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Copy file name to clipboardExpand all lines: codes/classical/matrices/unitary/clifford_group.yml
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detail: 'Stabilizer states on \(n\) qubits form 3-designs on complex projective spaces \(\mathbb{C}P^{2^n}\) \cite{arxiv:1510.02767}. The \hyperref[topic:clifford]{Clifford group} is a unitary 2-design \cite{arxiv:quant-ph/0103098} and a 3-design \cite[Thm. 1.6(B)]{arxiv:math/0502080}\cite[pg. 191]{doi:10.1007/3-540-30731-1}\cite{arxiv:1510.02619,arxiv:1510.02769} on \(U(2^n)\). The \([[2m,2m-2,2]]\) code when \(2m\) is a multiple of four obstructs the Clifford group from being a 4-design \cite{arxiv:1609.08172}.'
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cousins:
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- code_id: qubit_stabilizer
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detail: 'Computing with \hyperref[topic:clifford]{Clifford gates}, Pauli measurements, and classical feedforward acting on stabilizer states only can be efficiently simulated on a classical computer by tracking stabilizer and logical generators, according to the \textit{Gottesman-Knill theorem} \cite{arxiv:quant-ph/9807006,manual:{E. Knill, private communication, ca. 1998}}. Stabilizer states can be mapped into the first lattice shell of a BW lattice over a cyclotomic field, while the \hyperref[topic:clifford]{Clifford group} is related to the symmetry group of the lattice \cite{arxiv:2404.17677}.'
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detail: 'Computing with \hyperref[topic:clifford]{Clifford gates}, Pauli measurements, and classical feedforward acting on stabilizer states only can be efficiently simulated on a classical computer by tracking stabilizer and logical generators, according to the \textit{Gottesman-Knill theorem} \cite{arxiv:quant-ph/9807006,preset:KnillPC98}. Stabilizer states can be mapped into the first lattice shell of a BW lattice over a cyclotomic field, while the \hyperref[topic:clifford]{Clifford group} is related to the symmetry group of the lattice \cite{arxiv:2404.17677}.'
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- code_id: kerdock
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detail: 'Kerdock codes correspond to cluster states, and the corresponding Clifford-group automorphisms of this set form a particular group \cite{doi:10.1112/S0024611597000403} that is a unitary 2-design on \(U(2^n)\) \cite{arxiv:1904.07842}. As such, cluster states form complex projective 2-designs on \(\mathbb{C}P^{2^n}\). These are useful in matrix-vector multiplication \cite{arxiv:2105.05879}.'
Copy file name to clipboardExpand all lines: codes/classical/q-ary_digits/ag/reed_solomon/generalized_reed_solomon.yml
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#introduced: '\cite{preset:MacSlo}'
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description: |
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An \([n,k,n-k+1]_q\) linear code that is a modification of the RS code where codeword polynomials are multiplied by additional prefactors \cite[Def. 15.3.19]{preset:HKSag}.
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An \([n,k,n-k+1]_q\) MDS code that is a modification of the RS code where codeword polynomials are multiplied by additional prefactors \cite[Def. 15.3.19]{preset:HKSag}.
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Each message \(\mu\) is encoded into a string of values of the corresponding polynomial \(f_\mu\) at the points \(\alpha_i\), multiplied by a corresponding nonzero factor \(v_i \in \mathbb{F}_q\),
Copy file name to clipboardExpand all lines: codes/classical/q-ary_digits/poset/poset.yml
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- 'See book \cite{doi:10.1007/978-3-319-93821-9} for more details.'
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#Codewords are q-ary strings, so q-ary primary parent
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#Distance metric is not Hamming, so no Hamming parent. Makes more sense since poset metric is a special case of the metric on symmetric spaces.
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relations:
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parents:
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- code_id: q-ary_digits_into_q-ary_digits
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- code_id: symmetric_space
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detail: 'Ordered Hamming space can be viewed as a finite symmetric space \cite{doi:10.4153/CJM-1999-017-5,arxiv:cs/0702033}\cite[Sec. 4.2.3]{arxiv:0909.4767}\cite[Table 3]{arxiv:1007.2905}.'
Copy file name to clipboardExpand all lines: codes/quantum/qubits/qubits_into_qubits.yml
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- 'If a qubit code \(Q\) of length \(n\) has compact subgroups \(N\triangleleft G\leq \mathrm{Aut}(Q)\) such that \(G/N\) is finite, non-Abelian, simple, and not \(A_5\), then \(n\) is at least the minimal permutation degree \(\mu(G/N)\) \cite[Thm. 1]{arxiv:2604.15111}.'
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general_gates:
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- |
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Computing with \hyperref[topic:clifford]{Clifford gates}, Pauli measurements, and classical feedforward acting on stabilizer states only can be efficiently simulated on a classical computer by tracking stabilizer and logical generators, according to the \textit{Gottesman-Knill theorem} \cite{arxiv:quant-ph/9807006,manual:{E. Knill, private communication, ca. 1998}}.
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Computing with \hyperref[topic:clifford]{Clifford gates}, Pauli measurements, and classical feedforward acting on stabilizer states only can be efficiently simulated on a classical computer by tracking stabilizer and logical generators, according to the \textit{Gottesman-Knill theorem} \cite{arxiv:quant-ph/9807006,preset:KnillPC98}.
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There is a canonical form for \hyperref[topic:clifford]{Clifford circuits} \cite{arxiv:2003.09412,arxiv:2408.15202} and many algorithms for simulating them \cite{arxiv:quant-ph/0406196,arxiv:1712.03554,arxiv:2301.02356}.
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Universal quantum computing can be achieved using \hyperref[topic:clifford]{Clifford gates} and a single type of \hyperref[topic:clifford]{non-Clifford} gate, such as the \(T\) gate \cite{arxiv:quant-ph/9503016}.
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More generally, the \textit{Solovay-Kitaev} theorem \cite{doi:10.1070/RM1997v052n06ABEH002155,doi:10.1090/gsm/047} states that any subset of gates that generates a dense subgroup of the full \(n\)-qubit gate group can be used to construct any gate to arbitrary accuracy (see \cite{arxiv:quant-ph/0505030}\cite[Appx. 3]{doi:10.1017/cbo9780511976667.019}). The task of approximating a desired gate by \hyperref[topic:clifford]{Clifford gates} and a fixed set of \hyperref[topic:clifford]{non-Clifford} gates is called \textit{gate compilation} or \textit{circuit synthesis}.
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# Stabilizer part also in clifford_group
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- code_id: clifford_group
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detail: |
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Computing with \hyperref[topic:clifford]{Clifford gates}, Pauli measurements, and classical feedforward acting on stabilizer states only can be efficiently simulated on a classical computer by tracking stabilizer and logical generators, according to the \textit{Gottesman-Knill theorem} \cite{arxiv:quant-ph/9807006,manual:{E. Knill, private communication, ca. 1998}}.
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Computing with \hyperref[topic:clifford]{Clifford gates}, Pauli measurements, and classical feedforward acting on stabilizer states only can be efficiently simulated on a classical computer by tracking stabilizer and logical generators, according to the \textit{Gottesman-Knill theorem} \cite{arxiv:quant-ph/9807006,preset:KnillPC98}.
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There is a canonical form for \hyperref[topic:clifford]{Clifford circuits} \cite{arxiv:2003.09412,arxiv:2408.15202} and many algorithms for simulating them \cite{arxiv:quant-ph/0406196,arxiv:1712.03554,arxiv:2301.02356}.
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Universal quantum computing can be achieved using \hyperref[topic:clifford]{Clifford gates} and a single type of \hyperref[topic:clifford]{non-Clifford} gate, such as the \(T\) gate \cite{arxiv:quant-ph/9503016}.
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More generally, the \textit{Solovay-Kitaev} theorem \cite{doi:10.1070/RM1997v052n06ABEH002155,doi:10.1090/gsm/047} states that any subset of gates that generates a dense subgroup of the full \(n\)-qubit gate group can be used to construct any gate to arbitrary accuracy (see \cite{arxiv:quant-ph/0505030}\cite[Appx. 3]{doi:10.1017/cbo9780511976667.019}). The task of approximating a desired gate by \hyperref[topic:clifford]{Clifford gates} and a fixed set of \hyperref[topic:clifford]{non-Clifford} gates is called \textit{gate compilation} or \textit{circuit synthesis}.
A stabilizer code obtained by morphing the \([[15,1,3]]\) quantum Reed-Muller code on a subset whose child code is the \([[8,3,2]]\) smallest interesting color code \cite{arxiv:2112.01446}.
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This is the smallest code that implements a fault-tolerant logical \(T\) gate using a diagonal depth-one Clifford circuit \cite{manual:{V. V. Albert, private communication, 2026}}.
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This is the smallest code that implements a fault-tolerant logical \(T\) gate using a diagonal depth-one Clifford circuit \cite{preset:AlbertPC26}.
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It is the smallest known stabilizer code with a fault-tolerant logical \(T\) gate, implemented via physical \(T\), \(T^{\dagger}\), and \(CCZ\) gates \cite{arxiv:2112.01446}.
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A stabilizer tableau for the code is \cite{arxiv:2403.13732}
Copy file name to clipboardExpand all lines: codes/quantum/qubits/small_distance/small/12/carbon.yml
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- code_id: bc_phantom
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detail: 'The carbon code is the B\&C phantom code obtained from the \([[3,1,2]]_4\) Galois-qudit code \cite{arxiv:2601.20927}.'
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- code_id: qubit_concatenated
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detail: 'The \([[12,2,4]]\) carbon code is based on Knill''s \(C_4/C_6\) scheme \cite{arxiv:2404.02280}. Using the concatenation convention presented here, it is a block concatenation with inner code \([[4,2,2]]\) and outer code \(C_6\).'
A rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it. This is the smallest code that implements a fault-tolerant logical \(S\) gate using a diagonal depth-one Clifford circuit \cite{manual:{V. V. Albert, private communication, 2026}}.
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A rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it. This is the smallest code that implements a fault-tolerant logical \(S\) gate using a diagonal depth-one Clifford circuit \cite{preset:AlbertPC26}.
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A stabilizer tableau for the code is given by \cite[ID 18]{preset:qiskit}
Copy file name to clipboardExpand all lines: codes/quantum/qubits/small_distance/small/7/qetc_7_2.yml
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- code_id: qetc
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- code_id: bare_7_1_3
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detail: 'The stabilizer group of the \([[7,2,2]]\) QETC, together with the logical-\(Z\) operator on the first logical qubit, generates the stabilizer group of a \([[7,1,3]]\) code \cite{arxiv:2310.10278} equivalent to the bare \([[7,1,3]]\) code \cite{manual:{V. V. Albert, private communication, 2026}}.'
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detail: 'The stabilizer group of the \([[7,2,2]]\) QETC, together with the logical-\(Z\) operator on the first logical qubit, generates the stabilizer group of a \([[7,1,3]]\) code \cite{arxiv:2310.10278} equivalent to the bare \([[7,1,3]]\) code \cite{preset:AlbertPC26}.'
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