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Copy file name to clipboardExpand all lines: codes/quantum/oscillators/stabilizer/lattice/gkp_concatenated.yml
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Recursively concatenating the \(C_6\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel \cite{arxiv:1712.00294}.
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Concatenating Abelian LP codes with GKP codes can surpass the CSS Hamming bound \cite{arxiv:2111.07029}.
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Particular families of GKP codes achieve the capacity of \hyperref[topic:ad]{AD} and amplification channels for some loss rates \cite{arxiv:2412.06715}.
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Concatenations of square-lattice GKP codes with Hermitian Galois-qudit codes achieves the capacity for all loss rates \cite{arxiv:2505.10499}.
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Concatenations of square-lattice GKP codes with Hermitian Galois-qudit codes achieve the capacity for all loss rates \cite{arxiv:2505.10499}.
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Concatenation of GKP codes with quantum polar codes achieves a rate against the displacement channel \cite{arxiv:2505.10499}.
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general_gates:
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- code_id: quantum_polar
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detail: 'Concatenation of GKP codes with quantum polar codes achieves a rate against the displacement channel \cite{arxiv:2505.10499}.'
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- code_id: stabilizer_over_gfqsq
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detail: 'Concatenations of square-lattice GKP codes with Hermitian Galois-qudit codes achieves the capacity for all loss rates \cite{arxiv:2505.10499}.'
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detail: 'Concatenations of square-lattice GKP codes with Hermitian Galois-qudit codes achieve the capacity for all loss rates \cite{arxiv:2505.10499}.'
Copy file name to clipboardExpand all lines: codes/quantum/qubits/nonstabilizer/movassagh_ouyang.yml
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An explicit code construction does exist for linear distance codes encoding one logical qubit using Radon's theorem \cite{doi:10.1007/BF01464231,doi:10.1007/978-1-4613-0039-7}.
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For finite rate codes, there is no rigorous proof that the construction algorithm succeeds, and approximate constructions are described instead.
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This family strictly generalizes CSS codes (because CSS codes come only from linear or selforthogonal classical codes). These codes can be shown to be realized as a subspace of the ground space of a (geometrically) local Hamiltonian.
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This family strictly generalizes CSS codes (because CSS codes come only from linear or self-orthogonal classical codes). These codes can be shown to be realized as a subspace of the ground space of a (geometrically) local Hamiltonian.
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protection: 'Let \(C \subset \{0,1,\dots,q-1\}^n\) be a classical code with distance \(d_x\). Let \(d_z\) satisfy \(q^n > 2 V_q(d_z-1) -1\), where \(V_q(r)\) is the volume of the \(q\)-ary Hamming ball of radius \(r\). Then the algorithm produces a quantum code with distance \(d = \text{min}(d_x,d_z)\). Asymptotically, the distance scales linearly with \(n\).'
Copy file name to clipboardExpand all lines: codes/quantum/qubits/qetc/qetc_7_2.yml
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Seven-qubit QETC that transmutes all single-qubit Pauli errors to logical phase errors.
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See \cite[Table 1]{arxiv:2310.10278} for its stabilizer generators.
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The stabilizer group of the \(((7,2))\) QETC, together with the logical-\(Z\) operator on the first logical qubit, generate the stabilizer group of one of the sixteen distinct \([[7,1,3]]\) codes \cite{arxiv:0709.1780}.
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The stabilizer group of the \(((7,2))\) QETC, together with the logical-\(Z\) operator on the first logical qubit, generates the stabilizer group of one of the sixteen distinct \([[7,1,3]]\) codes \cite{arxiv:0709.1780}.
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relations:
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parents:
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- code_id: qetc
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cousins:
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- code_id: small_distance_qubit_stabilizer
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detail: 'The stabilizer group of the \(((7,2))\) QETC, together with the logical-\(Z\) operator on the first logical qubit, generate the stabilizer group of one of the sixteen distinct \([[7,1,3]]\) codes \cite{arxiv:0709.1780}.'
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detail: 'The stabilizer group of the \(((7,2))\) QETC, together with the logical-\(Z\) operator on the first logical qubit, generates the stabilizer group of one of the sixteen distinct \([[7,1,3]]\) codes \cite{arxiv:0709.1780}.'
Copy file name to clipboardExpand all lines: codes/quantum/qubits/stabilizer/fracton/layer.yml
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introduced: '\cite{arxiv:2309.16503}'
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description: |
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Member of a family of qubit QLDPC CSS codes with stabilizer generator weights \(\leq 6\) that are obtained by coupling layers of 2D surface code according to the Tanner graph of a QLDPC code (or a more general qubit stabilizer code).
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Member of a family of qubit QLDPC CSS codes with stabilizer generator weights \(\leq 6\) that are obtained by coupling layers of 2D surface codes according to the Tanner graph of a QLDPC code (or a more general qubit stabilizer code).
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Geometric locality is maintained because, instead of being concatenated, each pair of parallel surface-code squares are fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery.
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features:
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rate: 'Code parameters on a cube, of \hyperref[topic:asymptotics]{order} \((10,40,4)\), achieve the 3D \hyperref[topic:bpt-bound]{BPT bound} when asymptotically good QLDPC codes are used in the construction.'
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rate: 'Code parameters on a cube, of \hyperref[topic:asymptotics]{order} \((10,40,4)\), achieve the 3D \hyperref[topic:bpt-bound]{BPT bound} when asymptotically good QLDPC codes are used in the construction.'
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decoders:
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- 'Decoders against stochastic and adversarial noise \cite{arxiv:2510.06659}.'
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- code_id: fracton
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detail: 'Layer codes are non-translation invariant 3D lattice stabilizer codes that can be viewed as fracton topological defect networks \cite{arxiv:2309.16503}.'
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- code_id: good_qldpc
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detail: 'Layer code parameters, of \hyperref[topic:asymptotics]{order} \((10,40,4)\), achieve the \hyperref[topic:bpt-bound]{BPT bound} in 3D when asymptotically good QLDPC codes are used in the construction.'
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detail: 'Layer code parameters, of \hyperref[topic:asymptotics]{order} \((10,40,4)\), achieve the \hyperref[topic:bpt-bound]{BPT bound} in 3D when asymptotically good QLDPC codes are used in the construction.'
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- code_id: qubit_concatenated
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detail: 'Each pair of surface-code squares in a layer code are fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery.'
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- code_id: self_correct
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detail: 'The energy barrier of excitations for layer codes constructed using asymptotically good QLDPC codes scales as \hyperref[topic:asymptotics]{order} \(\Theta{n^{1/3}}\) \cite{arxiv:2309.16503}. Layer codes are partially self-correcting quantum memories \cite{arxiv:2510.06659,arxiv:2510.09218}. Layer codes constructed from random CSS codes have near-optimal scaling of code parameters and a polynomial energy barrier, exhibiting behavior consistent with partial selfcorrection \cite{arxiv:2510.06659}.'
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detail: 'The energy barrier of excitations for layer codes constructed using asymptotically good QLDPC codes scales as \hyperref[topic:asymptotics]{order} \(\Theta(n^{1/3})\) \cite{arxiv:2309.16503}. Layer codes are partially self-correcting quantum memories \cite{arxiv:2510.06659,arxiv:2510.09218}. Layer codes constructed from random CSS codes have near-optimal scaling of code parameters and a polynomial energy barrier, exhibiting behavior consistent with partial self-correction \cite{arxiv:2510.06659}.'
Copy file name to clipboardExpand all lines: codes/quantum/qubits/stabilizer/qldpc/qldpc.yml
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Member of a family of \([[n,k,d]]\) qubit stabilizer codes for which the number of sites participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant \(w\) as \(n\to\infty\).
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The code can be denoted by \([[n,k,d,w]]\).
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Sometimes, the two parameters are explicitly stated: each site of an an \((l,w)\)\textit{-regular qubit QLDPC code} is acted on by \(\leq l\) generators of weight \(\leq w\).
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Sometimes, the two parameters are explicitly stated: each site of an \((l,w)\)\textit{-regular qubit QLDPC code} is acted on by \(\leq l\) generators of weight \(\leq w\).
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Qubit QLDPC codes can correct many stochastic errors far beyond the distance, which may not scale as favorably.
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Together with more accurate, faster, and easier-to-parallelize measurements than those of general stabilizer codes, this property makes QLDPC codes interesting in practice.
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- 'Iterative error estimation based on the MIN-SUM and SUM-PRODUCT algorithms \cite{arxiv:quant-ph/0502086}.'
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- 'Quantum belief propagation (BP) decoder \cite{arxiv:0706.4094,arxiv:0708.1337,arxiv:0801.1241} is a quantum version of the classical BP decoder, but performance suffers due to degeneracy \cite{arxiv:2012.15297}.
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Various post-processing algorithms have been proposed (see below and also Refs. \cite{doi:10.1109/MILCOM58377.2023.10356284,doi:10.1109/ICASSP48485.2024.10446153}).'
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- 'BP-OSD decoder, scaling as \(O(n^3)\), adds a post-processing step based on ordered statistics decoding (OSD) to the belief propogation (BP) decoder \cite{arxiv:1904.02703}.'
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- 'BP-OSD decoder, scaling as \(O(n^3)\), adds a post-processing step based on ordered statistics decoding (OSD) to the belief propagation (BP) decoder \cite{arxiv:1904.02703}.'
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- 'Neural network BP decoders \cite{arxiv:1811.07835,arxiv:2212.10245} and GNN decoders \cite{arxiv:2307.01241,arxiv:2310.17758} for qubit codes.'
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- 'Partially and fully decoupled BP decoders, which use the decoupling representation, yield improvements against depolarizing noise \cite{arxiv:2305.17505}.'
Copy file name to clipboardExpand all lines: codes/quantum/qubits/subsystem/qldpc/bbs/bacon_shor/bacon_shor_9.yml
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- '\(2.02 \times 10^{-5}\) \hyperref[topic:computational-threshold]{concatenated threshold} for the recursively concatenated code \cite{arxiv:0805.4213}.'
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realizations:
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- 'Trapped-ion qubits: state preparation, logical measurement, and syndrome extraction (deferred to the end) demonstrated on a 13-qubit device by M. Cetina and C. Monroe groups \cite{arxiv:2009.11482}.'
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- 'Neutral atom arrays: repeated error correction demonstrated on a device by Atom Computing \cite{arxiv:2411.11822}.'
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realizations:
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- 'Trapped-ion qubits: state preparation, logical measurement, and syndrome extraction (deferred to the end) demonstrated on a 13-qubit device by M. Cetina and C. Monroe groups \cite{arxiv:2009.11482}.'
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- 'Neutral atom arrays: repeated error correction demonstrated on a device by Atom Computing \cite{arxiv:2411.11822}.'
Copy file name to clipboardExpand all lines: codes/quantum/qubits/subsystem/subsystem_spacetime_circuit.yml
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The subsystem code can be made geometrically local at the cost of more ancilla qubits \cite{arxiv:1411.3334}.
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features:
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rate: 'The spacetime circuit code construction is used to show the existance of spatially local subsystem codes that nearly saturate the \hyperref[topic:subsystem-bt-bound]{subsystem BT bound} \cite{arxiv:1411.3334}.'
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rate: 'The spacetime circuit code construction is used to show the existence of spatially local subsystem codes that nearly saturate the \hyperref[topic:subsystem-bt-bound]{subsystem BT bound} \cite{arxiv:1411.3334}.'
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fault_tolerance:
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- 'Fault-tolerant measurement gadget that is a modification based on the DiVincenzo-Shor cat-state method \cite{arxiv:quant-ph/9605011, arxiv:quant-ph/9605031}.'
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cousins:
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- code_id: spacetime_circuit
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detail: 'Spacetime circuit codes can yield subsystem spacetime circuit codes by \hyperref[topic:gauging-out]{gauging out} a subgroup of the logical \hyperref[topic:pauli]{Pauli group} which causes trivial faults in the corresponding \hyperref[topic:clifford]{Clifford circuit}.
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This construction is used to show the existance of geometrically local subsystem codes that nearly saturate the \hyperref[topic:subsystem-bt-bound]{subsystem BT bound} \cite{arxiv:1411.3334}.'
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This construction is used to show the existence of geometrically local subsystem codes that nearly saturate the \hyperref[topic:subsystem-bt-bound]{subsystem BT bound} \cite{arxiv:1411.3334}.'
Copy file name to clipboardExpand all lines: codes/quantum/qudits_galois/stabilizer/duadic/galois_quad_residue.yml
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For \(q\) not divisible by \(n\), its distance satisfies \(d^2-d+1 \geq n\) when \(n \equiv 3\) modulo 4 \cite[Thm. 40]{arxiv:quant-ph/0508070} and \(d \geq \sqrt{n}\) when \(n\equiv 1\) modulo 4 \cite[Thm. 41]{arxiv:quant-ph/0508070}.
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features:
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transversal_gates:
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transversal_gates:
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- 'Qubit quantum QR codes admit transversal implementations of the \hyperref[topic:clifford]{single-qubit Clifford group} \cite{arxiv:2408.12752}. They yield a family of high-distance triorthogonal codes \cite{arxiv:2408.12752} via the doubling transformation \cite{arxiv:1509.03239}; such codes admit transversal implementations of the \(T\) gate.'
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