quantum GV bound

codes/classical/groups/group_linear.yml

@@ -20,6 +20,7 @@ relations:
     - code_id: group_orbit
       detail: 'The set of codewords of a linear code over \(G\) can be thought of as an orbit of a particular codeword under the group formed by the code.
       However, group orbit codes do not have to be linear \cite[Remark 8.4.3]{preset:EricZin}.'
+    - code_id: block
   cousins:
     - code_id: group_gkp
 

codes/classical/q-ary_digits/q-ary_digits_into_q-ary_digits.yml

@@ -60,6 +60,10 @@ protection: |
   A code detects errors on up to \(d-1\) coordinates, corrects erasure errors on up to \(d-1\) coordinates, and corrects general errors on up to \(\left\lfloor (d-1)/2 \right\rfloor\) coordinates.
   Often, the relative distance \(\delta=d/n\) is used to compare codes of different lengths.
 
+  \subsection{Noise channels}
+
+  Noise channels include the symmetric noise channel, asymmetric noise channels \cite{manual:{Varshamov, R. R. "Some features of linear codes that correct asymmetric errors." Soviet Physics Doklady. Vol. 9. 1965.},doi:10.1109/TIT.1973.1054954,doi:10.1016/S0019-9958(79)90329-2,manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.},doi:10.1142/6400}, and insertion/deletion noise.
+
   \subsection{Weight enumerator and four fundamental parameters}
   \begin{defterm}{Weight enumerator}
   \label{topic:weight-enumerator}
@@ -80,10 +84,7 @@ protection: |
   Other types of weight enumerators includes the Hamming weight enumerator, Lee weight enumerator, joint weight enumerator, split weight enumerator, and biweight enumerator \cite{preset:MacSlo}.
   \end{defterm}
 
-  \subsection{Noise channels}
-  Noise channels include the symmetric noise channel, asymmetric noise channels \cite{manual:{Varshamov, R. R. "Some features of linear codes that correct asymmetric errors." Soviet Physics Doklady. Vol. 9. 1965.},doi:10.1109/TIT.1973.1054954,doi:10.1016/S0019-9958(79)90329-2,manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.},doi:10.1142/6400}, and insertion/deletion noise.
-
-  \subsection{Bounds}
+  \subsection{Bounds on code parameters}
   Bounds on the parameters of an \((n,K,d)_q\) code include the Hamming a.k.a. sphere-packing bound, Singleton bound, Gilbert-Varshamov (GV) bound, Griesmer bound, Plotkin bound, and various linear programming (LP) bounds; see \cite{preset:HKSbasics}.
   A code whose parameters attain the Hamming bound (Singleton bound, Griesmer bound, Delsarte LP bound) is called a perfect code (an MDS code, a Griesmer code, an \hyperref[code:univ_opt_q-ary]{LP universally optimal code}).
 
@@ -94,11 +95,13 @@ protection: |
   K\sum_{j=0}^{d-1}{n \choose j}(q-1)^{j}\geq q^{n}~.
   \end{align}
   In other words, if the left-hand side of the above is less than or equal to the right-hand side, then a code with such parameters exists.
-  The GV bound gives rise to the \textit{asymptotic GV bound} (i.e., GV bound in the \(n\to\infty\) limit), expressed in terms of the achievable rate \(R\) and relative distance \(\delta\),
+  The GV bound gives rise to the \textit{asymptotic GV bound} (i.e., GV bound in the \(n\to\infty\) limit), expressed in terms of the maximum achievable rate \(R\) and relative distance \(\delta\),
   \begin{align}
-    R\geq1-h_{q}(\delta)~,
+    R\geq 1-h_{q}(\delta)~,
   \end{align}
-  where \(h_q\) is the \(q\)-ary entropy function.
+  where \(h_q\) is the \(q\)-ary entropy function,
+  \begin{align}
+    h_{q}(\delta)=-\delta\log_{q}\frac{\delta}{q-1}-(1-\delta)\log_{q}(1-\delta)~.
   \end{defterm}
 
 features:

codes/quantum/properties/block/quantum_mds.yml

@@ -14,11 +14,11 @@ introduced: '\cite{arxiv:quant-ph/9604034,arxiv:quant-ph/9702031,arxiv:quant-ph/
 description: |
   An \(((n,K,d))\) code constructed out of \(q\)-dimensional qubits or Galois qudits is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound is satisfied.
 
-  In other words, the quantum Singleton
+  In other words, the quantum Singleton bound \cite{arxiv:quant-ph/9604034,arxiv:quant-ph/9702031,arxiv:quant-ph/9703048}
   \begin{align}
   K \leq q^{n-2(d-1)}
   \end{align}
-  becomes an equality.
+  becomes an equality for such codes.
   When \(K = q^k\) for some integer \(k\), the above reduces to \(2(d-1) \leq n-k\).
   Such codes are \hyperref[topic:quantum-weight-enumerator]{pure} \cite{arxiv:quant-ph/9703048}; see also \cite{manual:{A. Winter, private communication (2019).}} mentioned in Ref. \cite{arxiv:1907.07733}.
   The length \(n\) of a quantum MSD code with distance \(d \geq 3\) is bounded by the qudit dimension, \(n \leq q^2 + d - 2\) \cite{arxiv:1907.07733}.

codes/quantum/properties/block/quantum_perfect.yml

@@ -10,11 +10,13 @@ name: 'Perfect quantum code'
 #introduced: ''
 
 description: |
-  A \hyperref[topic:degeneracy]{non-degenerate} code constructed out of \(q\)-dimensional qudits and having parameters \(((n,K,2t+1))\) is perfect if \(n\), \(K\), \(t\), and \(q\) are such that the quantum Hamming bound \cite{arxiv:quant-ph/9602022}
+  A \hyperref[topic:degeneracy]{non-degenerate} code constructed out of \(q\)-dimensional qudits and having parameters \(((n,K,2t+1))\) is perfect if \(n\), \(K\), \(t\), and \(q\) are such that the quantum Hamming bound is satisfied,
+  
+  In other words, the quantum Hamming bound \cite{arxiv:quant-ph/9602022}
   \begin{align}
   \sum_{j=0}^{t}(q^2-1)^{j}{n \choose j}\leq q^{n}/K
   \end{align}
-  becomes an equality.
+  becomes an equality for such codes.
   For example, for a qubit \(q=2\) code with one logical qubit (\(K=2\)) and \(t=1\), the bound becomes \(3n+1 \leq 2^{n-1}\).
   The bound can be saturated only at certain \(n\).
 

codes/quantum/properties/stabilizer/random_stabilizer.yml

@@ -16,7 +16,7 @@ description: |
   Since stabilizer encoders are Clifford circuits, such codes can be thought of as arising from random Clifford circuits.
 
 features:
-  rate: 'Random qubit stabilizer codes achieve the quantum GV bound \cite{arXiv:quant-ph/9605005,arXiv:quant-ph/9705052}; see notes \cite{preset:PreskillNotes}.
+  rate: 'Random qubit stabilizer codes achieve the \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{arXiv:quant-ph/9605005,arXiv:quant-ph/9705052}; see notes \cite{preset:PreskillNotes}.
   In fact, sampling random CSS codes is sufficient \cite{arxiv:quant-ph/9512032}.'
 
 relations:

codes/quantum/qubits/qubits_into_qubits.yml

@@ -46,14 +46,14 @@ protection: |
   Codes whose distance is greater than the diagonal distance are \hyperref[topic:degeneracy]{degenerate}.
   \hyperref[topic:degeneracy]{Degenerate} codes admit undetectable Pauli errors (i.e., errors whose projection into the codespace is nonzero) of weight less than the code distance (i.e., the projection satisfies the \term{Knill-Laflamme conditions}).
 
+  \subsection{Noise channels}
+
   A quantum channel whose Kraus operators are Pauli strings is called a \textit{Pauli channel}, and such channels are typically more tractable than general, non-Pauli channels.
   Relevant Pauli channels include dephasing noise and depolarizing noise (a.k.a. Werner-Holevo channel \cite{arxiv:quant-ph/0203003}).
   Relevant non-Pauli channels are \hyperref[topic:ad]{AD} noise, erasure (which maps all qubit states into a third state \(|e\rangle\) outside of the qubit Hilbert space), and biased erasure (in which case only the \(|1\rangle\) qubit state is mapped to \(|e\rangle\)).
   Noise can be correlated in space or in time, with the latter being an example of a non-Markovian phenomenon \cite{arxiv:quant-ph/0505153,arxiv:2012.01894}.
 
-  \subsection{Bounds on code parameters}
-  Bounds on code performance include the quantum Singleton bound \cite{arxiv:quant-ph/9604034,arxiv:quant-ph/9702031,arxiv:quant-ph/9703048}, quantum Hamming bound \cite{arxiv:quant-ph/9602022}, and quantum GV bound \cite{arxiv:quant-ph/9602022}.
-  Linear programming bounds also exist \cite{arxiv:quant-ph/9611001,arxiv:quant-ph/9709049}.
+  \subsection{Quantum weight enumerators}
 
   \begin{defterm}{Quantum weight enumerator}
   \label{topic:quantum-weight-enumerator}
@@ -65,6 +65,8 @@ protection: |
     \end{split}
     \end{align}
   where \(\Pi\) is the code projection, and where the sum is over the Pauli group modulo the subgroup of phases (hence, the dagger below is necessary in case the coset representative is not Hermitian).
+  \end{defterm}
+
   The dual quantum weight enumerator is
     \begin{align}
     \begin{split}
@@ -73,14 +75,15 @@ protection: |
     \end{split}
     \end{align}
   The weight enumerator and its dual satisfy the \textit{quantum MacWilliams identity} \cite{arxiv:quant-ph/9610040}; see \cite[Ch. 7]{preset:GottesmanBook}.
+  It gives rise to quantum linear programming (LP) bounds \cite{arxiv:quant-ph/9611001,arxiv:quant-ph/9709049}; see the book \cite{preset:GottesmanBook}.
+
   The distance \(d\) of a code is the smallest \(j=d\) at which \(A_j \neq B_j\) \cite{arxiv:quant-ph/9906126}.
   Such a code is called \textit{pure} if \(A_j = B_j = 0\) for all \(j < d\); otherwise, the code is called \textit{impure}.
   \hyperref[topic:degeneracy]{Degeneracy} is sufficient but not necessary for impurity \cite{preset:GottesmanBook}.
+
   Other types of quantum weight enumerators are the Rains \textit{shadow enumerators} \cite{arxiv:quant-ph/9611001} (see also \cite{arxiv:quant-ph/0406063}).
   There are techniques to compute them for general codes \cite{arxiv:2308.05152}.
   These notions can be generalized to qudit codes and other error bases \cite{doi:10.1016/j.aam.2020.102085,arxiv:2211.02756,arxiv:2308.05152}.
-  \end{defterm}
-
 
 features:
   rate: 'Exact two-way assisted capacities have been obtained for the erasure and dephasing channels \cite{arxiv:1510.08863}.'

codes/quantum/qubits/stabilizer/qubit_css.yml

@@ -181,9 +181,9 @@ relations:
       detail: 'CSS codes for which \(C_X=C_Z \equiv C\) are called \textit{self-orthogonal} or \textit{homogeneous} \cite{doi:10.1109/TCOMM.2022.3231879} since \(C^{\perp} \subseteq C\).
       The stabilizer group of such codes is invariant under the Hadamard gate exchanging \(X\) and \(Z\).'
     - code_id: alternant
-      detail: 'Alternant codes used in the CSS construction yield quantum codes that asymptotically achieve the quantum GV bound \cite{doi:10.1109/TIT.2022.3201239}.'
+      detail: 'Alternant codes used in the CSS construction yield quantum codes that asymptotically achieve the \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{doi:10.1109/TIT.2022.3201239}.'
     - code_id: random_stabilizer
-      detail: 'Random CSS codes asymptotically achieve linear distance with high probability, achieving the quantum GV bound \cite{arxiv:quant-ph/9512032}.'
+      detail: 'Random CSS codes asymptotically achieve linear distance with high probability, achieving the \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{arxiv:quant-ph/9512032}.'
     - code_id: ag
       detail: 'Algebraic geometry codes can be plugged into the CSS construction to yield asymptotically good quantum codes \cite{arxiv:quant-ph/0006061,arxiv:quant-ph/0107129}.
       However, such codes are not QLDPC.'

codes/quantum/qudits/qudits_into_qudits.yml

@@ -37,6 +37,10 @@ protection: |
 
   The Pauli error set is a unitary basis for linear operators on the multi-qudit Hilbert space that is orthonormal under the Hilbert-Schmidt inner product; it is a \hyperref[topic:nice-error-basis]{nice error basis}. The distance associated with this set is often the minimum weight of a qudit Pauli string that implements a nontrivial logical operation in the code.
 
+  \subsection{Bounds on code parameters}
+  Bounds on code performance include the quantum Singleton bound, quantum Hamming bound, \hyperref[topic:quantum-gv-bound]{quantum GV bound}, various quantum linear programming (LP) bounds \cite{arxiv:quant-ph/9611001,arxiv:quant-ph/9709049} (see the book \cite{preset:GottesmanBook}), and other bounds \cite{doi:10.1109/TIT.2005.862086}.
+  A code whose parameters attain the quantum Hamming bound (quantum Singleton bound) is called a perfect quantum code (a quantum MDS code).
+
 
 features:
   general_gates:

codes/quantum/qudits_galois/galois_into_galois.yml

@@ -40,6 +40,23 @@ protection: |
 
   The Galois-qudit Pauli error set is a unitary basis for linear operators on the multi-qudit Hilbert space that is orthonormal under the Hilbert-Schmidt inner product; it is a \hyperref[topic:nice-error-basis]{nice error basis}. The distance associated with this set is often the minimum weight of a Galois qudit Pauli string that implements a nontrivial logical operation in the code.
 
+  \subsection{Bounds on code parameters}
+  Bounds on code performance include the quantum Singleton bound, quantum Hamming bound, \hyperref[topic:quantum-gv-bound]{quantum GV bound}, various quantum linear programming (LP) bounds \cite{arxiv:quant-ph/9611001,arxiv:quant-ph/9709049} (see the book \cite{preset:GottesmanBook}), and other bounds \cite{doi:10.1109/TIT.2005.862086}.
+  A code whose parameters attain the quantum Hamming bound (quantum Singleton bound) is called a perfect quantum code (a quantum MDS code).
+
+  \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 
+  \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}
+  The quantum GV bound gives rise to the \textit{asymptotic quantum GV bound} (i.e., quantum GV bound in the \(n\to\infty\) limit), expressed in terms of the maximum achievable rate \(R\) and relative distance \(\delta\),
+  \begin{align}
+    R\geq 1-\delta\log_q(q+1) - h_{q}(\delta)~,
+  \end{align}
+  where \(h_q\) is the \hyperref[topic:gv-bound]{\(q\)-ary entropy function}.
+  \end{defterm}
+
 
 features:
   general_gates:

codes/quantum/qudits_galois/stabilizer/evaluation/binary_quantum_goppa.yml

@@ -27,7 +27,7 @@ description: |
   that \(H\) is a subgroup of \(G\), we can construct a classical Goppa code \(C(D,G)\), where \(D\) is the sum of all \(P_i\). Using \(C(D,G)\), we can construct a \([[n,k,d]]_q\) quantum stabilizer code such that
   \(k  = \text{dim} G - \text{dim}(G-P_1 - \cdots - P_n - \sigma P_1 - \cdots  - \sigma P_n) - n~.\)
 
-protection: 'Protects against weight \(t\) errors where \( 0 < t \leq  \lfloor \frac{d^*-g-1}{2} \rfloor \) where \( d^* = \text{deg} G + 2 -2g \) and \(g\) is the genus of the function field and \(d \geq n - \lfloor \frac{deg G}{2} \rfloor\).'
+protection: 'Protects against weight \(t\) errors where \( 0 < t \leq  \lfloor \frac{d^*-g-1}{2} \rfloor \) where \( d^* = \text{deg} G + 2 -2g \) and \(g\) is the genus of the function field and \(d \geq n - \lfloor \frac{deg G}{2} \rfloor\). Such codes can exceed the \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{doi:10.1007/s11128-006-0047-9}.'
 
 features:
   encoders:

codes/quantum/qudits_galois/stabilizer/evaluation/rs/galois_grs.yml

@@ -15,7 +15,7 @@ description: |
   True \(q\)-Galois-qudit stabilizer code constructed from generalized Reed-Solomon (GRS) codes via either the Hermitian construction \cite{arxiv:1311.3009,doi:10.1142/S0219749919500060,doi:10.1109/TIT.2010.2054174} or the Galois-qudit CSS construction \cite{arxiv:quant-ph/9906129,arxiv:0812.4514}.
 
 features:
-  rate: 'Concatenations of quantum GRS codes and random stabilizer codes can achieve the quantum GV bound \cite{arxiv:1004.1127}.'
+  rate: 'Concatenations of quantum GRS codes and random stabilizer codes can achieve the \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{arxiv:1004.1127}.'
 
 relations:
   parents:
@@ -30,9 +30,9 @@ relations:
     - code_id: stabilizer_over_gfqsq
       detail: 'Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction.'
     - code_id: quantum_concatenated
-      detail: 'Concatenations of quantum GRS codes and random stabilizer codes can achieve the quantum GV bound \cite{arxiv:1004.1127}.'
+      detail: 'Concatenations of quantum GRS codes and random stabilizer codes can achieve the \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{arxiv:1004.1127}.'
     - code_id: random_stabilizer
-      detail: 'Concatenations of quantum GRS codes and random stabilizer codes can achieve the quantum GV bound \cite{arxiv:1004.1127}.'
+      detail: 'Concatenations of quantum GRS codes and random stabilizer codes can achieve the \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{arxiv:1004.1127}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qudits_galois/stabilizer/galois_stabilizer.yml

@@ -70,6 +70,7 @@ notes:
   - 'Tables of bounds and examples of Galois-qudit stabilizer codes for various \(n\) and \(k\), based on algorithms developed in Refs. \cite{doi:10.1007/978-3-540-37634-7_13,arxiv:2405.15057}, are maintained by M. Grassl at this \href{http://codetables.markus-grassl.de/}{website}. A modular-qudit stabilizer code with composite dimension \(q\) contains a subcode that is isomorphic to a \(p\)-dimensional prime-qudit stabilizer code for every prime factor \(p\) of \(q\), and the distance of the full stabilizer code is upper bound by the distance of this subcode \cite{manual:{Markus Grassl, private communication, 2024}}.'
   - 'The number of Galois-qudit stabilizer codes was determined in Ref. \cite{arxiv:quant-ph/0602001}.'
   - 'See Quantum Codes qudit stabilizer database, maintained by N. Aydin, P. Liu, and B. Yoshino, at this \href{http://quantumcodes.info/}{website}.'
+  - 'Review of nonbinary stabilizer codes \cite{doi:10.1201/9781584889007-18}.'
 
 relations:
   parents: