0\mathrm{Tr}(\rho)\) for some constant state \(\rho_0\), which is equivalent to the \term{Knill-Laflamme conditions} \cite{arxiv:0811.1621,arxiv:0907.5391}.
   The logical and physical dimensions are related to the channel rank for non-degenerate codes via the quantum packing bound \cite{arxiv:1007.3655}. 
 
+  Exact correctability can also be expressed using the \hyperref[topic:coherent-information]{coherent information}.
+
+  \begin{defterm}{Coherent information}
+  \label{topic:coherent-information}
+  Given a bipartite state \(\rho_{RQ}\), the \hyperref[topic:coherent-information]{coherent information} in subsystem \(Q\) is
+  \begin{align}
+    I_{c}(\rho_{RQ})=S(\rho_{Q})-S(\rho_{RQ})~.
+  \end{align}
+  For a channel \(\mathcal{E}:L\to Q\) and a pure input state \(\rho\) on \(R\otimes L\), the \hyperref[topic:coherent-information]{coherent information} of the channel is
+  \begin{align}
+    I_{c}(\mathcal{E},\rho)=S(\mathcal{E}(\rho_{L}))-S((\mathrm{id}\otimes\mathcal{E})(\rho))~.
+  \end{align}
+  \hyperref[topic:coherent-information]{Coherent information} cannot increase under further processing of the output, a statement known as the \textit{quantum data processing inequality} \cite{arxiv:quant-ph/9604022,arxiv:quant-ph/9604034,arxiv:quant-ph/9702031,arxiv:quant-ph/9707023,arxiv:quant-ph/0304007,preset:GottesmanBook}.
+  \end{defterm}
+
+  Exact correctability is equivalent to preservation of \hyperref[topic:coherent-information]{coherent information}: a channel \(\mathcal{E}\) is exactly correctable on a code iff the \hyperref[topic:coherent-information]{coherent information} after encoding and noise is the same as that of the logical input for every pure input state, and it is enough to check this on a maximally entangled state between the logical system and a reference \cite{arxiv:quant-ph/9604022,arxiv:quant-ph/9702031,preset:GottesmanBook}.
+
+
 features:
+  rate: |
+    The quantum channel capacity, i.e., the regularized \hyperref[topic:coherent-information]{coherent information}, is the highest rate of quantum information transmission through a quantum channel with arbitrarily small error rate \cite{arxiv:quant-ph/9604015,preset:ShorMSRI,arxiv:quant-ph/0304127}. 
+    In other words, the capacity formula implies that one can achieve a transmission rate
+    \(r\) over a quantum channel \(\mathcal{E}\) iff, for sufficiently large \(n\), \(m=\lfloor r n \rfloor\),
+    and for all \(\epsilon>0\),
+    \begin{align}
+      ||\mathcal{D}\mathcal{E}\mathcal{U}-I^{\otimes m}||_1\leq \epsilon
+    \end{align}
+    for some encoding channel \(\mathcal{U}\) and some recovery channel \(\mathcal{D}\).
+    The quantum capacity \(Q\) of \(\mathcal{E}\) is defined as the supremum over \(n\) of achievable transmission rates \cite{doi:10.1017/9781316848142}.
+    See \cite[Ch. 24]{arxiv:1106.1445} for definitions and a history.     
+    
+    The fault-tolerant capacity is the capacity for the more general case where the encoding and decoding maps are also assumed to undergo noise \cite{arxiv:2009.07161}.
+
+    Doeblin coefficients \cite{preset:Doeblin37} for quantum channels have been studied \cite{arxiv:2309.08475}.
+
   decoders:
     - 'The operation \(\cal{D}\) in the definition of this code is called the decoder. However, the term \textit{decoder} can sometimes be used for the unencoder \(\cal{U}\) (i.e., the inverse of the encoder), which does not correct errors.'
     - 'There are several recovery maps which work for noise that is not exactly correctable; see \hyperref[code:approximate_qecc]{AQECC} entry.'
     - 'QECCs are useful \cite{arxiv:1507.07072} for the mean king''s measurement problem \cite{doi:10.1103/PhysRevLett.58.1385}.'
     - 'Protection can be implemented via \textit{autonomous QEC} (a.k.a. continuous QEC or continuous-time QEC) \cite{doi:10.1098/rspa.1998.0165,arxiv:quant-ph/9912104,arxiv:quant-ph/0110111,arxiv:quant-ph/0501038,arxiv:quant-ph/0511221} via, e.g., reservoir engineering \cite{doi:10.1103/PhysRevLett.77.4728}; see review \cite{arxiv:1311.2485}. There are analogues of the \term{Knill-Laflamme conditions} for autonomous QEC \cite{arxiv:1711.02999,arxiv:2103.05007}, and it has been adapted to non-Markovian noise \cite{arxiv:0705.2342}. Information-theoretic bounds have been derived for open-loop control \cite{arxiv:quant-ph/0409187}. Machine learning can be used to optimize autonomous QEC encoding and recovery \cite{arxiv:2506.21707}.'
 
   code_capacity_threshold:
-    - 'Coherent information of the state under the action of a noise channel can be used to estimate the optimal threshold \cite{arxiv:2312.06664}.'
+    - '\hyperref[topic:coherent-information]{Coherent information} of the state under the action of a noise channel can be used to estimate the optimal threshold \cite{arxiv:2312.06664}.'
 
 
 relations:
   parents:
     - code_id: qecc
+      detail: 'Finite-dimensional QECCs are a special case of quantum error-correcting codes, which can also include infinite-dimensional codes such as bosonic codes. The Knill-Laflamme conditions and the notion of correctability can be extended to infinite-dimensional codes.'
   cousins:
     - code_id: ecc_finite
       detail: 'Finite-dimensional QECCs are quantum analogues of finite-dimensional classical ECCs.'

codes/quantum/properties/quantum_concatenated.yml

@@ -26,6 +26,8 @@ features:
   decoders:
     - 'Standard decoding proceeds in the reverse order: first decode the outer code blocks and then use the resulting data to decode the inner code.'
     - 'Maximum-likelihood decoding can be formulated as contraction of a tree tensor network, yielding exact decoders that improve on minimum-distance decoding for recursively concatenated block codes \cite{arxiv:quant-ph/0606126,arxiv:1312.4578}.'
+  fault_tolerance:
+    - 'Recursive concatenation is the standard route to threshold-theorem constructions: if one level of a fault-tolerant simulation suppresses logical error below the physical error rate, then further concatenation suppresses it rapidly, yielding the family of protocols used in polylogarithmic-overhead threshold theorems \cite[Chs. 10 and 14]{preset:GottesmanBook}.'
 
 notes:
   - 'See the book \cite{preset:GottesmanBook} for an introduction.'

codes/quantum/qubits/small_distance/small/15/stab_15_1_3.yml

@@ -17,6 +17,7 @@ alternative_names:
 
 description: |
   A \([[15,1,3]]\) quantum Reed-Muller code that is most easily thought of as a tetrahedral 3D color code.
+  It can be constructed as a CSS code from the \([15,5,8]\) punctured Reed-Muller code and its even subcode, which explains its transversal \(T^\dagger\) gate \cite{preset:GottesmanBook}.
 
   This code contains 15 qubits, represented by four vertices, four face centers, six edge centers, and one body center.
   The tetrahedron is cellulated into four identical polyhedron cells by connecting the body center to all four face centers, where each face center is then connected by three adjacent edge centers.
@@ -45,6 +46,7 @@ description: |
     X & I & X & I & X & I & X & I & X & I & X & I & X & I & X
   \end{smallmatrix}~.
   \end{align}
+  The logical \(\ket{\overline{0}}\) and \(\ket{\overline{1}}\) states are superpositions of computational-basis words of weight \(0 \bmod 8\) and \(7 \bmod 8\).
 
 features:
   encoders:

codes/quantum/qubits/small_distance/small/5/stab_5_1_3.yml

@@ -74,6 +74,7 @@ features:
     - 'Syndrome extraction circuit using only CNOT-SWAP gates \cite{arxiv:2207.13356}.'
     - 'Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived \cite{arxiv:2203.01706}.'
     - 'Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed \cite{arxiv:2409.13681}.'
+    - 'Ideal transversal computational-basis measurement distinguishes logical basis states by the parity of the outcome string, but this is not a fault-tolerant measurement gadget because a single faulty measurement bit can flip the decoded logical outcome \cite{preset:GottesmanBook}.'
 
   fault_tolerance:
     - 'Pieceable fault-tolerant CZ, CNOT, and \(CCZ\) gates \cite{arxiv:1603.03948}.'

codes/quantum/qubits/stabilizer/qubit_css.yml

@@ -131,6 +131,7 @@ features:
     - 'Fault-tolerant CNOT gate using generalized lattice surgery \cite{arxiv:2505.01370}.'
   fault_tolerance:
     - 'Steane error correction \cite{arxiv:quant-ph/9611027}, where fault-tolerance is ensured by preparing ancillary encoded states and extracting syndromes via \(CNOT\) gates.'
+    - 'Transversal computational-basis measurement followed by classical decoding is a fault-tolerant gadget for logical measurement of all encoded qubits \cite{preset:GottesmanBook}.'
     - 'Fault-tolerant error correction and logical measurements using flag qubits for distance-three cyclic CSS codes \cite{arxiv:1803.09758}. Parallel syndrome extraction for distance-three codes can be done fault-tolerantly using one flag qubit \cite{arxiv:2208.00581}. \hyperref[topic:effective-distance]{Distance-preserving} flag fault-tolerant error correction can be done using lookup tables for small codes \cite{arxiv:2306.12862}.'
     - 'Homomorphic gadgets fault-tolerant measurement unify Steane and Shor error correction \cite{arxiv:2211.03625}.'
     - 'A fault-tolerant error-correction protocol using \(O(d\log d)\) syndrome measurements can be applied to any CSS code with distance \(d \geq \Omega(n^{\alpha})\) for any \(\alpha > 0\) \cite{arxiv:2002.05180}.'