diff --git a/codes/quantum/oecc.yml b/codes/quantum/oecc.yml index c9147577a..e9410a9a1 100644 --- a/codes/quantum/oecc.yml +++ b/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}.' diff --git a/codes/quantum/properties/approximate_qecc.yml b/codes/quantum/properties/approximate_qecc.yml index c321177f1..d45c2e64c 100644 --- a/codes/quantum/properties/approximate_qecc.yml +++ b/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}. diff --git a/codes/quantum/properties/qecc.yml b/codes/quantum/properties/qecc.yml index 1531d76be..6522a345f 100644 --- a/codes/quantum/properties/qecc.yml +++ b/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}.' diff --git a/codes/quantum/qubits/ea_stabilizer/eastab.yml b/codes/quantum/qubits/ea_stabilizer/eastab.yml index 30e59a6d2..50c9a70ae 100644 --- a/codes/quantum/qubits/ea_stabilizer/eastab.yml +++ b/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: diff --git a/codes/quantum/qubits/small_distance/small/5/stab_5_1_3.yml b/codes/quantum/qubits/small_distance/small/5/stab_5_1_3.yml index 957622b71..2efad557f 100644 --- a/codes/quantum/qubits/small_distance/small/5/stab_5_1_3.yml +++ b/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}.' diff --git a/codes/quantum/qubits/small_distance/small/7/steane/steane.yml b/codes/quantum/qubits/small_distance/small/7/steane/steane.yml index 3015443f6..fcb028e3c 100644 --- a/codes/quantum/qubits/small_distance/small/7/steane/steane.yml +++ b/codes/quantum/qubits/small_distance/small/7/steane/steane.yml @@ -58,6 +58,7 @@ 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: @@ -65,6 +66,7 @@ features: 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}.' diff --git a/codes/quantum/qubits/stabilizer/mbqc/cluster_state.yml b/codes/quantum/qubits/stabilizer/mbqc/cluster_state.yml index b5d6f3b76..6bc7a8fd9 100644 --- a/codes/quantum/qubits/stabilizer/mbqc/cluster_state.yml +++ b/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.' diff --git a/codes/quantum/qubits/stabilizer/mbqc/rbh.yml b/codes/quantum/qubits/stabilizer/mbqc/rbh.yml index 57224604a..f82bdca2a 100644 --- a/codes/quantum/qubits/stabilizer/mbqc/rbh.yml +++ b/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}.' diff --git a/codes/quantum/qubits/stabilizer/qubit_css.yml b/codes/quantum/qubits/stabilizer/qubit_css.yml index 5b819ae88..5e1bd6537 100644 --- a/codes/quantum/qubits/stabilizer/qubit_css.yml +++ b/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}.' diff --git a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml b/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml index 4673c5ce1..dba9e872f 100644 --- a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml +++ b/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}.' diff --git a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface.yml b/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface.yml index 5d18e208e..a0cf44926 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface.yml +++ b/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}.' diff --git a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/toric/toric.yml b/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/toric/toric.yml index ef1804b10..31caa2c84 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/toric/toric.yml +++ b/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}.' diff --git a/codes/quantum/qudits_galois/nonstabilizer/galois_non_stabilizer.yml b/codes/quantum/qudits_galois/nonstabilizer/galois_non_stabilizer.yml index e2a75b9be..f5e9440ab 100644 --- a/codes/quantum/qudits_galois/nonstabilizer/galois_non_stabilizer.yml +++ b/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: