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valbert4 committed May 6, 2024
1 parent 33e38e1 commit 719d030
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3 changes: 3 additions & 0 deletions codes/quantum/properties/block/single_shot.yml
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Expand Up @@ -56,6 +56,9 @@ relations:
cousins:
- code_id: self_correct
detail: 'The presence of an energy barrier (i.e., confinement) is sufficient for a code to be single shot, and is also conjectured to be necessary for a code to be a self-correcting memory.'
- code_id: hypergraph_product
detail: 'Two-fold application of the hypergraph product to a pair of binary linear codes yields single-shot QLDPC codes that exploit redundancy in their stabilizer generators \cite{arxiv:1805.09271}.'
# Called homological but for us that is between quantum codes


# Begin Entry Meta Information
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Expand Up @@ -97,7 +97,7 @@ relations:
- code_id: galois_quad_residue
detail: 'The Steane code is a qubit quantum QR code \cite{arxiv:0712.0103,arxiv:1907.01393}.'
- code_id: data_syndrome
detail: 'There exists a set of stabilizer generators for the Steane code that make it a DS code \cite{arxiv:1409.2559}.'
detail: 'There exists a set of stabilizer generators for the Steane code that make it a QDS code \cite{arxiv:1409.2559}.'
- code_id: pg_qldpc
detail: 'The Steane code is the \(m=1\) member of the \([[2^{2m}+2^{m}+1,1,>2^{m}]]\) PG-QLDPC code family that is constructed from codes corresponding to lines and affine charts in \(PG(2,2^m)\) via the CSS construction \cite[Def. 4.9]{arxiv:1512.07081}.'
- code_id: group_representation
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28 changes: 14 additions & 14 deletions codes/quantum/qubits/stabilizer/data_syndrome.yml
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Expand Up @@ -7,8 +7,8 @@ code_id: data_syndrome
physical: qubits
logical: qubits

name: 'Quantum data-syndrome (DS) code'
short_name: 'DS code'
name: 'Quantum data-syndrome (QDS) code'
short_name: 'QDS code'
introduced: '\cite{arxiv:1409.2559,doi:10.1109/ISIT.2014.6874892,doi:10.1109/ISIT.2015.7282628,doi:10.1109/ISIT.2016.7541704,arxiv:1907.01393}'

description: |
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protection: |
Protects against both physical qubit and syndrome measurement errors.
Quantum Singleton bounds, quantum Hamming bounds, and \hyperref[topic:quantum-weight-enumerator]{quantum MacWilliams identities} can be extended to DS codes.
Single-error-correcting DS codes stemming from \hyperref[topic:quantum-weight-enumerator]{impure} stabilizer codes must satisfy a variant of the quantum Hamming bound \cite{arxiv:2302.01527}.
Quantum Singleton bounds, quantum Hamming bounds, and \hyperref[topic:quantum-weight-enumerator]{quantum MacWilliams identities} can be extended to QDS codes.
Single-error-correcting QDS codes stemming from \hyperref[topic:quantum-weight-enumerator]{impure} stabilizer codes must satisfy a variant of the quantum Hamming bound \cite{arxiv:2302.01527}.
features:
decoders:
- 'Syndrome errors are decoded using redundant stabilizer measurements.'

fault_tolerance:
- '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 \(\ell\) times, can be recast as a DS code whose underlying matrix \(A\) is the identity matrix \(I_m\) repeated \(\ell\) times \cite{arxiv:1907.01393}.'
- '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 \(\ell\) times, can be recast as a QDS code whose underlying matrix \(A\) is the identity matrix \(I_m\) repeated \(\ell\) times \cite{arxiv:1907.01393}.'


relations:
parents:
- code_id: qubit_stabilizer
detail: 'Quantum DS codes are stabilizer codes whose stabilizer generators encode extra redundancy (via a linear binary code) so as to protect from syndrome measurement errors.'
detail: 'QDS codes are stabilizer codes whose stabilizer generators encode extra redundancy (via a linear binary code) so as to protect from syndrome measurement errors.'
cousins:
- code_id: quantum_hamming_css
detail: 'Codes such as the quantum Hamming code can be expanded to DS codes using almost any good binary linear code because their stabilizer generators all have the same weight \cite{arxiv:1907.01393}.'
detail: 'Codes such as the quantum Hamming code can be expanded to QDS codes using almost any good binary linear code because their stabilizer generators all have the same weight \cite{arxiv:1907.01393}.'
- code_id: quantum_mds
detail: 'The quantum Singleton bound can be extended to DS codes \cite{arxiv:1907.01393}.'
detail: 'The quantum Singleton bound can be extended to QDS codes \cite{arxiv:1907.01393}.'
- code_id: quantum_perfect
detail: 'The quantum Hamming bound can be extended to DS codes \cite{arxiv:1907.01393}.'
detail: 'The quantum Hamming bound can be extended to QDS codes \cite{arxiv:1907.01393}.'
- code_id: binary_linear
detail: 'The DS code construction employs a particular binary linear code to provide protection against syndrome measurement errors.'
detail: 'The QDS code construction employs a particular binary linear code to provide protection against syndrome measurement errors.'
- code_id: quantum_convolutional
detail: 'The DS code framework has been extended to quantum convolutional codes \cite{arxiv:1902.07395}.'
detail: 'The QDS code framework has been extended to quantum convolutional codes \cite{arxiv:1902.07395}.'
- code_id: single_shot
detail: 'DS codes contain redundancy in their stabilizer generators so as to protect from syndrome measurement errors.'
detail: 'QDS codes contain redundancy in their stabilizer generators so as to protect from syndrome measurement errors.'
- code_id: bch
detail: 'Primitive narrow-sense BCH codes can be used as the syndrome measurement codes of a DS code \cite{arxiv:2311.16044}.
detail: 'Primitive narrow-sense BCH codes can be used as the syndrome measurement codes of a QDS code \cite{arxiv:2311.16044}.
This construction requires fewer measurements than a previous general construction \cite{arxiv:1409.2559}.'
- code_id: qubit_subsystem
- code_id: subsystem_stabilizer
detail: 'The DS construction can be extended to qubit subsystem codes \cite{arxiv:2302.01527}.'


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2 changes: 1 addition & 1 deletion codes/quantum/qubits/stabilizer/magic/qubit_golay.yml
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Expand Up @@ -66,7 +66,7 @@ relations:
- code_id: galois_quad_residue
detail: 'The Golay code is a qubit quantum QR code \cite{arxiv:0712.0103,arxiv:1907.01393}.'
- code_id: data_syndrome
detail: 'There exists a set of stabilizer generators for the qubit Golay code that make it a DS code \cite{arxiv:1907.01393}.'
detail: 'There exists a set of stabilizer generators for the qubit Golay code that make it a QDS code \cite{arxiv:1907.01393}.'
cousins:
- code_id: golay
detail: 'The qubit Golay code is a CSS code constructed with the Golay code.'
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cousins:
- code_id: q-ary_quad_residue
- code_id: data_syndrome
detail: 'A family of qubit quantum QR codes can be made into DS codes \cite[Thm. 14]{arxiv:1907.01393}.'
detail: 'A family of qubit quantum QR codes can be made into QDS codes \cite[Thm. 14]{arxiv:1907.01393}.'
- code_id: quantum_mds
detail: 'Almost all quantum QR codes for prime-dimensional qudits are quantum MDS \cite[Corr. 11]{arxiv:quant-ph/9703048}.'

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