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Copy file name to clipboardExpand all lines: codes/quantum/qubits/small_distance/small/7/steane/steane.yml
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- 'Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity \cite{arxiv:2402.17761}.'
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- 'Parity-check encoding with flag-bridge qubits on a square lattice connectivity \cite{arxiv:2504.01083}.'
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transversal_gates:
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- 'The \hyperref[topic:clifford]{single-qubit Clifford group} \cite{arxiv:quant-ph/9605011,arxiv:0706.1382}.'
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- 'The \hyperref[topic:clifford]{single-qubit Clifford group} \cite{arxiv:quant-ph/9605011,arxiv:0706.1382}. More generally, \(k\) copies of the Steane code form a \([[7k,k,3]]\) code that admits a \(k\)-fold transversal implementation of the full \hyperref[topic:clifford]{Clifford group} on all \(k\) logical qubits, showing the tightness of a no-go theorem that requires at least \(k\)-fold transversal gadgets for the full Clifford group \cite{arxiv:2602.13395}.'
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general_gates:
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- 'Fault-tolerant approximations of arbitrary single-qubit gates \cite{arxiv:quant-ph/0411206,arxiv:quant-ph/0506126}.'
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- 'A fault-tolerant universal gate set can be done via \hyperref[topic:code-switching]{code switching} between the Steane code and the \([[15,1,3]]\) code \cite{arxiv:1509.03239,arxiv:1304.3709,arxiv:1403.2734,arxiv:1703.03860,arxiv:2210.14074}.'
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- 'A fault-tolerant universal gate set can be done via \hyperref[topic:code-switching]{code switching} between the Steane code and the \([[10,1,2]]\) code \cite{arxiv:2403.13732}.'
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- 'A fault-tolerant logical \(T\) gate can be obtained by encoding the Steane code''s seven physical qubits into the seven logical qubits of a \([[63,7,3]]\) outer quantum divisible CSS code preserved by transversal \(T^\dagger\) \cite{arxiv:2204.13176}.'
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- 'Fault-tolerant logical zero and magic state preparation \cite{doi:10.1038/srep19578}.
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Magic-state preparation converts unbiased noise into biased noise \cite{arxiv:2401.10982}.'
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- '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}.'
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- 'Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity \cite{arxiv:2402.17761}.'
- 'Syndrome measurement can be done with ancillary flag qubits \cite{arxiv:1705.02329} or with no extra qubits \cite{doi:10.1088/2058-9565/abc6f4}.
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The depth of syndrome extraction circuits can be lowered by using past syndrome values \cite{arxiv:2305.00784}.'
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- 'Syndrome measurement can be done with ancillary flag qubits \cite{arxiv:1705.02329} or with no extra qubits \cite{doi:10.1088/2058-9565/abc6f4}. The depth of syndrome extraction circuits can be lowered by using past syndrome values \cite{arxiv:2305.00784}.'
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- 'Computation of ground-state energy of the hydrogen molecule \cite{arxiv:2505.09133}.'
Copy file name to clipboardExpand all lines: codes/quantum/qubits/stabilizer/qubit_stabilizer.yml
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- 'Computation can be sped up substantially for codes that admit transversal measurements of logical \(X\) and \(Z\) \cite{arxiv:1210.4626}.'
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- 'Diagonal transversal \hyperref[topic:clifford]{Clifford gates} on multiple code blocks must form one of six families of matrix groups: \(O(\ell,\mathbb{F}_2)\) generically, \(U(\ell,\mathbb{F}_4)\) for Hermitian qubit codes, \(GL(\ell,\mathbb{F}_2)\) for non-self-dual CSS codes, \(O( \ell, \mathbb{F}_2[x]/(x^2) )\) for self-dual non-CSS codes, \(U(\ell,R_8)\) for intermediate semi-self-dual CSS or self-dual semi-CSS cases, and \(Sp(2\ell,\mathbb{F}_2)\) for self-dual CSS codes \cite{arxiv:2507.10519}. There are two \([[8,1,3]]\) self-dual non-CSS codes; see QECDB \cite{preset:QECDB}.'
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- 'An entangling diagonal transversal two-qubit \hyperref[topic:clifford]{Clifford gate} exists only for codes equivalent to CSS or self-dual codes; canonical representatives are the inter-block CNOT in the CSS case and a \(Y\)-controlled-\(Y\) gate, locally Clifford equivalent to \(CZ\), in the self-dual case \cite{arxiv:2507.10519}.'
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- 'Implementing the full Clifford group on \(k\) logical qubits requires \(k\)-fold transversal gates. In particular, stabilizer codes cannot admit a transversal implementation of the Clifford group on more than one logical qubit \cite{arxiv:2602.13395}.'
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- 'Implementing the full Clifford group on \(k\) logical qubits requires \(k\)-fold transversal gates; in particular, stabilizer codes cannot admit a transversal (1-fold) implementation of the full \hyperref[topic:clifford]{Clifford group} on more than one logical qubit, a fold-transversal (2-fold) implementation on more than two logical qubits, or a code-automorphism implementation on more than one logical qubit \cite{arxiv:2602.13395}. These bounds are tight: \(k\) copies of the \([[7,1,3]]\) Steane code form a \([[7k,k,3]]\) code admitting a \(k\)-fold transversal \hyperref[topic:clifford]{Clifford group} \cite{arxiv:2602.13395}.'
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general_gates:
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- 'Logical \hyperref[topic:clifford]{Clifford gates} can be performed by physical \hyperref[topic:clifford]{Clifford circuits} that permute logical Pauli operators \cite{arxiv:1803.06987}.'
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