~

codes/classical/analog/lattice/points_into_lattices.yml

@@ -51,7 +51,7 @@ protection: |
 
   The \textit{lattice quantizer problem} is to find a lattice whose \textit{fundamental Voronoi cell} \(\Pi\), the Voronoi cell at the origin, has the smallest possible normalized second moment,
   \begin{align}
-    G(\Pi)=\frac{\frac{1}{n}\int_{\Pi}x\cdot x\,dx}{\text{Vol}(\Pi)^{1+2/n}}\,.
+    G(\Pi)=\frac{\frac{1}{n}\int_{\Pi}x\cdot x\,\textnormal{d}x}{\text{Vol}(\Pi)^{1+2/n}}\,.
   \end{align}
   Higher-dimensional lattices yield quantizers with lower normalized second moments than the 1D integer lattice \cite{manual:{P. L. Zador, Development and evaluation of procedures for quantiZing multivariate distributions, Ph.D. Dissertation, Stanford Univ., 1963},doi:10.1109/TIT.1982.1056490}.
 

codes/classical/properties/block/universally_optimal/t-designs.yml

@@ -18,7 +18,7 @@ description: |
 
   As such, a design can be used to determine the average of degree-\(\leq t\) polynomials \(p\) over \(X\),
   \begin{align}
-    \int_{X}dxp(x)={\textstyle \frac{1}{|D|}}\sum_{x\in D}p(x)~,
+    \int_{X}\textnormal{d}xp(x)={\textstyle \frac{1}{|D|}}\sum_{x\in D}p(x)~,
   \end{align}
   where the integral is over \(X\) (given some measure \(d x\)), while the sum is over the design \(D\subset X\).
   A \textit{weighted design} is a design for which each term \(p(x)\) in the above sum must be multiplied by a weight \(w(x)\) in order to be equal to the left-hand side.

codes/quantum/groups/rotors/stabilizer/css/zero_pi.yml

@@ -23,7 +23,7 @@ description: |
   An alternative codeword basis in terms of angular position states is
   \begin{align}
   \begin{split}
-    |\overline{+}\rangle&=\intop_{U(1)}d\phi\left|\phi,\phi\right\rangle \\|\overline{-}\rangle&=\intop_{U(1)}d\phi\left|\phi,\phi+\pi\right\rangle~.
+    |\overline{+}\rangle&=\intop_{U(1)}\textnormal{d}\phi\left|\phi,\phi\right\rangle \\|\overline{-}\rangle&=\intop_{U(1)}\textnormal{d}\phi\left|\phi,\phi+\pi\right\rangle~.
   \end{split}
   \end{align}
 

codes/quantum/oscillators/tiger/tiger.yml

@@ -25,7 +25,7 @@ description: |
 
   Using multi-index notation, a projected coherent state can be written in two ways,
   \begin{align}
-    |\boldsymbol{\alpha}\rangle_{\boldsymbol{\Delta}}^{H}&\propto\int d\boldsymbol{\phi}e^{i\boldsymbol{\phi}(H\hat{\mathbf{n}}-\boldsymbol{\Delta})}|\boldsymbol{\alpha}\rangle\\&\propto\sum_{H\mathbf{n}=\boldsymbol{\Delta}}\frac{\boldsymbol{\alpha}^{\mathbf{n}}}{\sqrt{\mathbf{n}!}}|\mathbf{n}\rangle~,
+    |\boldsymbol{\alpha}\rangle_{\boldsymbol{\Delta}}^{H}&\propto\int \textnormal{d}\boldsymbol{\phi}e^{i\boldsymbol{\phi}(H\hat{\mathbf{n}}-\boldsymbol{\Delta})}|\boldsymbol{\alpha}\rangle\\&\propto\sum_{H\mathbf{n}=\boldsymbol{\Delta}}\frac{\boldsymbol{\alpha}^{\mathbf{n}}}{\sqrt{\mathbf{n}!}}|\mathbf{n}\rangle~,
   \end{align}
   where \(\boldsymbol{\alpha}\) is a complex vector, \(\boldsymbol{\Delta}\) is an integer vector, and \(\boldsymbol{\phi}\) is a vector of phases iterating over the elements of the group generated by \(H\).
   Tiger codewords are of the above form, and their phase-space values \(\boldsymbol{\alpha}\) lie on a torus embedded in the complex sphere of fixed-energy coherent coherent states, satisfying \(|\alpha_j|^2 = 1\).

codes/quantum/oscillators/uncategorized/penrose.yml

@@ -14,7 +14,7 @@ description: |
 
   Letting \(|T\rangle\) be a Penrose tiling, the codeword corresponding to this tiling is a superposition of all points in the tiling's orbit under all Euclidean transformations,
   \begin{align}
-    |\overline{T}\rangle=\int dg|gT\rangle~,
+    |\overline{T}\rangle=\int \textnormal{d}g|gT\rangle~,
   \end{align}
   where \(g\) is a Euclidean transformation.
 

codes/quantum/properties/block/quantum_perfect.yml

@@ -33,7 +33,7 @@ description: |
 protection: |
   Perfect codes have been classified.
   For qubits (\(q=2\)), the only nontrivial codes are the stabilizer code family \([[(4^r-1)/3, (4^r-1)/3 - 2r, 3]]\) for \(r \geq 2\), obtained from Hamming codes over \(GF(4)\) via the Hermitian construction \cite{arxiv:quant-ph/9607027,arxiv:quant-ph/9608006}.
-  For qudits, the corresponding family is the \([[\frac{q^{2r}-1}{q^{2}-1},q^{n-2r},3]]\) family of quantum twisted codes \cite{arxiv:0907.0049,doi:10.1002/(SICI)1520-6610(2000)8:3<174::AID-JCD3>3.0.CO;2-T}.
+  For qudits, the corresponding family is the \([[\frac{q^{2r}-1}{q^{2}-1},q^{n-2r},3]]_q\) family of quantum twisted codes \cite{arxiv:0907.0049,doi:10.1002/(SICI)1520-6610(2000)8:3<174::AID-JCD3>3.0.CO;2-T}.
 
 # The trivial code (\(k=n\)) is also perfect.
 

codes/quantum/qubits/nonstabilizer/self_complementary.yml

@@ -28,7 +28,7 @@ relations:
       detail: 'Self-complementary quantum codes consisting of computational basis states whose bitstrings are sufficiently spaced apart correct at least one \hyperref[topic:ad]{AD} error \cite[Thm. 2.5]{arxiv:0712.2586}\cite[Thm. 2]{arxiv:0907.5149}.'
   cousins:
     - code_id: bits_into_bits
-      detail: 'A binary code is called \textit{self-complementary} if, for each codeword \(c\), its negation \(\overline{c}\) is also a codeword.'
+      detail: 'A binary code is called \textit{self-complementary} if, for each codeword \(c\), its negation \(\overline{c}\) is also a codeword. Any self-complementary \((n,K,d > 1)\) classical code yields an \(((n,K/2,2))\) self-complementary quantum code whose quantum codewords are superpositions of the classical codewords and their complements \cite[Lemma 1]{arxiv:quant-ph/0701065}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/qubits_into_qubits.yml

@@ -132,7 +132,7 @@ features:
     - 'The decoder determining the most likely error given a noise channel is called the \textit{maximum probability error} (MPE) decoder. For few-qubit codes (\(n\) is small), MPE decoding can be based by creating a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used.'
     - '\begin{defterm}{Effective distance and hook errors}
       \label{topic:effective-distance}
-      Decoders are characterized by an effective distance (a.k.a. \textit{circuit-level distance}), the minimum number of faulty operations during syndrome measurement that is required to make an undetectable error. A code is \textit{distance-preserving} if it admits a decoder whose circuit-level distance is equal to the code distance. A particularly dangerous class of syndrome measurement circuit faults are \textit{hook errors}, which are faults that cause more than one data-qubit error \cite{arxiv:quant-ph/0110143}. Hook errors occur at specific places in a syndrome extraction circuit and can sometimes be removed by re-ordering the gates of the circuit. If not, the use of \textit{flag qubits} (see \cite{preset:GottesmanBook}) to detect hook errors may be necessary to yield fault-tolerant decoders. 
+      Decoders are characterized by an effective distance (a.k.a. \textit{circuit-level distance}), the minimum number of faulty operations during syndrome measurement that is required to make an undetectable error. A code is \textit{distance-preserving} if it admits a decoder whose circuit-level distance is equal to the code distance. A particularly dangerous class of syndrome measurement circuit faults are \textit{hook errors}, which are ancilla faults that cause more than one data-qubit error \cite{arxiv:quant-ph/0110143}. Hook errors occur at specific places in a syndrome extraction circuit and can sometimes be removed by re-ordering the gates of the circuit. If not, the use of \textit{flag qubits} (see \cite{preset:GottesmanBook}) to detect hook errors may be necessary to yield fault-tolerant decoders. 
       \end{defterm}'
   fault_tolerance:
     - 'There are lower bounds on the overhead of fault-tolerant QEC in terms of the capacity of the noise channel \cite{arxiv:2202.00119}. A more stringent bound applies to geometrically local QEC due to the fact that locality constrains the growth of the entanglement that is needed for protection \cite{arxiv:2302.04317}.'

codes/quantum/qudits_galois/stabilizer/bch/quantum_twisted.yml

@@ -22,7 +22,7 @@ relations:
     - code_id: stabilizer_over_gfqsq
   cousins:
     - code_id: quantum_perfect
-      detail: 'The \([[\frac{q^{2r}-1}{q^{2}-1},q^{n-2r},3]]\) family of quantum twisted codes are the only perfect Galois-qudit codes \cite{arxiv:0907.0049,doi:10.1002/(SICI)1520-6610(2000)8:3<174::AID-JCD3>3.0.CO;2-T}.'
+      detail: 'The \([[\frac{q^{2r}-1}{q^{2}-1},q^{n-2r},3]]_q\) family of quantum twisted codes are the only perfect Galois-qudit codes \cite{arxiv:0907.0049,doi:10.1002/(SICI)1520-6610(2000)8:3<174::AID-JCD3>3.0.CO;2-T}.'
     - code_id: twisted_bch