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qubit_8_1_3
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valbert4 committed Jan 7, 2025
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3 changes: 3 additions & 0 deletions codes/quantum/properties/block/symmetric/quantum_cyclic.yml
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protection: 'Cyclic symmetry guarantees that if a single subsystem is protected against some noise, then all other subsystems are also.'

features:
encoders:
- 'Linear feedback shift registers \cite{arxiv:quant-ph/9910061}.'
decoders:
- 'Linear feedback shift registers \cite{arxiv:quant-ph/9910061}.'
- 'Adapted from the Berlekamp decoding algorithm for classical BCH codes \cite{arxiv:1007.1697}.'

notes:
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10 changes: 0 additions & 10 deletions codes/quantum/properties/hamiltonian/constant_excitation.yml
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For spin-\(S\) codes, this generalizes to \(H=\sum_i J_z^{(i)}\), where \(J_z\) is the spin-\(S\) \(Z\)-operator.
For bosonic codes, such as Fock-state codes, codewords are often in an eigenspace with eigenvalue \(N>0\) of the \textit{total excitation} or \textit{energy Hamiltonian}, \(H=\sum_i \hat{n}_i\).
One of the first such codes \cite{arxiv:quant-ph/9603022} is a \(((8,1,3))\) qubit code, with codewords
\begin{align}
\begin{split}
|\overline{0}\rangle&=(|00001111\rangle+|11101000\rangle−|10010110\rangle−|01110001\rangle\\&+|11010100\rangle+|00110011\rangle+|01001101\rangle+|10101010\rangle)/\sqrt{8}\\
|\overline{1}\rangle&=X^{\otimes8}|\overline{0}\rangle~.
\end{split}
\end{align}
Each logical state is a superposition of computational basis states with four excitations.
protection: |
Fock-state CE codes are protected from identical \hyperref[topic:ad]{AD} acting on all modes because the damping acts on all codewords identically \cite{arxiv:quant-ph/9704002,doi:10.1103/PhysRevA.56.1114}.
The all-zero \hyperref[topic:ad]{AD} Kraus operator acts identically on every state and so can be exactly correctable in the case of Fock-state CE codes.
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2 changes: 1 addition & 1 deletion codes/quantum/properties/hamiltonian/self_correct.yml
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Self-correcting quantum memories currently exist in four and higher dimensions, with their existence in three dimensions being an open question.
For similar reasons as the classical Ising model, the four-dimensional toric code is a self-correcting quantum memory due to an \hyperref[topic:asymptotics]{order} \(O(n)\) energy cost of creating a logical error \cite{arxiv:quant-ph/0110143,arxiv:0811.0033}.
On the other hand, the 2D toric code is not thermally stable \cite{arxiv:quant-ph/0702102,arxiv:0709.2717,arxiv:0810.4584,arxiv:0911.3843} because its string-like logical operators anti-commite with stabilizer generators supported only at their ends, and thus have a constant energy cost of creation.
On the other hand, the 2D toric code is not thermally stable \cite{arxiv:quant-ph/0702102,arxiv:0709.2717,arxiv:0810.4584,arxiv:0911.3843} because its string-like logical operators anti-commute with stabilizer generators supported only at their ends, and thus have a constant energy cost of creation.
There is a general upper bound on the relaxation rate of a qubit stabilizer or qubit subsystem stabilizer quantum memory interacting with a Markovian environment \cite{arxiv:0907.2807}.
An \(n\)-dependent energy barrier to creating all logical errors is likely necessary for a thermally stable memory, having been shown as such for a large class of 2D topological phases \cite{arxiv:0810.3557,arxiv:1412.2858,arxiv:1601.01324,arxiv:2107.01628}.
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1 change: 1 addition & 0 deletions codes/quantum/properties/qecc_finite.yml
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\end{defterm}
A channel \(\mathcal{E}\) is correctable if \(\mathcal{E}^C(\rho)=\rho_0\mathrm{Tr}(\rho)\) for some constant state \(\rho_0\), which is equivalent to the \term{Knill-Laflamme conditions} \cite{arxiv:0811.1621,arxiv:0907.5391}.
The logical and physical dimensions are related to the channel rank for non-degenerate codes via the quantum packing bound \cite{arxiv:1007.3655}.
features:
rate: 'One can achieve a transmission rate
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4 changes: 2 additions & 2 deletions codes/quantum/qubits/qubits_into_qubits.yml
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- 'Non-Clifford gates are typically more difficult to implement than Clifford gates and so are treated as a resource. Optimizing T-gate count in circuit synthesis is \(NP\)-hard \cite{arxiv:2310.05958} and can be done using various procedures \cite{arxiv:1303.2042,arxiv:1308.4134,arxiv:1601.07363,arxiv:1710.07345,arxiv:1712.01557,arxiv:2110.10292}, e.g., \textit{ZX calculus} (a.k.a. Penrose spin calculus) \cite{arxiv:1903.10477,arxiv:1911.09039,arxiv:2004.05164,arxiv:2109.01076} or reinforcement learning \cite{arxiv:2402.14396}.
There is an optimal asymptotic scaling of the number of T gates needed to prepare an arbitrary state \cite{arxiv:1812.00954,arxiv:2411.04790}.
Decompositions in terms of Toffoli and Hadamard gates \cite{arxiv:quant-ph/0205115} as well as cosine-sine gates also exist \cite{arxiv:quant-ph/0404089}. Gate errors in circuit synthesis can sometimes add up destructively \cite{arxiv:1612.01011}.
There is a threshold against depolarizing noise for any single-qubit gate that determines if the gate enables universal quantum computation \cite{arxiv:0907.3189}.'
There is a threshold against depolarizing noise for any single-qubit gate that determines if the gate enables universal quantum computation \cite{arxiv:0907.3189,arxiv:1011.2497}.'
- '\begin{defterm}{Clifford hierarchy}
\label{topic:clifford-hierarchy}
The Clifford hierarchy \cite{arxiv:quant-ph/9908010,arxiv:1608.06596,arxiv:1902.04022,arXiv:2212.05398,arxiv:2410.11818} is a tower of gate sets which includes Pauli and Clifford gates at its first two levels, and non-Clifford gates at higher levels.
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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}
- 'There is a threshold against depolarizing noise for any single-qubit gate that determines if the gate enables universal quantum computation \cite{arxiv:0907.3189}.'
- 'There is a threshold against depolarizing noise for any single-qubit gate that determines if the gate enables universal quantum computation \cite{arxiv:0907.3189,arxiv:1011.2497}.'
- '\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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38 changes: 38 additions & 0 deletions codes/quantum/qubits/small_distance/small/8/qubit_8_1_3.yml
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#######################################################
## This is a code entry in the error correction zoo. ##
## https://github.com/errorcorrectionzoo ##
#######################################################

code_id: qubit_8_1_3
physical: qubits
logical: qubits

name: '\(((8,1,3))\) Plenio-Vedral-Knight CE code'
introduced: '\cite{arxiv:quant-ph/9603022}'

description: |
An eight-qubit qubit code that is the first CE code.
Each logical state is a superposition of computational basis states with four excitations.
Admits codewords of the form
\begin{align}
\begin{split}
|\overline{0}\rangle&=(|00001111\rangle+|11101000\rangle−|10010110\rangle−|01110001\rangle\\&+|11010100\rangle+|00110011\rangle+|01001101\rangle+|10101010\rangle)/\sqrt{8}\\
|\overline{1}\rangle&=X^{\otimes8}|\overline{0}\rangle~.
\end{split}
\end{align}
relations:
parents:
- code_id: qubits_into_qubits
- code_id: constant_excitation
- code_id: small_distance_quantum


# Begin Entry Meta Information
_meta:
changelog:
- user_id: VictorVAlbert
date: '2024-02-13'
3 changes: 3 additions & 0 deletions codes/quantum/qudits/qudits_into_qudits.yml
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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.'
threshold:
- 'There is a threshold against depolarizing noise for any modular-qudit gate that determines if the gate is non-Clifford \cite{arxiv:1011.2497}.'


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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3 changes: 2 additions & 1 deletion codes/quantum/qudits_galois/galois_into_galois.yml
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Codes can be denoted as \(((n,K))_q\) or \(((n,K,d))_q\), whenever the code''s distance \(d\) is defined.
This notation differentiates between Galois-qudit and \(((n,K,d))_{\mathbb{Z}_q}\) modular-qudit codes, although the same notation is usually used for both.'
There exists an analogue of the Wigner function for Galois qudits \cite{arxiv:quant-ph/0401155}, and Galois-qudit stabilizer states correspond to the set of states with positive Wigner functions \cite{arxiv:quant-ph/0401155}.
protection: |
\subsection{Galois-qudit Pauli-string error basis}
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notes:
- 'Introduction to Galois qudits by \href{https://ethz.ch/content/vp/en/conferences/2014/qec/05_thursday/dab6ca18-7453-4197-aaaa-8b1964ece714.html}{Gottesman}.'
- 'Wigner function for Galois qudits \cite{arxiv:quant-ph/0401155}.'
- 'Julia \href{https://github.com/esabo/CodingTheory}{CodingTheory} framework by E. Sabo.'

relations:
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