Merge branch 'main' of https://github.com/errorcorrectionzoo/eczoo_data

codes/quantum/oscillators/stabilizer/lattice/gkp.yml

@@ -72,6 +72,8 @@ relations:
       detail: 'Because square-lattice GKP error states are parameterized by two modular (i.e., periodic) variables of position and momentum, measuring one of the GKP stabilizers constrains the oscillator Hilbert space into that of a rotor.'
     - code_id: hypercubic
       detail: 'GKP codewords, when written in terms of coherent states, form a square lattice in phase space.'
+    - code_id: fusion
+      detail: 'GKP states can be used to perform computation in a fusion-based encoding \cite{manual:{Doherty, A., Gimeno-segovia, M., Litinski, D., Nickerson, N., Pant, M., Rudolph, T. and Sparrow, C., Psiquantum, Corp., 2024. GENERATION AND MEASUREMENT OF ENTANGLED SYSTEMS OF PHOTONIC GKP QUBITS. U.S. Patent Application 18/273,753.}}.'
 
     #- code_id: toric_GKP
       #detail: 'It is a concatenation that replace the physical qubits in toric code by GKP codes. Each physical qubit is replaced by a harmonic oscillator in GKP state \cite{doi:10.1103/PhysRevA.99.032344}. By using the GKP error information, the toric code threshold is improved from \(10%\) to \(14%\).'

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

@@ -100,7 +100,7 @@ features:
     - 'Sliding-window decoding \cite{arxiv:2311.03307}.'
     - 'Closed-branch decoder \cite{arxiv:2402.01532}.'
     - 'BP with guided decimation guessing (GDG) sliding-window decoder for CSS qubit codes \cite{arxiv:2403.18901}.'
-    - 'Performing \(d\) syndrome extraction rounds obtains an effective distance of \(d\) for a qubit QLDPC code \cite{arxiv:1310.2984}.'
+    - 'Performing \(d\) syndrome extraction rounds obtains an \hyperref[topic:effective-distance]{effective distance} of \(d\) for a qubit QLDPC code \cite{arxiv:1310.2984}.'
     - 'Fault-tolerant constant-depth encoder and unencoder \cite{arxiv:2408.06299}.'
     - 'BP plus ordered Tanner forest (BP+OTF) almost-linear time decoder \cite{arxiv:2409.01440}.'
   fault_tolerance:

codes/quantum/qubits/qubits_into_qubits.yml

@@ -124,9 +124,11 @@ features:
     - 'Arbitrary \(n\)-qubit circuits can be implemented fault-tolerantly in a 3D architecture using \(O(n^{3/2}\log^3 n)\) qubits, and in a 2D architecture using only \(O(n^2 \log^3 n)\) qubits \cite{arxiv:2402.13863}.'
   decoders:
     - 'Incorporating faulty syndrome measurements can be done using the \textit{phenomenological noise model}, which simulates errors during syndrome extraction by flipping some of the bits of the measured syndrome bitstring. In the more involved \textit{circuit-level noise model}, every component of the syndrome extraction circuit can be faulty.'
-    - '\textit{Hook errors} are syndrome measurement circuit 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.'
     - '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.'
-    - 'Decoders are characterized by an \textit{effective distance} or \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.'
+    - '\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. 
+      \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}.'
     - 'Arbitrary \(n\)-qubit circuits can be implemented fault-tolerantly in a 3D architecture using \(O(n^{3/2}\log^3 n)\) qubits, and in a 2D architecture using only \(O(n^2 \log^3 n)\) qubits \cite{arxiv:2402.13863}.'

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

@@ -41,17 +41,20 @@ features:
     - 'Hadamard (up to logical SWAP gates) and control-\(Z\) on all logical qubits \cite{arxiv:2204.10812}.'
     - 'Patch-transversal gates inherited from the automorphism group of the underlying classical codes \cite[Appx. D]{arxiv:2309.11719}.'
   decoders:
+    - 'Single-ancilla syndrome extraction circuits do not admit \hyperref[topic:effective-distance]{hook errors} \cite{arxiv:2409.02193}.'
     - 'ReShape decoder that uses minimum weight decoders for the classical codes used in the hypergraph construction \cite{arxiv:2105.02370}.'
     - '2D geometrically local syndrome extraction circuits with depth \hyperref[topic:asymptotics]{order} \(O(\sqrt{n})\) using \hyperref[topic:asymptotics]{order} \(O(n)\) ancilla qubits \cite{arxiv:2109.14599}.'
     - 'Improved BP-OSD decoder \cite{arxiv:2206.03122}.'
     - 'Erasure-correction can be implemented approximately with \(O(n^2)\) operations with quantum generalizations \cite{arxiv:2208.01002} of the peeling and pruned peeling decoders \cite{doi:10.1109/18.910575}, with a probabilistic version running in \(O(n^{1.5})\) operations.'
-    - 'Syndrome measurements are distance-preserving because syndrome extraction circuits can be designed to avoid hook errors \cite{arxiv:2308.15520}.'
+    - 'Syndrome measurements are \hyperref[topic:effective-distance]{distance-preserving} because syndrome extraction circuits can be designed to avoid \hyperref[topic:effective-distance]{hook errors} \cite{arxiv:2308.15520}.'
     - 'Generalization \cite{arxiv:2310.07868} of Viderman''s algorithm for expander codes \cite{doi:10.1145/2493252.2493255}.'
   general_gates:
     - 'Code deformation techniques yield Clifford gates \cite{arxiv:1909.07424}.'
   code_capacity_threshold:
     - 'Some thresholds were determined in Ref. \cite{arxiv:1208.2317}.'
     - 'Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model \cite{arxiv:1804.01950}. For example, a threshold of \(7\%\) was obtained under independent \(X\) and \(Z\) noise for codes obtained from random \((3,4)\)-regular Gallager codes.'
+  fault_tolerance:
+    - 'Single-ancilla syndrome extraction circuits do not admit \hyperref[topic:effective-distance]{hook errors} \cite{arxiv:2409.02193}.'
   threshold:
     - 'Circuit-level noise: \(0.1\%\) with all-to-all connected syndrome extraction circuits \cite{arxiv:2109.14599} and DiVincenzo-Aliferis syndrome extraction circuits \cite{arxiv:quant-ph/0607047} combined with non-local gates \cite{arxiv:2409.05818}.
     No threshold observed above physical noise rates at or above \(10^{-6}\) using 2D geometrically local syndrome extraction circuits.'

codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/lp/qcga.yml

@@ -36,7 +36,7 @@ features:
 
   decoders:
     - 'Syndrome extraction circuit requires seven layers of CNOT gates regardless of code length. BP-OSD decoder \cite{arxiv:1904.02703} has been extended \cite{arxiv:2308.07915} to account for measurement errors (i.e., the circuit-based noise model \cite{arxiv:0803.0272}).'
-    - 'Random and optimized syndrome extraction schedules from Ref. \cite{arxiv:2308.07915} are not distance preserving.'
+    - 'Random and optimized syndrome extraction schedules from Ref. \cite{arxiv:2308.07915} are not \hyperref[topic:effective-distance]{distance-preserving}.'
     - 'Some long-range check operators can be measured less frequently than others \cite{arxiv:2404.17676}.'
     - 'Syndrome extraction circuits called \textit{morphing circuits} \cite{arxiv:2407.16336}, generalizing circuits for the color code \cite{arxiv:2312.08813}.'
 

codes/quantum/qubits/stabilizer/qubit_css.yml

@@ -136,7 +136,7 @@ features:
     - 'Steane error correction \cite{arxiv:quant-ph/9611027}, where fault-tolerance is ensured by preparing ancillary encoded states and extracting syndromes via \(CNOT\) gates.'
     - '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}.
-    Distance-preserving flag fault-tolerant error correction can be done using lookup tables for small codes \cite{arxiv:2306.12862}.
+    \hyperref[topic:effective-distance]{Distance-preserving} flag fault-tolerant error correction can be done using lookup tables for small codes \cite{arxiv:2306.12862}.
     Any self-dual CSS code with bounded-weight stabilizer generators admits flag fault-tolerant syndrome extraction \cite{arxiv:1708.02246}.'
     - '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}.'

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

@@ -56,7 +56,7 @@ relations:
     - code_id: doubled_color
       detail: 'The \([[15,1,3]]\) code can be viewed as a (gauge-fixed) doubled color code obtained from the Steane code via the doubling transformation \cite{arxiv:1509.03239}.'
     - code_id: steane
-      detail: 'The \([[15,1,3]]\) code can be viewed as a (gauge-fixed) doubled color code obtained from the Steane code via the doubling transformation \cite{arxiv:1509.03239}. 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,arxiv:2306.17686}. The \([[105,1]]\) concatenation of the \([[15,1,3]]\) and Steane codes allows for a universal gate set consisting of gates that are transversal w.r.t. to two different partitions \cite{arxiv:1309.3310,arxiv:1710.07256}.'
+      detail: 'The \([[15,1,3]]\) code can be viewed as a (gauge-fixed) doubled color code obtained from the Steane code via the doubling transformation \cite{arxiv:1509.03239}. 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,arxiv:2306.17686}. An \([[105,1,3]]\) alternative concatenation of the \([[15,1,3]]\) and Steane codes allows for a universal gate set consisting of gates that are close to transversal \cite{arxiv:1309.3310,arxiv:1710.07256}.'
     - code_id: concatenated_steane
       detail: 'The \([[105,1]]\) concatenation of the \([[15,1,3]]\) and Steane codes allows for a universal gate set consisting of gates that are transversal w.r.t. to two different partitions \cite{arxiv:1309.3310,arxiv:1710.07256}.'
     - code_id: qubit_concatenated

codes/quantum/qubits/stabilizer/topological/color/color.yml

@@ -32,7 +32,7 @@ features:
   transversal_gates:
     - 'Some color codes on \(D\)-dimensional lattices can transversally implement a gate at the \((D-1)\)st level of the \term{Clifford hierarchy} in the form of a \(Z\)-rotation by angle \(-\pi/2^D\) \cite[Fig. 3]{arxiv:1410.0069}.'
   decoders:
-    - 'In contrast to the surface code, the color code can suffer from unremovable hook errors due to the specifics of its syndrome extraction circuits.
+    - 'In contrast to the surface code, the color code can suffer from unremovable \hyperref[topic:effective-distance]{hook errors} due to the specifics of its syndrome extraction circuits.
     Fault-tolerant decoders thus have to utilize additional flag qubits.'
   fault_tolerance:
     - 'The 6D color code is a self-correcting quantum memory and admits fault-tolerant universal gate set in 7D \cite{arxiv:0907.5228}.'

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

@@ -32,13 +32,13 @@ protection: 'The \([[L^2,1,L]]\) variant \cite{arxiv:quant-ph/0703272} includes
 
 features:
   decoders:
-    - 'Only certain syndrome extraction schedules are distance-preserving \cite{arxiv:1404.3747}.'
+    - 'Only certain syndrome extraction schedules are \hyperref[topic:effective-distance]{distance-preserving} \cite{arxiv:1404.3747}.'
     - 'Local neural-network using 3D convolutions, combined with a separate global decoder \cite{arxiv:2208.01178}.'
     - 'Iterative CNOT decoder \cite{arxiv:2407.20976}.'
   threshold:
     - 'Thresholds for various amounts of erasure, Pauli, and measurement noise are known \cite{arxiv:2408.00829}.'
   fault_tolerance:
-    - 'A particular choice of CNOT gates during syndrome extraction is required to avoid hook errors and be fault-tolerant to syndrome qubit errors \cite{arxiv:quant-ph/0110143,arxiv:1208.0928,arxiv:1404.3747}.'
+    - 'A particular choice of CNOT gates during syndrome extraction is required to avoid \hyperref[topic:effective-distance]{hook errors} and be fault-tolerant to syndrome qubit errors \cite{arxiv:quant-ph/0110143,arxiv:1208.0928,arxiv:1404.3747}.'
 
 
 relations:

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

@@ -92,7 +92,7 @@ features:
   decoders:
     - 'Using data from multiple syndrome measurements prior to decoding allows for correcting syndrome measurement errors.
     The surface code requires \hyperref[topic:asymptotics]{order} \(O(d)\) extraction rounds in order to gain a reliable estimate.
-    Syndrome measurements are distance-preserving because syndrome extraction circuits can be designed to avoid hook errors \cite{arxiv:quant-ph/0110143}.'
+    Syndrome measurements are \hyperref[topic:effective-distance]{distance-preserving} because syndrome extraction circuits can be designed to avoid \hyperref[topic:effective-distance]{hook errors} \cite{arxiv:quant-ph/0110143}.'
     - 'Syndrome extraction circuits consist of CNOT gates and ancillary measurements since this is a stabilizer code \cite{arxiv:1208.0928}. Measurement schedules can be optimized using spacetime circuit codes to yield the \textit{3CX surface code} \cite{arxiv:2302.02192}. Schedules can also be optimized via ZX calculus \cite{doi:10.1007/978-3-540-70583-3_25,arxiv:0906.4725}. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed \cite{arxiv:2007.00307,arxiv:2206.12780,arxiv:2310.12981}.'
     - 'Expanding diamonds decoder correcting errors of some maximum fractal dimension \cite{manual:{Andrew Landahl, private communication, 2023}}.
     The sub-threshold failure probability scales as \((p/p_{\text{th}})^{d^\beta}\), where \(p_{\text{th}}\) is the threshold and \(\beta = \log_3 2\).'

codes/quantum/qubits/stabilizer/topological/surface/non-css/triangle_surface.yml

@@ -84,10 +84,10 @@ features:
     - 'The symmetry of triangle codes allows for fault-tolerant measurement and encoding in any Pauli basis \cite{arxiv:1612.04795}.'
     - 'A non-fault-tolerant curcuit initializes the triangle code.
     To guarantee fault-tolerance, postselection is performed on trivial measurements of the syndrome and of the logical Pauli, depending on the basis of the logical states \cite{arxiv:1612.04795}.'
-    - 'Making syndrome extraction fault tolerant requires a specific ordering of syndrome measurements so as to avoid hook errors \cite{arxiv:1612.04795}.'
+    - 'Making syndrome extraction fault tolerant requires a specific ordering of syndrome measurements so as to avoid \hyperref[topic:effective-distance]{hook errors} \cite{arxiv:1612.04795}.'
 
 # , instead of using the conventional ordering of coupling to the loop after coupling to the plaquette, the extraction has the coupling to the loop interwoven with the coupling to the plaquette.
-# These are equivalent in circuits, but makes a difference to the fault-tolerance since the latter can detect hook errors and the former cannot
+# These are equivalent in circuits, but makes a difference to the fault-tolerance since the latter can detect \hyperref[topic:effective-distance]{hook errors} and the former cannot
 
   code_capacity_threshold:
     - '\(10\%\) under either bit-flip or bit-phase noise for ideal syndrome measurements.

codes/quantum/qudits_galois/stabilizer/qldpc/distance_balanced.yml

@@ -8,7 +8,7 @@ physical: galois
 logical: galois
 
 name: 'Distance-balanced code'
-introduced: '\cite{arxiv:1611.03790,arxiv:2004.07935}'
+introduced: '\cite{arxiv:1611.03790,arxiv:2102.10030,arxiv:2004.07935}'
 
 description: |
   Galois-qudit CSS code constructed from a CSS code and a classical code using a distance-balancing procedure based on a generalized homological product.
@@ -20,6 +20,14 @@ description: |
   A related procedure called \textit{weight reduction} \cite{arxiv:1611.03790,arxiv:2402.05228} takes in a CSS stabilizer code and outputs a longer CSS code that admits a set of stabilizer generators whose weight is independent of the number of qubits \(n\).
   \end{defterm}
 
+features:
+  decoders:
+    - 'The effective distance of single-ancilla syndrome extraction QLDPC code circuits can be preserved under weight reduction \cite{arxiv:2409.02193}. The distance balancing technique of Ref. \cite{arxiv:2004.07935} preserves the \hyperref[topic:effective-distance]{effective distance} of single-ancilla syndrome extraction circuits \cite{arxiv:2409.02193}.'
+  fault_tolerance:
+    - 'Single-ancilla syndrome extraction circuits that, for the most part, preserve the \hyperref[topic:effective-distance]{effective distance} of weight-reduced qLDPC codes \cite{arxiv:2409.02193}. The distance balancing technique of Ref. \cite{arxiv:2004.07935} preserves \hyperref[topic:effective-distance]{effective distance} \cite{arxiv:2409.02193}.'
+
+
+
 relations:
   parents:
     - code_id: galois_css
@@ -31,7 +39,7 @@ relations:
     - code_id: gkp-cluster-state
       detail: '\hyperref[topic:weight-reduction]{Weight reduction} has been studied in the context of GKP CV-cluster-state codes \cite{arxiv:2402.05228}.'
     - code_id: qldpc
-      detail: 'Lattice surgery techniques for QLDPC codes \cite{arxiv:2110.10794,arxiv:2308.08648} utilize \hyperref[topic:weight-reduction]{weight reduction}.'
+      detail: 'Lattice surgery techniques for QLDPC codes \cite{arxiv:2110.10794,arxiv:2308.08648} utilize \hyperref[topic:weight-reduction]{weight reduction}. Single-ancilla syndrome extraction circuits that, for the most part, preserve the \hyperref[topic:effective-distance]{effective distance} of weight-reduced qLDPC codes \cite{arxiv:2409.02193}.'
 
 
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