Gottesman refs

codes/quantum/oecc.yml

@@ -27,13 +27,15 @@ protection: |
   \Pi E^{\dagger}_a E_b \Pi = I_{\mathsf{A}} \otimes g_{ab}^{\mathsf{B}}
   \end{align}
   where \(\Pi\) is a projector onto the codespace \(\mathsf{C}\), and \(g_{ab}^{\mathsf{B}}\) is an arbitrary operator on the gauge subsystem.
-  These have also been studied in the presence of continuous noise \cite{arxiv:0806.3145}.
+  These have been studied in the presence of continuous noise \cite{arxiv:0806.3145}.
 
   A \textit{unitarily correctable subsystem} is a subsystem code whose encoded information can be recovered via a unitary, i.e., in a measurement-free way \cite{arxiv:quant-ph/0608045} (see also \cite{arxiv:quant-ph/9609015}). For unital noise channels, such codes are related to the multiplicative domain of the channel \cite{arxiv:0811.0947}.
 
 features:
   encoders:
     - 'Subsystem QECCs are robust to initialization errors \cite{arxiv:0709.3533}.'
+  decoders:
+    - 'Petz recovery map is shown to be near-optimal for certain subsystem codes \cite{arxiv:1202.5139}.'
 
 realizations:
   - 'A two-qubit unitarily correctable subsystem code recovery has been realized in an optical system \cite{arxiv:0909.1584}.'

codes/quantum/properties/approximate_qecc.yml

@@ -160,7 +160,7 @@ features:
     - 'The \textit{Cafaro recovery map} \cite{arxiv:1308.4582} can be obtained for noise Kraus operators if there exists a basis of error words with respect to which the uncorrectable piece in the Knill-Laflamme conditions is diagonal; see Ref. \cite{arxiv:2406.02444}.
     The map recovers information perfectly for strictly correctable noise.'
     - 'The \textit{Petz recovery map} a.k.a. the \textit{transpose map} \cite{doi:10.1007/BF01212345,doi:10.1093/qmath/39.1.97,arxiv:1810.03150}, a quantum channel determined by the codespace and noise channel, yields an infidelity of recovery that is at most twice away from the infidelity of the best possible recovery \cite{arxiv:quant-ph/0004088}. 
-    The fidelity can be expressed exactly as a function of the \term{Knill-Laflamme conditions} \cite[Thm. 1]{arxiv:2401.02022}, and it can be used to derive a generalization of the \term{Knill-Laflamme conditions} for approximate QECCs \cite{arxiv:0909.0931}.
+    The fidelity can be expressed exactly as a function of the \term{Knill-Laflamme conditions} \cite[Thm. 1]{arxiv:2401.02022}, and it can be used to derive a generalization of the \term{Knill-Laflamme conditions} for approximate QECCs \cite{arxiv:0909.0931,arxiv:1202.5139}.
     Satisfaction of the \term{Knill-Laflamme conditions} is sufficient but not necessary for the Petz recovery map to be the optimal recovery, and a necessary and sufficient condition has been derived \cite{arxiv:2410.23622}.
     The infidelity of a modified Petz recovery map under erasure can be bounded using the conditional mutual information via the \textit{approximate Petz theorem} \cite{arxiv:1410.0664,arxiv:1509.07127,arxiv:1610.06169}.
     In the case of topological codes, the Petz infidelity is related to the topological entanglement entropy \cite{arxiv:2408.00857}.

codes/quantum/properties/qecc.yml

@@ -63,7 +63,7 @@ notes:
   - 'See Refs. \cite{arxiv:quant-ph/9712048,arxiv:quant-ph/0004072,doi:10.1090/gsm/047,doi:10.1017/CBO9780511976667,arxiv:quant-ph/0507174,arxiv:quant-ph/0612185,arxiv:0904.2557,arxiv:0905.2794,arxiv:1302.3428,doi:10.1103/RevModPhys.88.041001,doi:10.1002/9783527805785.ch1,arxiv:1508.03695,arxiv:1907.11157,preset:PreskillNotes,arxiv:1910.03672,doi:10.1002/9781119790327.ch10,arxiv:2407.12737} for overviews of quantum error correction.'
   - 'See Refs. \cite{doi:10.1017/CBO9781139034807,doi:10.1201/b15868,preset:GottesmanBook} for books on quantum error correction.'
   - 'See video tutorials by \href{https://www.youtube.com/watch?v=_ls3KczZL2c}{V. V. Albert}, \href{https://www.youtube.com/watch?v=uD69GCYF9Zg}{S. M. Girvin}, \href{https://www.youtube.com/watch?v=buIbd_aXAHw}{P. Shor}, \href{https://www.youtube.com/watch?v=Je7sVJGKMgU}{B. Terhal}, and \href{https://www.youtube.com/watch?v=mcwpe8iJ5uo}{J. Wright}.'
-  - 'Quantum error correction was initially claimed not to be theoretically possible \cite{arxiv:hep-th/9406058,doi:10.1098/rsta.1995.0106}.'
+  - 'Quantum error correction was initially claimed not to be theoretically possible \cite{arxiv:hep-th/9406058,doi:10.1098/rsta.1995.0106} and has been criticized since \cite{arxiv:1310.8457}.'
   - 'Resource-theoretic interpretations of quantum error correction have been developed, including those that think of codes together with recovery operations as superchannels (a.k.a. quantum combs or bipartite operations) \cite{arxiv:1105.4464,arxiv:1210.4722,arxiv:1406.7142,arxiv:2405.17567,arxiv:2409.09416}.'
 
 

codes/quantum/qubits/ea_stabilizer/eastab.yml

@@ -20,6 +20,7 @@ description: |
   An \([[n,k+e;e]]\) EA stabilizer code can be constructed from an ordinary \([[n,k]]\) stabilizer code with check matrix \(H=(A|B)\), where the required number of ebits is \(e = \text{rank}(AB^T+BA^T)\) \cite{arxiv:0804.1404}.
 
 protection: |
+  Ancillary shared entanglement is assumed to be perfect, but this assumption can be relaxed \cite{arxiv:1302.5081}.
   There are quantum Griesmer \cite{doi:10.1007/s11128-015-1143-5} and Plotkin \cite{doi:10.1103/PhysRevA.87.032309} bounds for EA qubit stabilizer codes.
 
 features:

codes/quantum/qubits/small_distance/small/5/stab_5_1_3.yml

@@ -39,6 +39,7 @@ features:
   encoders:
     - 'Nine single- and two-qubit unitaries, six of which are CNOT gates \cite{arxiv:quant-ph/0410004}.'
     - 'Four generalized control gates, four Hadamard, and one \(Z\) gate \cite[Fig. 10.16]{doi:10.1201/9781420012293}.'
+    - 'Evolution under stabilizer Hamiltonian \cite{arxiv:1301.4796}.'
     - 'Four CNOT and five CPHASE gates \cite{arxiv:1509.01239}.'
     - 'Reinforcement learning encoding circuits \cite{arxiv:2402.17761}.'
     - 'Fault-tolerant logical one and logical minus state preparation in all-to-all and 2D grid connectivity \cite{arxiv:2402.17761}.'

codes/quantum/qubits/small_distance/small/7/steane/steane.yml

@@ -58,13 +58,15 @@ protection: 'The Steane code is a distance 3 code. It detects errors on 2 qubits
 features:
   encoders:
     - 'Nine CNOT and four Hadamard gates (\cite{doi:10.1201/9781420012293}, Fig. 10.14).'
+    - 'Evolution under stabilizer Hamiltonian \cite{arxiv:1301.4796}.'
     - 'Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity \cite{arxiv:2402.17761}.'
 
   transversal_gates:
     - 'The \hyperref[topic:clifford]{single-qubit Clifford group} \cite{arxiv:quant-ph/9605011,arxiv:0706.1382}.'
 
   general_gates:
     - 'Fault-tolerant approximations of arbitrary single-qubit gates \cite{arxiv:quant-ph/0411206,arxiv:quant-ph/0506126}.'
+    - 'Non-fault-tolerant \(T\) gate \cite{arxiv:1303.4291}.'
     - 'Fault-tolerant logical zero and magic state preparation \cite{doi:10.1038/srep19578}. Magic-state preparation converts unbiased noise into biased noise \cite{arxiv:2401.10982}.'
     - 'Pieceable fault-tolerant CCZ gate \cite{arxiv:1603.03948}.'
 

codes/quantum/qubits/stabilizer/mbqc/cluster_state.yml

@@ -90,7 +90,7 @@ features:
 # So information is effectively stored near the boundary of the 2D slice. Measuring slice by slice will teleport it to the next slice.
 
 realizations:
-  - 'Quantum computation with cluster states has been realized in the polarizations of photons \cite{arxiv:quant-ph/0503126,arxiv:0906.2233}.'
+  - 'Polarizations of photons: quantum computation \cite{arxiv:quant-ph/0503126,arxiv:0906.2233} and single-qubit error correction on an 8-qubit cluster state \cite{arxiv:1202.5459}.'
 
 notes:
   - 'See Refs. \cite{arxiv:quant-ph/0602096,doi:10.1002/9783527635283} for a review of cluster states and their applications.'

codes/quantum/qubits/stabilizer/mbqc/rbh.yml

@@ -44,6 +44,7 @@ features:
     - '\(24.9\%\) under erasure noise \cite{arxiv:1005.2456}.'
     - 'Concatenation of the RBH code with small codes such as the \([[2,1,1]]\) repetition code, \([[4,1,1,2]]\) subsystem code, or Steane code can improve thresholds \cite{arxiv:2209.09390}.'
 
+
 notes:
   - 'Introduction to MBQC protocols with the RBH state \cite{arxiv:1504.01444}.'
 

codes/quantum/qubits/stabilizer/qubit_css.yml

@@ -159,7 +159,7 @@ realizations:
 
 notes:
   - 'See Refs. \cite{arxiv:quant-ph/9605021,doi:10.1017/CBO9780511976667,preset:PreskillNotes,preset:GottesmanBook} for simple examples of CSS codes.'
-  - 'Introduction to \ref{topic:CSS-to-homology-correspondence} by \href{https://www.youtube.com/watch?v=SeLpWg_8qlc}{M. Hastings}; see also the book \cite{arxiv:1504.01444}.'
+  - 'Introduction to \ref{topic:CSS-to-homology-correspondence} by \href{https://www.youtube.com/watch?v=SeLpWg_8qlc}{M. Hastings}; see also Refs. \cite{arxiv:1310.5376,arxiv:1504.01444,manual:{D. Browne, \href{https://sites.google.com/site/danbrowneucl/teaching/lectures-on-topological-codes-and-quantum-computation}{Lecture notes}}}.'
   - 'Entanglement purification protocols with qubit CSS codes are related to quantum key distribution (QKD) \cite{arxiv:quant-ph/0003004}.'
   - 'Qubit CSS codes can be used in quantum repeaters \cite{arxiv:0809.3629}.'
 

codes/quantum/qubits/stabilizer/qubit_stabilizer.yml

@@ -98,6 +98,7 @@ description: |
 
   Qubit stabilizer states can be expressed in terms of linear and quadratic functions over \(\mathbb{Z}_2^n\) \cite{arxiv:quant-ph/0304125,arxiv:quant-ph/0408190,arxiv:0811.0898}.
   There are efficient ways to compute their inner products and other functions \cite{arxiv:1711.07848}.
+  The overlap between a stabilizer state and any \(n\)-qubit product state is at most \(2/2^d\) \cite[Thm. 2]{arxiv:2405.01332}.
   Alternative representations include the \textit{decoupling representation}, in which Pauli strings are represented as vectors over \(GF(2)\) using three bits \cite{arxiv:2305.17505}.
 
 
@@ -227,7 +228,6 @@ notes:
   - 'Tables of bounds and examples of stabilizer codes for various \(n\) and \(k\), based on algorithms developed in Ref. \cite{doi:10.1007/978-3-540-37634-7_13}, are maintained by M. Grassl at this \href{https://codetables.markus-grassl.de/}{website}. A Magma implementation exists at this \href{https://magma.maths.usyd.edu.au/magma/handbook/text/1976}{website}.'
   - 'See Quantum Codes qubit stabilizer database, maintained by N. Aydin, P. Liu, and B. Yoshino \cite{arxiv:2106.12065,arxiv:2108.03567}, at this \href{https://quantumcodes.info/}{website}.'
   - 'Entanglement purification protocols with qubit stabilizer codes are related to quantum key distribution (QKD) \cite{arxiv:quant-ph/0209091}. There is a correspondence between stabilizer codes and bilocal Clifford entanglement distillation circuits \cite{arxiv:2303.11465}. Purification protocols using two-way classical channels can exceed the quantum Hamming and quantum Singleton bounds \cite{arxiv:quant-ph/0310097}.'
-  - 'The overlap between any stabilizer codeword and any \(n\)-qubit product state is at most \(2/2^d\) \cite[Thm. 2]{arxiv:2405.01332}.'
   - 'Qubit stabilizer codes can be used to estimate physical Pauli noise up to their \hyperref[topic:quantum-weight-enumerator]{pure distance} \cite{arxiv:2107.14252}, and logical Pauli noise for any correctable physical noise \cite{arxiv:2209.09267}.'
   - 'The stabilizer formalism has been gamified \cite{arxiv:2405.06795}.'
   - 'Codes can be found via genetic algorithms \cite{arxiv:2409.13017}.'

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

@@ -101,13 +101,13 @@ features:
     - '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 what is known as 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}, with the scheme in Ref. \cite{arxiv:2206.12780} being a special case of DWR.'
     - '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 \(\beta = \log_3 2\).'
-    - 'Minimum weight perfect-matching (MWPM) \cite{arxiv:quant-ph/0110143,arxiv:1307.1740} (based on work by Edmonds on finding a matching in a graph \cite{doi:10.4153/CJM-1965-045-4,doi:10.6028/jres.069B.013}), which takes time up to polynomial in \(n\) for the surface code.
+    - 'Minimum weight perfect-matching (MWPM) \cite{arxiv:quant-ph/0110143,arxiv:1202.5602,arxiv:1307.1740} (based on work by Edmonds on finding a matching in a graph \cite{doi:10.4153/CJM-1965-045-4,doi:10.6028/jres.069B.013}), which takes time up to polynomial in \(n\) for the surface code.
     For the case of the surface code, minimum-weight decoding reduces to MWPM \cite{arxiv:quant-ph/0110143,doi:10.4153/CJM-1965-045-4,doi:10.1088/0305-4470/15/2/033}. MWPM solves the MPE decoding problem exactly for independent \(X\) and \(Z\) noise. MPE decoding is \(NP\)-hard in general for the surface code \cite{arxiv:2309.10331}. PyMatching is a Python software library for implementing MWPM \cite{arxiv:2105.13082}.'
     - 'Bravyi-Suchara-Vargo (BSV) tensor network decoder \cite{arxiv:1405.4883} approximately solves the ML decoding problem under independent \(X,Z\) noise for the surface code and takes time of \hyperref[topic:asymptotics]{order} \(O(n^2)\) \cite{arxiv:1405.4883}.
     ML decoding \cite{arxiv:quant-ph/0110143} is \(\#P\)-hard in general for the surface code \cite{arxiv:2309.10331}.'
     - 'Union-find decoder \cite{arxiv:1709.06218} uses the \textit{union-find data structure} \cite{doi:10.1145/364099.364331,doi:10.1137/0202024,doi:10.1145/62.2160}, solving the MPE decoding problem exactly for low-weight errors under depolarizing noise. A subsequent modification utilizes the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) \cite{arxiv:2107.13589}. Belief union find is a combination of belief-propagation and union-find \cite{arxiv:2203.04948}.
     Strictly local (as opposed to partially local) union find \cite{arxiv:2305.18534} has a worst-case runtime of \hyperref[topic:asymptotics]{order} \(O(d^3)\) in the distance \(d\).'
-    - 'Modified MWPM decoders: pipeline MWPM (accounting for correlations between events) \cite{arxiv:1310.0863,arxiv:2205.09828}; modification tailored to asymmetric noise \cite{arxiv:1812.01505}; parity blossom MWPM and fusion blossom MWPM \cite{arxiv:2305.08307}, a modification utilizing the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) \cite{arxiv:2107.13589}; belief perfect matching (a combination of belief-propagation and MWPM) \cite{arxiv:2203.04948}; spanning tree matching (STM) and rapid-fire (RFire) decoders \cite{arxiv:2405.01151}; ordered decoding based on MWPM \cite{arxiv:2408.01393}. Combining, or \textit{harmonizing}, various decoders can improve performance \cite{arxiv:2401.12434}. One such example is the Libra decoder \cite{arxiv:2408.12135}, a combination of MWPM decoders combined with matching synthesis.'
+    - 'Modified MWPM decoders: topological code Autotune \cite{arxiv:1202.6111}; pipeline MWPM (accounting for correlations between events) \cite{arxiv:1310.0863,arxiv:2205.09828}; modification tailored to asymmetric noise \cite{arxiv:1812.01505}; parity blossom MWPM and fusion blossom MWPM \cite{arxiv:2305.08307}, a modification utilizing the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) \cite{arxiv:2107.13589}; belief perfect matching (a combination of belief-propagation and MWPM) \cite{arxiv:2203.04948}; spanning tree matching (STM) and rapid-fire (RFire) decoders \cite{arxiv:2405.01151}; ordered decoding based on MWPM \cite{arxiv:2408.01393}. Combining, or \textit{harmonizing}, various decoders can improve performance \cite{arxiv:2401.12434}. One such example is the Libra decoder \cite{arxiv:2408.12135}, a combination of MWPM decoders combined with matching synthesis.'
     - 'Renormalization group (RG) \cite{arxiv:0911.0581,arxiv:1304.6100,arxiv:1411.3028}; see Ref. \cite{arxiv:1310.2393} for the planar surface code.'
     - 'Linear-time ML erasure decoder \cite{arxiv:1703.01517}.'
     - 'Linear-time decoder for general noise, including coherent noise and correlated noise \cite{arxiv:1801.01879}.'

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

@@ -55,7 +55,7 @@ features:
     The threshold under ML decoding corresponds to the value of a critical point of the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) \cite{arxiv:1202.1852} (see also APS Physics viewpoint \cite{doi:10.1103/Physics.5.50}).'
     - 'Erasure noise: \(50\%\) for square tiling \cite{arxiv:0904.3556,arxiv:0912.1159}.
     There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition \cite{arxiv:1810.09621}.'
-    - 'Correlated noise: \(10.04(6)\%\) under mildly correlated bit-flip noise \cite{arxiv:1809.10704}.'
+    - 'Correlated noise: the threshold under ML decoding corresponds to the value of a critical point of a particular random-bond Ising model (RBIM) \cite{arxiv:1209.2157,arxiv:1304.2975}. A threshold of \(10.04(6)\%\) under mildly correlated bit-flip noise is obtained in Ref. \cite{arxiv:1809.10704}.'
     # The latter work considers \(X\)-type noise only, but this is equivalent since \(X\)- and \(Z\)-type noise is corrected independently.
     - 'The toric code has a \hyperref[topic:measurement-threshold]{measurement threshold} of one \cite{arxiv:2402.00145}.'
     - 'Coherent noise: the threshold under ML decoding corresponds to the value of a critical point of a particular random-bond Ising model (RBIM) called the complex-coupled Ashkin-Teller model \cite{arxiv:2410.22436,arxiv:2411.05785}. Another statistical mechanical mapping has been studied for \(X\)-type noise channels interpolating between coherent and incoherent noise \cite{arxiv:2412.21055}.'

codes/quantum/qudits_galois/nonstabilizer/galois_non_stabilizer.yml

@@ -8,7 +8,7 @@ physical: galois
 logical: galois
 
 name: 'Galois-qudit USt code'
-introduced: '\cite{arxiv:quant-ph/9703002,arxiv:quant-ph/9703016,arxiv:quant-ph/9710031,arxiv:quant-ph/0210097,arxiv:0801.2144}'
+introduced: '\cite{arxiv:quant-ph/9703002,arxiv:quant-ph/9703016,arxiv:quant-ph/9710031,arxiv:quant-ph/0210097,arxiv:0801.2144,arxiv:1208.4907}'
 # First ref discusses unions of general codes
 
 alternative_names: