Add tile codes and barbell codes (open-boundary qLDPC codes on a 2D lattice)

codes/quantum/qubits/stabilizer/qldpc/barbell.yml

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+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: barbell
+physical: qubits
+logical: qubits
+
+name: 'Barbell code'
+short_name: 'Barbell'
+introduced: '\cite{arxiv:2606.06062}'
+
+description: |
+  Member of a family of high-rate, two-dimensionally local CSS \hyperref[code:qldpc]{QLDPC} codes designed so that all two-qubit interactions of the QEC cycle are native to a fixed-connectivity superconducting chip layout whose hardware complexity stays constant as the code distance grows. Barbell codes are a special case of tile codes \cite{arxiv:2504.09171,arxiv:2504.08887}, with the \(X\)- and \(Z\)-type stabilizers tailored to a \textit{six-qubit star lattice plus near-local coupler} (6QSL+NLC) architecture, also called the \textit{Barbell architecture}.
+
+  Each \(X\)-check qubit is joined to a partner \(Z\)-check qubit by a single near-local coupler, and the two checks' data-qubit neighborhoods together form a \textit{barbell}. The corresponding pair of \(X\)- and \(Z\)-type stabilizers (of weight up to eight in the family studied) is supported on the six hexagonal cells of this barbell. Syndrome information is read out using superdense syndrome extraction \cite{arxiv:2312.08813}: a Bell pair is prepared on the two check qubits via the near-local coupler, the data qubits are entangled, and a Bell measurement is performed. For weight-\(w\) stabilizers this yields a QEC cycle of depth \(w+4\) (depth 12 for weight-8 stabilizers). Because all near-local couplers are parallel and of equal, distance-independent length, they route within a single additional hardware tier without air bridges.
+
+  The weight-8 family studied numerically contains the \([[242,16,8]]\), \([[288,16,9]]\), \([[338,16,10]]\), \([[392,16,11]]\), and \([[450,16,14]]\) codes, with \(n=2(L+D)(M+D)\) for an \(L\times M\) lattice and bulk-stabilizer box parameter \(D\); families with weight-6 and weight-10 bulk stabilizers, and with \(k=7\) and \(k=8\), are also given \cite{arxiv:2606.06062}. With fewer than 30 data qubits per logical qubit, barbell codes offer up to an eightfold reduction in qubit overhead relative to the rotated \hyperref[code:rotated_surface]{surface code} at comparable distance, and the weight-10 family improves the code-parameter efficiency \(kd^2/n\) by up to a factor of eight over surface-code patches.
+
+protection: |
+  Protects against Pauli noise. Under uniform depolarizing circuit-level noise, a distance-14 weight-8 barbell code reaches the teraquop regime (logical error rate below \(10^{-12}\) per round, enabling several trillion QEC cycles) at physical error rates above \(10^{-4}\). At a physical error rate of \(10^{-3}\), a distance-11 barbell code and a distance-5 rotated surface code use the same 400 data qubits to encode 16 logical qubits, but the barbell code achieves a per-round logical error rate of \(8.8\times 10^{-7}\) versus \(9.6\times 10^{-4}\) for the surface code \cite{arxiv:2606.06062}.
+
+features:
+  rate: 'High-rate QLDPC family. The weight-8 examples encode \(k=16\) logical qubits using fewer than 30 data qubits per logical qubit, an up-to-eightfold reduction in physical-qubit overhead compared with the rotated surface code of the same distance.'
+
+  decoders:
+    - 'Superdense syndrome-extraction circuit \cite{arxiv:2312.08813} of depth \(w+4\) (depth 12 for weight-8 stabilizers), using one near-local coupler per barbell to prepare and later measure a Bell pair on the paired \(X\)- and \(Z\)-check qubits. A Pauli-frame correction tracked in software relates the measurement outcomes to the stabilizer syndromes.'
+    - 'Relay-BP decoder \cite{arxiv:2506.01779} applied to detector error models constructed with Stim \cite{arxiv:2103.02202}; \(X\)- and \(Z\)-type errors are decoded separately.'
+
+  fault_tolerance:
+    - 'Fault-tolerant logical multi-qubit Pauli measurements, both within a single patch and between two patches, are realized with the planar tile-code protocol of \cite{arxiv:2506.18061} without additional connectivity requirements. The per-round logical error rate of a distance-8 logical \(ZZ\) measurement is only slightly higher than that of the corresponding memory experiment \cite{arxiv:2606.06062}.'
+
+notes:
+  - 'The 6QSL+NLC (Barbell) architecture is composed only of experimentally demonstrated components: a six-qubit star lattice of qubits coupled to central elements by tunable couplers \cite{arxiv:2503.12869}, supplemented by near-local couplers connecting check-qubit pairs. Its hardware-complexity metric is \(C_{\text{hw}}\approx 1.65\) in the framework of \cite{arxiv:2507.23011}, compared with \(C_{\text{hw}}=1\) for the surface code and \(C_{\text{hw}}>3\) for the \([[144,12,12]]\) bivariate bicycle code.'
+  - 'Logical error curves are extrapolated by fitting \(\mathrm{LER}(p)=p^{d/2}\exp(c_0+c_1 p + c_2 p^2)\) \cite{arxiv:2606.06062}.'
+  - 'Logical operators and derived automorphisms of the underlying tile codes are studied in \cite{arxiv:2511.14589}.'
+
+relations:
+  parents:
+    - code_id: tile
+      detail: 'Barbell codes are a special case of tile codes whose \(X\)- and \(Z\)-type stabilizers are tailored for measurement on the six-qubit star lattice plus near-local coupler (Barbell) architecture, with near-local couplers used exclusively to pair \(X\)- and \(Z\)-check qubits.'
+  cousins:
+    - code_id: 2d_stabilizer
+      detail: 'Barbell codes are 2D lattice stabilizer codes derived from tile codes by adding ancilla check qubits, translating the qubit positions, and embedding the qubits in the connectivity graph of the six-qubit star lattice plus near-local coupler architecture.'
+    - code_id: rotated_surface
+      detail: 'Barbell codes match the rotated surface code in logical performance per QEC round at comparable distance while reducing the physical-qubit overhead by up to a factor of eight \cite{arxiv:2606.06062}.'
+    - code_id: qcga
+      detail: 'Bivariate bicycle (BB) codes are the closest QLDPC comparison. Unlike BB codes, whose longest near-local coupler scales with the code distance and whose routing requires multiple tiers (hardware complexity \(C_{\text{hw}}>3\) for the \([[144,12,12]]\) code), the near-local couplers of a barbell code are parallel, equal-length, and distance-independent, fitting into a single additional tier.'
+    - code_id: 2d_color
+      detail: 'Barbell syndrome extraction adapts the superdense syndrome-extraction circuits originally developed to implement color codes on a square grid \cite{arxiv:2312.08813}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: TomaszAndrzejewski
+      date: '2026-06-11'

codes/quantum/qubits/stabilizer/qldpc/tile.yml

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+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: tile
+physical: qubits
+logical: qubits
+
+name: 'Tile code'
+introduced: '\cite{arxiv:2504.09171,arxiv:2504.08887}'
+
+description: |
+  Member of a family of two-dimensionally local CSS \hyperref[code:qldpc]{QLDPC} codes on a planar square lattice with open boundaries, generalizing the \hyperref[code:surface]{surface code} by allowing stabilizer generators of weight greater than four whose supports are confined to a box of fixed size \((D+1)\times(D+1)\) rather than to nearest-neighbor edges only. Data qubits sit on the edges of the lattice and stabilizer checks (\textit{tiles}) are placed at its vertices in a translationally invariant fashion in the bulk.
+
+  A tile code is specified by a \textit{local structure} -- a pair of \(X\)- and \(Z\)-type stabilizer tiles confined to a box of size \((D+1)\times(D+1)\) and chosen so that any relative placement of an \(X\)- and a \(Z\)-tile has even overlap (ensuring commutativity) -- together with a \textit{global structure}, a rectangular layout of bulk-stabilizer positions. Boundary stabilizers are then added on the four sides, after which qubits not supported by any stabilizer and stabilizers with empty support are removed. Under mild conditions (the tiles are not confined to a strip and give rise to topological order), the construction yields an \([[n,k,d]]\) code with logical dimension \(k=2D^2\).
+
+  With a \(3\times 3\) box and weight-6 stabilizers, the construction yields a \([[288,8,12]]\) code (first found in \cite{arxiv:2504.08887}), a fourfold improvement in the code-parameter efficiency \(kd^2/n\) over the rotated surface code; weight-8 stabilizers in a \(3\times 3\) box give a \([[288,8,14]]\) code, and weight-8 stabilizers in a \(4\times 4\) box give \([[288,18,13]]\) and \([[512,18,19]]\) codes, the latter a more-than-twelvefold gain in \(kd^2/n\) over the rotated surface code \cite{arxiv:2504.09171}. The surface code, hypergraph-product codes, and the open-boundary (planar) versions of bivariate bicycle codes are recovered as special cases.
+
+protection: |
+  Protects against Pauli noise. Code distance scales with the lattice dimensions provided the stabilizer tiles satisfy the topological-order condition. Distances of explicit examples are estimated with a probabilistic algorithm and then confirmed exactly using integer linear programming \cite{arxiv:2504.09171}.
+
+features:
+  rate: 'Logical dimension \(k=2D^2\) for stabilizer tiles confined to a \((D+1)\times(D+1)\) box and satisfying the topological-order condition. The code-parameter efficiency \(kd^2/n\) reaches 12.7 for the \([[512,18,19]]\) code, more than twelve times that of the rotated surface code \cite{arxiv:2504.09171}.'
+
+  decoders:
+    - 'The QEC cycle developed for bivariate bicycle codes \cite{arxiv:2308.07915} can be adapted to tile codes; an explicit superconducting realization using superdense syndrome extraction is given by the barbell-code construction \cite{arxiv:2606.06062}.'
+
+  fault_tolerance:
+    - 'Fault-tolerant logical multi-qubit Pauli measurements, both within a single patch and between two patches, can be performed without additional connectivity requirements via the protocol of \cite{arxiv:2506.18061}.'
+    - 'Derived automorphisms \(T_x\) and \(T_y\) implement products of logical CNOT gates fault-tolerantly and with low overhead by extending the lattice on one side and shrinking it on the other; their action on the logical space is multiplication by \(x\) or \(y\) on \(\mathbb{F}_2[x^{\pm},y^{\pm}]/(f,g)\) \cite{arxiv:2511.14589}. This operation is trivial for the surface code.'
+
+notes:
+  - 'The logical operators of a tile code admit a canonical symplectic basis localized along the lattice boundary and generated by a cellular automaton with \(2D^2\) update rules, making them amenable to lattice-surgery techniques \cite{arxiv:2511.14589}.'
+  - 'Tile codes arise as derived (higher) global sections of a shifted Koszul complex on \(\mathbb{P}^1\times\mathbb{P}^1\) associated with a pair of Laurent polynomials \(f,g\) specifying the stabilizer tiles; the logical space is isomorphic to \(\mathbb{F}_2[x^{\pm},y^{\pm}]/(f,g)\), so the logical dimension equals the number of intersection points of the zero sets of \(f\) and \(g\) (Bezout''s theorem). The framework extends to higher dimensions, e.g. a four-dimensional \([[486,24,d]]\) tile code with \(10\leq d\leq 15\) \cite{arxiv:2511.14589}.'
+  - 'Hardware layouts of tile codes on multilayer superconducting hardware outperform the other QLDPC proposals studied in \cite{arxiv:2507.23011} with respect to average coupler length and number of hardware components (through-silicon vias, bump bonds, and routing tiers).'
+
+relations:
+  parents:
+    - code_id: qldpc
+      detail: 'Tile codes are two-dimensionally local CSS QLDPC codes on a planar lattice with open boundaries, generalizing the surface code by allowing higher-weight checks confined to a fixed-size box.'
+    - code_id: qubit_css
+  cousins:
+    - code_id: surface
+      detail: 'Tile codes generalize the surface code; the (rotated and unrotated) surface code is recovered by choosing the surface-code tiles together with an appropriate layout.'
+    - code_id: rotated_surface
+      detail: 'The most efficient tile codes outperform the rotated surface code in the efficiency \(kd^2/n\) by factors of up to more than 12 \cite{arxiv:2504.09171}.'
+    - code_id: qcga
+      detail: 'Tile codes are open-boundary (planar) analogues of bivariate bicycle (BB) codes, which arise by tiling a torus with stabilizer tiles; tile codes instead use open boundaries and retain true \(O(1)\)-locality on a planar 2D lattice. The logical dimension of a BB code is determined by the period of the same cellular automaton that generates tile-code logical operators \cite{arxiv:2511.14589}.'
+    - code_id: hypergraph_product
+      detail: 'Pruned hypergraph-product constructions of planar BB codes are recovered by the tile-code construction using an unrotated bulk-stabilizer layout \cite{arxiv:2504.09171}.'
+    - code_id: lacross
+      detail: 'Hypergraph-product La-Cross codes are instances of tile codes whose stabilizers are induced by univariate polynomials \cite{arxiv:2511.14589}.'
+    - code_id: 2d_stabilizer
+      detail: 'Tile codes are translationally invariant (in the bulk) 2D lattice stabilizer codes with open boundaries.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: TomaszAndrzejewski
+      date: '2026-06-11'

users/users_db.yml

@@ -708,3 +708,7 @@
   githubusername: LeonidPryadko
   pageurl: "https://faculty.ucr.edu/~leonid/"
   avatarurl: "builtinavatar:LeonidPryadko"
+
+- user_id: TomaszAndrzejewski
+  name: 'Tomasz Andrzejewski'
+  githubusername: TomaszAnd