~

codes/classical/bits/ltc/gs-ltc.yml

@@ -11,7 +11,7 @@ name: 'Goldreich-Sudan code'
 introduced: '\cite{doi:10.1145/1162349.1162351}'
 
 description: |
-  Locally testable \([n,k,d]\) code with \(n = k^{1+O(1/u)}\) and distance of order \(\Omega(n)\) for query complexity \(u\). The same work also presented a probabilistic construction of codes of size \(k^{1+o(1)}\).
+  Locally testable \([n,k,d]\) code with \(n = k^{1+O(1/u)}\) and distance of \hyperref[topic:asymptotics]{order} \(\Omega(n)\) for query complexity \(u\). The same work also presented a probabilistic construction of codes of size \(k^{1+o(1)}\).
 
 relations:
   parents:

codes/classical/bits/tanner/regular_tanner/regular_ldpc/expander.yml

@@ -29,7 +29,7 @@ features:
 
   decoders:
     - 'Decoding can be done in \(O(n)\) runtime using a greedy \textit{flip decoder} \cite{doi:10.1109/18.556667} (see also \cite{doi:10.1109/TIT.2006.887523}). The algorithm consists of flipping a bit of the received word if it will result in a greater number of satisfied parity checks. This is repeated until a codeword is reached.'
-    - '''Find erasures and Decode'' a.k.a. Viderman''s algorithm correcting order \(\Omega(n)\) errors in \hyperref[topic:asymptotics]{order} \(O(n)\) time \cite{doi:10.1145/2493252.2493255}.'
+    - '''Find erasures and Decode'' a.k.a. Viderman''s algorithm correcting \hyperref[topic:asymptotics]{order} \(\Omega(n)\) errors in \hyperref[topic:asymptotics]{order} \(O(n)\) time \cite{doi:10.1145/2493252.2493255}.'
 
   fault_tolerance:
     - 'The flip decoding algorithm is fault tolerant against parity check errors \cite{doi:10.1109/18.556668}; see also \href{http://courses.csail.mit.edu/6.440/spring08/index.html}{course notes} by M. Sudan.'

codes/classical/q-ary_digits/convolutional/turbo.yml

@@ -16,7 +16,7 @@ description: |
   The choice of interleaver is important to the code design \cite{doi:10.1109/ICC.1995.525138,doi:10.1109/49.924867}.
 
 protection: |
-  Parallel concatenated turbo codes have typical minimum distance with upper bound \(O(\log(n))\). Truhachev, Lentmacher, and Zignagirov produced a sequence of turbo codes with minimum distance of order \(\Theta(\log(n))\). \cite{doi:10.1109/ISIT.2002.1023356}.
+  Parallel concatenated turbo codes have typical minimum distance with upper bound \(O(\log(n))\). Truhachev, Lentmacher, and Zignagirov produced a sequence of turbo codes with minimum distance of \hyperref[topic:asymptotics]{order} \(\Theta(\log(n))\). \cite{doi:10.1109/ISIT.2002.1023356}.
   Various bounds on code parameters exist \cite{doi:10.1109/TIT.2002.1003833,doi:10.1109/26.681398}.
 
 features:

codes/classical/q-ary_digits/wozencraft.yml

@@ -16,7 +16,7 @@ description: |
   Each \([2k,k]_q\) code is defined by a parameter \(\alpha \in GF(q^k)\) and consists of codewords \((x,\alpha x)\) for each message \(x \in GF(q^k)\), where each element of \(GF(q^k)\) is expressed over \(GF(q)^k\) using a fixed basis.
 
 protection: |
-  Bounds and constructions with order \(\Omega(\sqrt{k})\) distance are provided in Ref. \cite{arxiv:2305.02484}.
+  Bounds and constructions with \hyperref[topic:asymptotics]{order} \(\Omega(\sqrt{k})\) distance are provided in Ref. \cite{arxiv:2305.02484}.
 
 features:
   rate: 'Meets the \hyperref[topic:gv-bound]{GV bound} for most choices of \(\alpha\).

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

@@ -108,7 +108,7 @@ features:
   \end{align}
   and a generalization of the formula to the non-orientable case can be found in Ref. \cite{arxiv:1612.07792}.'
   encoders:
-    - 'A depth of 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.
+    - '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}.'
   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}.'

codes/quantum/properties/stabilizer/qldpc/qldpc.yml

@@ -46,7 +46,7 @@ description: |
       \(\Theta(n)\) & \(\Theta(n)\) & \hyperref[code:dhlv]{Dinur-Hsieh-Lin-Vidick}
       }
       \end{cells}
-      \caption{Notable QLDPC codes; \(c\) is a positive integer.}
+      \caption{Notable QLDPC codes and their \hyperref[topic:asymptotics]{asymptotic scaling}; \(c\) is a positive integer.}
       \label{table:qldpc-codes}
     \end{table}
 
@@ -67,7 +67,7 @@ features:
     Asymptotic scaling of \(k\) and \(d\) with \(n\) depends heavily on the code construction.
     Bounds generalizing the \hyperref[topic:bpt-bound]{BPT bound} to QLDPC codes depend on the separation profile of the code's underlying connectivity graph \cite{arxiv:2106.00765,arxiv:2307.03283}.
     A constant relative minimum distance can be achieved only for graphs that contain expanders \cite{arxiv:2106.00765}.
-    Conversely, a code with parameters \(k\) and \(d\) requires a graph with order \(\Omega(d)\) edges of length of order \(\Omega(d/n^{1/D})\) \cite{arxiv:2109.10982}.
+    Conversely, a code with parameters \(k\) and \(d\) requires a graph with \hyperref[topic:asymptotics]{order} \(\Omega(d)\) edges of length of \hyperref[topic:asymptotics]{order} \(\Omega(d/n^{1/D})\) \cite{arxiv:2109.10982}.
     Random QLDPC codes found by solving certain constraint satisfaction problems (CSPs) practically achieve the capacity of the erasure channel \cite{arxiv:2207.03562}.
 
     Qubit QLDPC codes cannot attain the capacity of the erasure channel \cite{arxiv:1205.7036}, but this capacity can be attained by code families with weight \(w = O(\text{polylog}n)\) \cite{arxiv:1703.00382}.

codes/quantum/qubits/nonstabilizer/circuit_to_hamiltonian.yml

@@ -12,13 +12,13 @@ introduced: '\cite{arxiv:1811.00277}'
 
 description: |
   Approximate qubit block code that forms the ground-state space of a frustration-free Hamiltonian with non-commuting terms.
-  Its distance and logical-qubit number are both of order \(\Omega(n/\log^5 n)\) \cite[Thm. 3.1]{arxiv:1811.00277}.
+  Its distance and logical-qubit number are both of \hyperref[topic:asymptotics]{order} \(\Omega(n/\log^5 n)\) \cite[Thm. 3.1]{arxiv:1811.00277}.
   The code is an approximate non-stabilizer QLWC code since the Hamiltonian consists of non-commuting weight-ten non-Pauli projectors, with each qubit acted on by \hyperref[topic:asymptotics]{order} \(O(\text{polylog}(n)\) projectors.
 
   The code is constructed by converting the encoding circuit of a Brown-Fawzi random Clifford-circuit code into a Hamiltonian using the spacetime circuit-to-Hamiltonian construction \cite{arxiv:quant-ph/0609067,arxiv:1311.6101} (a generalization of the Feynman-Kitaev clock construction \cite{doi:10.1090/gsm/047}).
   The ground-state subspace of this Hamiltonian is the \(\epsilon\)-approximate code with infidelity of recovery \(\epsilon = O(1/\text{polylog}(n))\).
 
-  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 \(D\).
+  Using Markov-chain techniques, the gap of the Hamiltonian can be proven to be of \hyperref[topic:asymptotics]{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}.
@@ -38,7 +38,7 @@ relations:
       detail: 'Circuit-to-Hamiltonian approximate codes form the ground-state space of a frustration-free non-commuting projector Hamiltonian whose projectors are constant weight, but such that each physical qubit is acted on by \hyperref[topic:asymptotics]{order} \(O(\text{polylog}(n))\) projectors.'
   cousins:
     - code_id: qlwc
-      detail: 'The circuit-to-Hamiltonian code construction yields approximate codes whose distance and logical-qubit number are both of order \(\Omega(n/\log^5 n)\) \cite[Thm. 3.1]{arxiv:1811.00277}.
+      detail: 'The circuit-to-Hamiltonian code construction yields approximate codes whose distance and logical-qubit number are both of \hyperref[topic:asymptotics]{order} \(\Omega(n/\log^5 n)\) \cite[Thm. 3.1]{arxiv:1811.00277}.
       These codes are approximate non-stabilizer QLWC codes since the Hamiltonian consists of non-commuting weight-ten non-Pauli projectors, with each qubit acted on by \hyperref[topic:asymptotics]{order} \(O(\text{polylog}(n)\) projectors.'
     - code_id: nonlocal_lowdepth
       detail: 'Circuit-to-Hamiltonian approximate codes are constructed by converting the encoding circuit of a Brown-Fawzi random Clifford-circuit code into a Hamiltonian using the spacetime circuit-to-Hamiltonian construction \cite{arxiv:quant-ph/0609067,arxiv:1311.6101}.'

codes/quantum/qubits/stabilizer/qldpc/concatenated/hierarchical.yml

@@ -16,7 +16,7 @@ description: |-
   The growing syndrome extraction circuit depth allows known bounds in the literature to be weakened \cite{arxiv:2109.14599,arxiv:2302.04317}.
 
 features:
-  rate: 'Rate vanishes as a function of order \(\Omega(1/\log(n)^2)\).'
+  rate: 'Rate vanishes as a function of \hyperref[topic:asymptotics]{order} \(\Omega(1/\log(n)^2)\).'
 
   decoders:
     - 'Decoding is performed as in a standard \hyperref[code:qubit_concatenated]{concatenated code} using a decoder for the inner code and outer code.

codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/fiber_bundle.yml

@@ -14,7 +14,7 @@ alternative_names:
   - 'Twisted product code'
 
 description: 'A CSS code constructed by combining one code as the base and another as the fiber of a fiber bundle.
-In particular, taking a random LDPC code as the base and a cyclic repetition code as the fiber yields, after distance balancing, a QLDPC code with distance of order \(\Omega(n^{3/5}\text{polylog}(n))\) and rate of order \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained.'
+In particular, taking a random LDPC code as the base and a cyclic repetition code as the fiber yields, after distance balancing, a QLDPC code with distance of \hyperref[topic:asymptotics]{order} \(\Omega(n^{3/5}\text{polylog}(n))\) and rate of \hyperref[topic:asymptotics]{order} \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained.'
 #In addition, weight reduction can be used to make this an \([[n,\Omega(n^{3/5}/\text{polylog}(n),\Omega(n^{3/5}/\text{polylog}(n))]]\) QLDPC code family.'
 
 #protection: 'Pauli errors up to weight \(\Omega(n^{3/5}\text{polylog}(n))\).'
@@ -36,7 +36,7 @@ relations:
     - code_id: distance_balanced
       detail: 'Fiber-bundle code constructions use distance balancing and weight reduction to increase distance.'
     - code_id: random_stabilizer
-      detail: 'Taking a random LDPC code as the base and a cyclic repetition code as the fiber yields, after distance balancing, a QLDPC code with distance of order \(\Omega(n^{3/5}\text{polylog}(n))\) and rate of order \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained.'
+      detail: 'Taking a random LDPC code as the base and a cyclic repetition code as the fiber yields, after distance balancing, a QLDPC code with distance of \hyperref[topic:asymptotics]{order} \(\Omega(n^{3/5}\text{polylog}(n))\) and rate of \hyperref[topic:asymptotics]{order} \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/quantum_expander.yml

@@ -12,16 +12,16 @@ introduced: '\cite{arxiv:1504.00822}'
 
 description: 'CSS codes constructed from a hypergraph product of bipartite expander graphs \cite{doi:10.1090/S0273-0979-06-01126-8} with bounded left and right vertex degrees. For every bipartite graph there is an associated matrix (the parity check matrix) with columns indexed by the left vertices, rows indexed by the right vertices, and 1 entries whenever a left and right vertex are connected. This matrix can serve as the parity check matrix of a classical code. Two bipartite expander graphs can be used to construct a quantum CSS code (the quantum expander code) by using the parity check matrix of one as \(X\) checks, and the parity check matrix of the other as \(Z\) checks.'
 
-protection: 'Pauli errors of weight \(\leq t\), distance scales as order \(\Omega(n^{1/2})\).'
+protection: 'Pauli errors of weight \(\leq t\), distance scales as \hyperref[topic:asymptotics]{order} \(\Omega(n^{1/2})\).'
 
 features:
   rate: '\([[n,k=\Theta(n),d=O(\sqrt{n})]]\) code with asymptotically constant rate.'
 
   decoders:
-    - 'Small set-flip linear-time decoder, which corrects order \(\Omega(n^{1/2})\) adversarial errors \cite{arxiv:1504.00822}.'
+    - 'Small set-flip linear-time decoder, which corrects \hyperref[topic:asymptotics]{order} \(\Omega(n^{1/2})\) adversarial errors \cite{arxiv:1504.00822}.'
     - 'Log-time decoder \cite{arxiv:1808.03821}.'
     - 'Constant-time decoder \cite{manual:{A. Grospellier. Constant time decoding of quantum expander codes and application to fault-tolerant quantum computation. PhD thesis, Inria Paris (2019).}}.'
-    - '2D geometrically local syndrome extraction circuits acting on a patch of \(N\) physical qubits have to be of depth of order \(\Omega(n/\sqrt{N})\) or deeper. More generally, there is a tradeoff between the depth \(D\) and width \(W\) of a syndrome extraction circuit, namely, \(D \geq n/\sqrt{W}\) \cite{arxiv:2109.14599}.'
+    - '2D geometrically local syndrome extraction circuits acting on a patch of \(N\) physical qubits have to be of depth of \hyperref[topic:asymptotics]{order} \(\Omega(n/\sqrt{N})\) or deeper. More generally, there is a tradeoff between the depth \(D\) and width \(W\) of a syndrome extraction circuit, namely, \(D \geq n/\sqrt{W}\) \cite{arxiv:2109.14599}.'
 
   fault_tolerance:
     - 'Fault-tolerance with constant overhead can be achieved \cite{arxiv:1808.03821}.'

codes/quantum/qubits/stabilizer/qldpc/homological/dlv.yml

@@ -15,8 +15,8 @@ description: |
   Member of a family of quantum locally testable codes constructed using cubical chain complexes, which are \(t\)-order extensions of the complexes underlying expander codes (\(t=1\)) and expander lifted-product codes (\(t=2\)).
 
   For \(t=4\), assuming a conjecture about random linear maps, there exists a family with linear dimension and inverse poly-logarithmic relative distance and soundness.
-  Applying weight reduction yields order \(\Omega(1/\text{polylog}n)\) soundness, distance, and dimension, but order \(\Theta(n)\) locality \cite[Table 4]{arxiv:2309.05541}.
-  Applying distance amplification and soundness amplification yields asymptotically constant soundness, order \(\Theta(n)\) distance, order \(\Theta(n)\) dimension, but poly-logarithmic locality \cite[Table 4]{arxiv:2309.05541}.
+  Applying weight reduction yields \hyperref[topic:asymptotics]{order} \(\Omega(1/\text{polylog}n)\) soundness, distance, and dimension, but \hyperref[topic:asymptotics]{order} \(\Theta(n)\) locality \cite[Table 4]{arxiv:2309.05541}.
+  Applying distance amplification and soundness amplification yields asymptotically constant soundness, \hyperref[topic:asymptotics]{order} \(\Theta(n)\) distance, \hyperref[topic:asymptotics]{order} \(\Theta(n)\) dimension, but poly-logarithmic locality \cite[Table 4]{arxiv:2309.05541}.
 
 
 relations:
@@ -25,7 +25,7 @@ relations:
   cousins:
     - code_id: qltc
       detail: 'DLV codes have linear dimension and inverse poly-logarithmic relative distance and soundness, assuming a conjecture about random linear maps \cite{arxiv:2402.07476}.
-      Applying distance amplification and soundness amplification yields asymptotically constant soundness, order \(\Theta(n)\) distance, order \(\Theta(n)\) dimension, but poly-logarithmic locality \cite[Table 4]{arxiv:2309.05541}.'
+      Applying distance amplification and soundness amplification yields asymptotically constant soundness, \hyperref[topic:asymptotics]{order} \(\Theta(n)\) distance, \hyperref[topic:asymptotics]{order} \(\Theta(n)\) dimension, but poly-logarithmic locality \cite[Table 4]{arxiv:2309.05541}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/stabilizer/qldpc/homological/ramanujan/ramanujan_tensor_product.yml

@@ -17,7 +17,7 @@ protection: 'Without distance balancing, a Ramanujan code can have \(d_X =\Omega
 
 features:
 
-  rate: 'For 2D Ramanujan complexes, the rate is of order \(\Omega(\sqrt{ \frac{1}{n \log n} })\), with minimum distance \(d = \Omega(\sqrt{n \log n}) \). For 3D, the rate is \( \Omega(\frac{1}{\sqrt{n}\log n}) \) with minimum distance \(d \geq \sqrt{n} \log n \).'
+  rate: 'For 2D Ramanujan complexes, the rate is of \hyperref[topic:asymptotics]{order} \(\Omega(\sqrt{ \frac{1}{n \log n} })\), with minimum distance \(d = \Omega(\sqrt{n \log n}) \). For 3D, the rate is \( \Omega(\frac{1}{\sqrt{n}\log n}) \) with minimum distance \(d \geq \sqrt{n} \log n \).'
 
   decoders:
     - 'For 2D simplicial complexes, cycle code decoder admitting a polynomial-time decoding algorithm can be used \cite{arxiv:2004.07935}.'

codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface.yml

@@ -38,7 +38,7 @@ features:
     - 'Graph-state based adaptive circuit \cite{arxiv:quant-ph/0703143,arxiv:1105.2111}.'
     - 'For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit \cite{arxiv:2002.00362,arxiv:2110.02020} or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates \cite{arxiv:0712.0348,arxiv:0806.4583,arxiv:1207.0253} (matching lower bounds \cite{arxiv:quant-ph/0603121,arxiv:quant-ph/0603114,arxiv:1810.03912}).'
     - 'Stabilizer measurement-based circuit of linear depth \cite{arxiv:quant-ph/0110143,arxiv:1404.2495}.'
-    - 'Any geometrically local unitary circuit on a lattice \(\Lambda\) that prepares a state whose energy density with respect to the surface code Hamiltonian is \(\epsilon\) must have depth of order \(\Omega(\min(\sqrt{|\Lambda|},1/\epsilon^{\frac{1-\alpha}{2}}))\) for any \(\alpha>0\) \cite{arxiv:2210.06796}.'
+    - 'Any geometrically local unitary circuit on a lattice \(\Lambda\) that prepares a state whose energy density with respect to the surface code Hamiltonian is \(\epsilon\) must have depth of \hyperref[topic:asymptotics]{order} \(\Omega(\min(\sqrt{|\Lambda|},1/\epsilon^{\frac{1-\alpha}{2}}))\) for any \(\alpha>0\) \cite{arxiv:2210.06796}.'
     - 'Single-shot state preparation \cite{arxiv:1904.01502}.'
     - 'Various techniques to generate lattices useful for particular architectures \cite{arxiv:2111.13729} or removing lattice defects \cite{arxiv:2211.08468,arxiv:2405.06941} exist.'
   #make one that you feel is applicable to a large and interesting class of codes