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valbert4 committed Mar 3, 2024
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8 changes: 2 additions & 6 deletions codes/quantum/qudits/qudits_into_qudits.yml
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features:
general_gates:
- 'Universal computing can be achieved using qudit Clifford gates \cite{arxiv:quant-ph/9802007,arxiv:quant-ph/0408190,arxiv:2008.00959} and a single type of non-Clifford gate, such as the \(T\) gate \cite{arxiv:1503.08800}.'
- 'Universal computing can be achieved using qudit Clifford gates \cite{arxiv:quant-ph/9802007,arxiv:quant-ph/0408190,arxiv:quant-ph/0512155,arxiv:2008.00959} and a single type of non-Clifford gate, such as the \(T\) gate \cite{arxiv:1503.08800}.'
- '\begin{defterm}{Qudit Clifford hierarchy}
\label{topic:qudit-clifford-hierarchy}
The modular-qudit Clifford hierarchy \cite{arXiv:quant-ph/9908010,arxiv:1608.06596} is a tower of gate sets which includes modular-qudit Pauli and modular-qudit Clifford gates at its first two levels, and non-Clifford qudit gates at higher levels.
The modular-qudit Clifford hierarchy \cite{arXiv:quant-ph/9908010,arxiv:1206.1598,arxiv:1408.1720,arxiv:1608.06596} is a tower of gate sets which includes modular-qudit Pauli and modular-qudit Clifford gates at its first two levels, and non-Clifford qudit gates at higher levels.
The \(k\)th level is defined recursively by
\begin{align}
C_k = \{ U | U P U^{\dagger} \in C_{k-1} \}~,
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\end{defterm}'
decoders:
- 'For few-qudit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the \textit{maximum-likelihood} (ML) decoder.'
# CITE qudit Clifford. Gottesman, Chaos, Solitons & Fractals 10, 1749 - 1758 (1999) + S. Clark, J. Phys. A-Math. Gen. 39, 2701 (2006) for qudit Clifford
# CITE for qudit Clifford hierarchy
# F. Pastawski and B. Yoshida, Phys. Rev. A 91 012305 (2015).
# M. Howard and J. Vala, Phys. Rev. A 86, 022316 (2012

notes:
- 'Weight distribution of a code depends on the average entanglement of codewords \cite{arXiv:quant-ph/0310137,arxiv:2209.07607}.'
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2 changes: 0 additions & 2 deletions codes/quantum/qudits/topological/qudit_surface.yml
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features:
decoders:
- 'Renormalization-group decoder \cite{arxiv:1311.4895,arxiv:1411.3028}.'
# More on RG: CITE H. Anwar, B.J. Brown, E.T. Campbell and D.E. Browne, New J. Phys. 16, 063038 (2014)
# RG: CITE G. Duclos-Cianci and D. Poulin, Quant. Inf. Comp. 9&10, 0721 (2014).

notes:
- 'The simplest \href{https://citizensciencegames.com/games/decodoku/}{Decodoku game} is based on the qudit surface code with \( q=10\). See related \href{https://github.com/quantumjim/qec_lectures}{Qiskit tutorial}.'
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3 changes: 1 addition & 2 deletions codes/quantum/qudits_galois/qldpc/galois_topological.yml
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name: 'Galois-qudit topological code'
introduced: '\cite{arxiv:quant-ph/0609070,doi:10.1109/CIG.2010.5592860,arxiv:1202.3338}'
# CITE P. Sarvepalli, IEEE ITW 1–5 (2010). Galois color? Has this been cited already?

alternative_names:
- '\(\mathbb{F}_q\)-qudit topological code'

description: |
Abelian topological code, such as a surface \cite{arxiv:quant-ph/0609070,arxiv:1202.3338} or color \cite{doi:10.1109/CIG.2010.5592860} code, constructed on lattices of Galois qudits.
Abelian topological code, such as a 2D surface \cite{arxiv:quant-ph/0609070,arxiv:1202.3338} or 2D color \cite{doi:10.1109/CIG.2010.5592860} code, constructed on lattices of Galois qudits.
relations:
parents:
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