~

codes/quantum/properties/asymmetric_qecc.yml

@@ -31,7 +31,7 @@ features:
     - 'A CNOT gate continuously connected to the identity cannot be noise-bias-preserving in finite dimensions \cite{arxiv:0806.0383}\cite[Appx. A]{arxiv:1904.09474}.'
 
   threshold:
-    - 'A lower bound on concatenated thresholds with CSS codes under biased noise \cite{arxiv:0710.1301}.'
+    - 'A lower bound on \hyperref[topic:computational-threshold]{concatenated thresholds} with CSS codes under biased noise \cite{arxiv:0710.1301}.'
 
   fault_tolerance:
     - 'Fault-tolerant noise-bias-preserving computation scheme \cite{arxiv:0806.0383}.'

codes/quantum/qubits/qubit_concatenated.yml

@@ -24,7 +24,10 @@ features:
     - 'The effective channel for a concatenation of codes is the composition of the codes'' effective channels \cite{arxiv:quant-ph/0206061}.'
     - 'Message passing algorithm for concatenated codes can be equivalent to ML decoding \cite{arxiv:quant-ph/0606126}.'
   threshold:
-    - 'The first methods to achieve a fault-tolerant \hyperref[topic:computational-threshold]{computational threshold} use concatenated qubit stabilizer codes \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}. Such methods require constant-space and polylogarithmic time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time \cite{arxiv:2207.08826}.'
+    - 'The first methods to achieve a fault-tolerant \hyperref[topic:computational-threshold]{computational threshold} use concatenated qubit stabilizer codes \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}. 
+    Such thresholds are called \hyperref[topic:computational-threshold]{concatenated thresholds},
+    These methods require constant-space and polylogarithmic time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time \cite{arxiv:2207.08826}.'
+
 
 
 relations:

codes/quantum/qubits/qubits_into_qubits.yml

@@ -120,10 +120,11 @@ features:
   threshold:
     - '\begin{defterm}{Computational threshold}
       \label{topic:computational-threshold}
-      A fault-tolerant computational threshold is the maximum noise rate in a noise model below which any logical computation of size \(M\) can be executed on a physical-qubit architecture to arbitrary accuracy and with an overhead of \hyperref[topic:asymptotics]{order} \(O(M\text{polylog}M)\).
-      The first methods to achieve a computational threshold use concatenated stabilizer codes \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}.
-      Such methods require constant-space and polylogarithmic-time overhead, but concatentions using quantum Hamming codes improve this to quasi-polylogarithmic time \cite{arxiv:2207.08826}.
-      Fault-tolerant computations with no notion of locality can be made local on a 2D or 3D geometry with minimial overhead \cite{arxiv:2402.13863}.
+      A fault-tolerant computational threshold is the maximum noise rate in a particular single-parameter noise model below which any logical computation of size \(M\) can be executed on a physical-qubit architecture to arbitrary accuracy and with an overhead of \hyperref[topic:asymptotics]{order} \(O(M\text{polylog}M)\).
+      The first methods to achieve a computational threshold use recursively concatenated stabilizer code families \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; such a threshold is called a \textit{concatenated threshold}.
+      Such methods require constant-space and polylogarithmic-time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time \cite{arxiv:2207.08826}.
+      Subsequently, thresholds were determined for infinite families of lattice stabilizer codes, starting with the toric code \cite{arxiv:quant-ph/0110143}; such a threshold is colloquially called a \textit{topological threshold}.
+      Fault-tolerant computations with no notion of locality can be made local on a 2D or 3D geometry with minimal overhead \cite{arxiv:2402.13863}.
       \end{defterm}'
     - '\begin{defterm}{Measurement threshold}
       \label{topic:measurement-threshold}

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

@@ -55,7 +55,7 @@ features:
     - 'Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived \cite{arxiv:2203.01706}.'
 
   threshold:
-    - 'Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.'
+    - 'Numerical study of \hyperref[topic:computational-threshold]{concatenated thresholds} of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.'
 
   fault_tolerance:
     - 'Pieceable fault-tolerant CZ, CNOT, and CCZ gates \cite{arxiv:1603.03948}.'

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

@@ -12,7 +12,7 @@ introduced: '\cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054}'
 
 description: |
   A member of the family of \([[7^m,1,3^m]]\) CSS codes, each of which is a recursive level-\(m\) concatenatenation of the Steane code.
-  This family is one of the first to admit a concatenated threshold \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054,arxiv:quant-ph/0207119,arxiv:quant-ph/0410047,arxiv:quant-ph/0604090}.
+  This family is one of the first to admit a \hyperref[topic:computational-threshold]{concatenated threshold} \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054,arxiv:quant-ph/0207119,arxiv:quant-ph/0410047,arxiv:quant-ph/0604090}.
 
 protection: 'Code performance against general Pauli channels has been worked out \cite{arxiv:quant-ph/0111003,arxiv:quant-ph/0206061}.'
 
@@ -23,9 +23,9 @@ features:
     fault_tolerance:
       - 'There exist fault-tolerant syndrome extraction protocols for the concatenated Steane code \cite{arxiv:2403.09978}.'
     code_capacity_threshold:
-      - 'This family is one of the first to admit a concatenated threshold \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054,arxiv:quant-ph/0207119,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}.'
+      - 'This family is one of the first to admit a \hyperref[topic:computational-threshold]{concatenated threshold} \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054,arxiv:quant-ph/0207119,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}.'
     threshold:
-      - 'Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}; see also \cite{arxiv:quant-ph/0406025}.'
+      - 'Numerical study of \hyperref[topic:computational-threshold]{concatenated thresholds} of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}; see also \cite{arxiv:quant-ph/0406025}.'
       - 'A \hyperref[topic:measurement-threshold]{measurement threshold} of one \cite{arxiv:2402.00145}.'
 
 relations:

codes/quantum/qubits/stabilizer/quantum_bch.yml

@@ -18,7 +18,7 @@ features:
     general_gates:
       - 'Magic-state distillation protocols \cite{arxiv:1709.02789}.'
     threshold:
-      - 'Semi-analytical estimates of concatenated thresholds \cite{arxiv:quant-ph/0207119}.
+      - 'Semi-analytical estimates of \hyperref[topic:computational-threshold]{concatenated thresholds} \cite{arxiv:quant-ph/0207119}.
       Qubit BCH codes are difficult to study numerically \cite{arxiv:0711.1556}.'
 
 notes:

codes/quantum/qubits/stabilizer/rm/quantum_hamming_css.yml

@@ -28,7 +28,7 @@ features:
     - 'Syndrome measurement can be done with two ancillary flag qubits \cite{arxiv:1705.02329}.'
     - 'Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads \cite{arxiv:2207.08826}.'
   threshold:
-    - 'Concatenated thresholds requiring constant-space and quasi-polylogarithmic time overhead \cite{arxiv:2207.08826}.'
+    - '\hyperref[topic:measurement-threshold]{Concatenated threshold} requiring constant-space and quasi-polylogarithmic time overhead \cite{arxiv:2207.08826}.'
 
 relations:
   parents:

codes/quantum/qubits/stabilizer/topological/color/3d_color/stab_15_1_3.yml

@@ -37,7 +37,7 @@ features:
     - 'Code is often used in magic-state distillation protocols because of its transversal \(T\) gate \cite{arxiv:quant-ph/0403025}.'
 
   threshold:
-    - 'Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.'
+    - 'Numerical study of \hyperref[topic:computational-threshold]{concatenated thresholds} of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.'
 
   fault_tolerance:
     - 'A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code \cite{arxiv:1304.3709,arxiv:1403.2734,arxiv:1703.03860,arxiv:2210.14074}.'

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

@@ -120,7 +120,7 @@ features:
     A threshold of \(0.41\%\) when concatenated with the \([[4,2,2]]\) code \cite{arxiv:1604.04062}.'
     - 'Phenomenological noise: \(3.3\%\) for square tiling \cite{arxiv:quant-ph/0401101}, and \(2.93(2)\%\) using several rounds of syndrome measurement \cite{arxiv:quant-ph/0207088}.'
     - 'Quasistatic phase damping and readout noise: \(2.85\%\) \cite{arxiv:2401.04530}.'
-    - 'Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.'
+    - 'Numerical study of \hyperref[topic:computational-threshold]{concatenated thresholds} of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.'
 
 
 realizations:

codes/quantum/qubits/subsystem/qldpc/bacon_shor/bacon_shor.yml

@@ -70,11 +70,11 @@ features:
   code_capacity_threshold:
     - 'The number of check operators scales sublinearly with system size, so the Bacon-Shor codes alone do not exhibit a topological threshold in the \(m_1,m_2 \to \infty\) limit \cite{arxiv:1903.03937}.
     However, a threshold can be obtained from concatenated Bacon-Shor codes that are further restricted to planar geometries, whose recovery circuit is a subset of a circuit used by a larger bona-fide Bacon-Shor code \cite{arxiv:2305.12046}.
-    This threshold differs from a concatenated threshold in that there are no long-range connectivity requirements.'
-    - 'Lower bounds for the concatenated threshold of various small Bacon-Shor codes are tabulated in \cite[Table I]{arxiv:quant-ph/0610063}.'
-    - '\(2.02 \times 10^{-5}\) concatenated threshold for the concatenated \([[9,1,3,3]]\) Bacon-Shor code \cite{arxiv:0805.4213}.'
+    This threshold differs from a \hyperref[topic:computational-threshold]{concatenated threshold} in that there are no long-range connectivity requirements.'
+    - 'Lower bounds for the \hyperref[topic:computational-threshold]{concatenated threshold} of various small Bacon-Shor codes are tabulated in \cite[Table I]{arxiv:quant-ph/0610063}.'
+    - '\(2.02 \times 10^{-5}\) \hyperref[topic:computational-threshold]{concatenated threshold} for the concatenated \([[9,1,3,3]]\) Bacon-Shor code \cite{arxiv:0805.4213}.'
   threshold:
-    - 'Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.'
+    - 'Numerical study of \hyperref[topic:computational-threshold]{concatenated thresholds} of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.'
     - 'The Bacon-Shor code has a \hyperref[topic:measurement-threshold]{measurement threshold} of zero \cite{arxiv:2402.00145}.'
   decoders:
     - 'Message passing for \([[9,1,3,3]]\) Bacon-Shor code \cite{arxiv:0806.2188}.'