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valbert4 committed Jan 3, 2025
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1 change: 1 addition & 0 deletions codes/quantum/properties/block/ampdamp.yml
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rate: 'The quantum capacity of the \hyperref[topic:ad]{AD} channel is \(\max\{0, \log \frac{1-\gamma}{\gamma}\} \) \cite{arxiv:quant-ph/0606132}.
Quantum capacities of the qubit \hyperref[topic:ad]{AD} channel are also determined \cite{arXiv:quant-ph/0405110,arXiv:1309.2219}, including of channels with memory \cite{arXiv:1207.5612,arXiv:1510.05313}. Capacities of qudit extensions have also been studied \cite{arxiv:2008.00477}.'


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- code_id: block_quantum
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notes:
- 'See Ref. \cite{arxiv:1708.06298} and corresponding \href{https://tp.nt.uni-siegen.de/ame/ame.html}{Table of AME states}.'
- 'Perfect tensors are useful for quantum secret sharing \cite{arxiv:1204.2289}.'


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3 changes: 2 additions & 1 deletion codes/quantum/qubits/qubit_concatenated.yml
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fault_tolerance:
- 'Fault-tolerant message passing between devices \cite{arxiv:2408.05260}.'
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 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}.'
- |
The first methods to achieve a \hyperref[topic:computational-threshold]{concatenated threshold} against local stochastic noise 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}.
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22 changes: 13 additions & 9 deletions codes/quantum/qubits/qubits_into_qubits.yml
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Semidefinite programming (SDP) hierarchies and a quantum Delsarte bound have been developed \cite{arxiv:2408.10323}.
features:
rate: 'Exact two-way assisted capacities have been obtained for the erasure and dephasing channels \cite{arxiv:1510.08863}.'
rate: 'Exact two-way assisted capacities have been obtained for the erasure and dephasing channels \cite{arxiv:1510.08863}. There are many bounds on the quantum capacity of the depolarizing channel (e.g., \cite{arxiv:quant-ph/0607039}); see review \cite{arxiv:1801.02019}.'
transversal_gates:
- 'A qubit code is \(U\)-\textit{quasi-transversal} if it can realize the logical gate \(U\) in the third level of the \term{Clifford hierarchy} using the physical gate \(C T^{\otimes n}\), where \(C\) is some Clifford gate \cite[Def. 4]{arxiv:1606.01904}.'
general_gates:
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- '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}.'
- 'Fault-tolerant gates can be done for any code supporting a transversal implementation of Pauli gates using generalized gate teleportation \cite{arxiv:2409.11616}.'
threshold:
- '\begin{defterm}{Computational threshold}
\label{topic:computational-threshold}
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}{Computational threshold}
\label{topic:computational-threshold}
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}.
Initially proven under local stochastic noise, the concatenated threshold theorem also holds for various types of non-Markovian noise \cite{arxiv:quant-ph/0402104,arxiv:quant-ph/0504218,arxiv:quant-ph/0510231}.
The resulting concatenated code is highly \hyperref[topic:degeneracy]{degenerate}, with all but an exponentially small fraction of generators having small weights.
Circuit and measurement designs have to take care of the few stabilizer generators with large weights in order to be fault tolerant.
Concatenated methods require constant-space and polylogarithmic-time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time \cite{arxiv:2207.08826,arxiv:2402.09606}, and concatenations of the Steane code and certain QLDPC codes further improve this to polylogarithmic time \cite{arxiv:2411.03683}.
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}
One can derive conditions quantifying how many random single-qubit measurements can be made without destroying the logical information \cite{arxiv:2402.00145}.
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6 changes: 6 additions & 0 deletions codes/quantum/qubits/small_distance/iceberg.yml
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Admits a basis such that each codeword is a superposition of a computational basis state labeled by an even-weight bitstring \(b\) and a state labeled by the negation of \(b\).
Its all-zero logical state is a conventional GHZ state.
Removing the \(Z\)-type generator expands the number of codewords to include superpositions of odd-weight bitstrings and their negations.
All of its automorphisms lie in the \hyperref[topic:clifford]{Clifford group} \cite[Thm. 13]{arxiv:quant-ph/9704043}.
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- 'Universal set of gates, each of which is supported on two qubits \cite{arxiv:2211.06703}.'
- 'Fault-tolerant Clifford Trotter circuits that are linear in \(k\) using flag qubits via a solve-and-stitch algorithm and application of a logical identity circuit \cite{arxiv:2404.11953}.'

decoders:
- 'The \([[2m,2m-2,2]]\) error-detecting code \cite{arxiv:quant-ph/0402067} and its relative the code with single stabilizer \(XX\cdots X\) \cite{arxiv:quant-ph/0302006} admit continuous-time QEC against single \hyperref[topic:ad]{AD} errors.'

fault_tolerance:
- 'Logical SWAP gates can be performed fault tolerantly using an ancilla qubit \cite[Sec. VII]{arxiv:quant-ph/9702029}.'
- 'Two-qubit fault-tolerant state preparation, error detection and projective measurements \cite{arxiv:1705.02329} (see also \cite{arxiv:2211.06703}).'
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detail: 'The \([[2m,2m-2,2]]\) error-detecting code is constructed via the CSS construction from an SPC code and its dual repetition code \cite[Sec. III]{arxiv:1803.06987}.'
- code_id: 4612_color
detail: 'The \([[2m,2m-2,2]]\) error-detecting code for \(m=4\) is a color code defined on a single octagon of the 6.6.6 or 4.6.12 tilings.'
- code_id: ampdamp
detail: 'The \([[2m,2m-2,2]]\) error-detecting code \cite{arxiv:quant-ph/0402067} and its relative the code with single stabilizer \(XX\cdots X\) \cite{arxiv:quant-ph/0302006} admit continuous-time QEC against single \hyperref[topic:ad]{AD} errors.'


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- 'Encoders of codes with polynomial distance encoders yield catastrophic errors, but code with bounded distance admit non-catastrophic encoders.'
decoders:
- 'Turbo decoder \cite[Sec. V]{arxiv:0712.2888}.'
- 'Modified decoder yield improvement over the memoryless depolarizing channel \cite{arxiv:1010.1256}.'
- 'Modified decoder yields improvement over the memoryless depolarizing channel \cite{arxiv:1010.1256}.'

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11 changes: 3 additions & 8 deletions codes/quantum/qubits/stabilizer/qubit_stabilizer.yml
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- 'Degenerate erasure decoder showing near ML decoding for various codes \cite{arxiv:2411.13509}.'

fault_tolerance:
- 'Gates in the \term{Clifford hierarchy} can be done using \textit{gate teleportation}, in which a gate can be obtained from a particular \textit{magic state} \cite{arxiv:quant-ph/9908010,arxiv:quant-ph/0002039}.
Such protocols can be made fault tolerant with the help of magic-state distillation \cite{arxiv:quant-ph/0403025}.'
- 'Gates in the \term{Clifford hierarchy} can be done using \textit{gate teleportation}, in which a gate can be obtained from a particular \textit{magic state} \cite{arxiv:quant-ph/9908010,arxiv:quant-ph/0002039}. Such protocols can be made fault tolerant with the help of magic-state distillation \cite{arxiv:quant-ph/0403025}. See review on magic-state distillation \cite{doi:10.1002/qua.24856}.'
- 'Logical Bell measurements can be done transversally, and thus fault tolerantly, by performing bitwise Bell measurements for each pair of qubits (with each member of the pair taken from one of the two code blocks) and processing the result.'
- 'With pieceable fault-tolerance, any \hyperref[topic:degeneracy]{non-degenerate} stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of non-transversal fault-tolerant gates \cite{arxiv:1603.03948}.'
- 'Shor error correction \cite{arxiv:quant-ph/9605011,arxiv:quant-ph/9605031}, in which fault tolerance against syndrome extraction errors is ensured by simply repeating syndrome measurements. A modification uses adaptive measurements \cite{arxiv:2208.05601}.'
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- '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:quant-ph/0110143,arxiv:1208.2317,arxiv:1311.7688,arxiv:1809.10704}. The AQEC relative entropy is related to the resulting threshold \cite{arxiv:2312.16991}.'

threshold:
- 'Computational thresholds against stochastic local noise can be achieved through repeated use of concatenatenation, and can rely on the same small code in every level \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9906129,arxiv:quant-ph/9705031,arxiv:quant-ph/0504218}. The resulting code is highly \hyperref[topic:degeneracy]{degenerate}, with all but an exponentially small fraction of generators having small weights.
Circuit and measurement designs have to take case of the few stabilizer generators with large weights in order to be fault tolerant.'
- 'Entanglement purification protocols with qubit stabilizer codes are related to quantum key distribution (QKD) \cite{arxiv:quant-ph/0209091}.'
- 'Certain operations can be implemented in a fault-tolerant version \cite{arxiv:0806.0875,arxiv:0904.2143} of holonomic quantum computation \cite{arxiv:quant-ph/9904011}.'

# and various \textit{gadgets} (encoders, gates, measurements, decoders, etc.)
Expand All @@ -208,11 +204,10 @@ notes:
- 'Introductions to stabilizer codes can be found in \cite{arxiv:quant-ph/9705052,preset:PreskillNotes,doi:10.1002/9783527618637.ch1}.'
- '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}.'
- 'Review on magic state distillation \cite{doi:10.1002/qua.24856}.'
- 'There is a correspondence between stabilizer codes and bilocal Clifford entanglement distillation circuits \cite{arxiv:2303.11465}.'
- '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}.'
- 'The overlap between any stabilizer codeword and any \(n\)-qubit product state is at most \(2/2^d\) \cite[Thm. 2]{arxiv:2405.01332}.'
- 'The stabilizer formalism has been gamified \cite{arxiv:2405.06795}.'
- 'Code can be found via genetic algorithms \cite{arxiv:2409.13017}.'
- 'Codes can be found via genetic algorithms \cite{arxiv:2409.13017}.'


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14 changes: 7 additions & 7 deletions codes/quantum/qubits/stabilizer/rm/quantum_hamming_css.yml
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- 'Efficient decoder \cite{arxiv:2207.08826}.'
fault_tolerance:
- '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 \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads \cite{arxiv:2207.08826}.'
- 'Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads \cite{arxiv:2207.08826,arxiv:2402.09606}.'
threshold:
- '\hyperref[topic:measurement-threshold]{Concatenated threshold} requiring constant-space and quasi-polylogarithmic time overhead \cite{arxiv:2207.08826}.'

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detail: 'Quantum Hamming codes result from applying the CSS construction to Hamming codes and their duals the simplex codes.'
- code_id: qubit_concatenated
detail: |
Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads \cite{arxiv:2207.08826}.
Quantum Hamming codes can also be concatened with surface codes \cite{arxiv:2407.16176}.
Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads \cite{arxiv:2207.08826,arxiv:2402.09606}.
Quantum Hamming codes can also be concatenated with surface codes \cite{arxiv:2407.16176}.
- code_id: stab_4_2_2
detail: 'Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads \cite{arxiv:2207.08826}.'
detail: 'Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads \cite{arxiv:2207.08826,arxiv:2402.09606}.'
- code_id: stab_6_2_2
detail: 'Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads \cite{arxiv:2207.08826}.'
detail: 'Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads \cite{arxiv:2207.08826,arxiv:2402.09606}.'
- code_id: surface
detail: 'Quantum Hamming codes can be concatened with surface codes \cite{arxiv:2407.16176}.'
detail: 'Quantum Hamming codes can be concatenated with surface codes \cite{arxiv:2407.16176}.'
# ### PhF -- merged with above qubit_concatenated cousin relationship. Remove following if that is ok.
#- code_id: qubit_concatenated
# detail: 'Quantum Hamming codes can be concatened with surface codes \cite{arxiv:2407.16176}.'
# detail: 'Quantum Hamming codes can be concatenated with surface codes \cite{arxiv:2407.16176}.'


# Begin Entry Meta Information
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This process can be viewed as an ungauging \cite{arxiv:1202.3120,arxiv:1605.01640,arxiv:1806.08679,arxiv:1806.08679} of certain symmetries.
cousins:
- code_id: surface
detail: 'The 2D color code is equivalent to multiple decoupled copies of the 2D surface code via a local constant-depth Clifford circuit \cite{arxiv:1007.4601,arxiv:1503.02065,arxiv:1804.00866}.
This process can be viewed as an ungauging \cite{arxiv:1202.3120,arxiv:1605.01640,arxiv:1806.08679,arxiv:1806.08679} of certain symmetries.
detail: 'The 2D color code is equivalent to multiple decoupled copies of the 2D surface code via a local constant-depth Clifford circuit \cite{arxiv:1007.4601,arxiv:1503.02065,arxiv:1804.00866} and has the same topological entanglement entropy \cite{arxiv:0809.4276}.
The convertion process can be viewed as an ungauging \cite{arxiv:1202.3120,arxiv:1605.01640,arxiv:1806.08679,arxiv:1806.08679} of certain symmetries.
Conversely, the 2D color code can \hyperref[topic:code-switching]{condense} to form the 2D surface code in nine different ways, i.e., by adding two body hopping terms along one of its three triangular directions to the stabilizer group and then taking the center of the resulting nonabelian group \cite{arxiv:2212.00042}.
Both the surface and 2D color codes can be constructed from two distinct types of lattices, namely, 4-valent and 3-valent 3-colorable lattices, respectively \cite{arxiv:1107.3502}.'
- code_id: 3d_color
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