diff --git a/codes/quantum/properties/block/symmetric/quantum_cyclic.yml b/codes/quantum/properties/block/symmetric/quantum_cyclic.yml index 84a2dd96c..377ec58f8 100644 --- a/codes/quantum/properties/block/symmetric/quantum_cyclic.yml +++ b/codes/quantum/properties/block/symmetric/quantum_cyclic.yml @@ -23,7 +23,10 @@ description: | 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: diff --git a/codes/quantum/properties/hamiltonian/constant_excitation.yml b/codes/quantum/properties/hamiltonian/constant_excitation.yml index 361682301..a9e3ba1b5 100644 --- a/codes/quantum/properties/hamiltonian/constant_excitation.yml +++ b/codes/quantum/properties/hamiltonian/constant_excitation.yml @@ -16,16 +16,6 @@ description: | 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. diff --git a/codes/quantum/properties/hamiltonian/self_correct.yml b/codes/quantum/properties/hamiltonian/self_correct.yml index 8514008e4..cf407d645 100644 --- a/codes/quantum/properties/hamiltonian/self_correct.yml +++ b/codes/quantum/properties/hamiltonian/self_correct.yml @@ -35,7 +35,7 @@ protection: | 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}. diff --git a/codes/quantum/properties/qecc_finite.yml b/codes/quantum/properties/qecc_finite.yml index 55b8726a0..557107d0e 100644 --- a/codes/quantum/properties/qecc_finite.yml +++ b/codes/quantum/properties/qecc_finite.yml @@ -75,6 +75,7 @@ protection: | \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 diff --git a/codes/quantum/qubits/qubits_into_qubits.yml b/codes/quantum/qubits/qubits_into_qubits.yml index 2a65d5f98..246e43846 100644 --- a/codes/quantum/qubits/qubits_into_qubits.yml +++ b/codes/quantum/qubits/qubits_into_qubits.yml @@ -116,7 +116,7 @@ features: - '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. @@ -152,7 +152,7 @@ features: 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}. diff --git a/codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/qubit_10_24_3.yml b/codes/quantum/qubits/small_distance/small/10/qubit_10_24_3.yml similarity index 100% rename from codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/qubit_10_24_3.yml rename to codes/quantum/qubits/small_distance/small/10/qubit_10_24_3.yml diff --git a/codes/quantum/qubits/small_distance/small/stab_10_1_2.yml b/codes/quantum/qubits/small_distance/small/10/stab_10_1_2.yml similarity index 100% rename from codes/quantum/qubits/small_distance/small/stab_10_1_2.yml rename to codes/quantum/qubits/small_distance/small/10/stab_10_1_2.yml diff --git a/codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/qubit_5_6_2.yml b/codes/quantum/qubits/small_distance/small/5/qubit_5_6_2.yml similarity index 100% rename from codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/qubit_5_6_2.yml rename to codes/quantum/qubits/small_distance/small/5/qubit_5_6_2.yml diff --git a/codes/quantum/qubits/small_distance/small/8/qubit_8_1_3.yml b/codes/quantum/qubits/small_distance/small/8/qubit_8_1_3.yml new file mode 100644 index 000000000..62e4e5af6 --- /dev/null +++ b/codes/quantum/qubits/small_distance/small/8/qubit_8_1_3.yml @@ -0,0 +1,38 @@ +####################################################### +## 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' diff --git a/codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/qubit_9_12_3.yml b/codes/quantum/qubits/small_distance/small/9/qubit_9_12_3.yml similarity index 100% rename from codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/qubit_9_12_3.yml rename to codes/quantum/qubits/small_distance/small/9/qubit_9_12_3.yml diff --git a/codes/quantum/qubits/small_distance/small/shor_nine.yml b/codes/quantum/qubits/small_distance/small/9/shor_nine.yml similarity index 100% rename from codes/quantum/qubits/small_distance/small/shor_nine.yml rename to codes/quantum/qubits/small_distance/small/9/shor_nine.yml diff --git a/codes/quantum/qudits/qudits_into_qudits.yml b/codes/quantum/qudits/qudits_into_qudits.yml index 1205de4d4..2edec2411 100644 --- a/codes/quantum/qudits/qudits_into_qudits.yml +++ b/codes/quantum/qudits/qudits_into_qudits.yml @@ -59,6 +59,9 @@ features: \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}.' diff --git a/codes/quantum/qudits_galois/galois_into_galois.yml b/codes/quantum/qudits_galois/galois_into_galois.yml index 9cb3d5e1b..db641a686 100644 --- a/codes/quantum/qudits_galois/galois_into_galois.yml +++ b/codes/quantum/qudits_galois/galois_into_galois.yml @@ -23,6 +23,8 @@ description: | 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} @@ -51,7 +53,6 @@ features: 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: