Open-Boundary Tile Codes

• Open-boundary tile codes are a class of planar CSS stabilizer codes that use repeated local tile patterns and boundary truncation to maintain commutation.
• They simplify hardware implementation by eliminating costly wraparound couplers and allowing flexible stabilizer support beyond nearest-neighbor interactions.
• Design tradeoffs balance logical efficiency and physical realization, positioning these codes as a promising frontier in planar qLDPC constructions.

Searching arXiv for papers on open-boundary tile codes and closely related planar qLDPC constructions. Open-boundary tile codes are planar CSS stabilizer codes with open boundaries, constructed by repeating local stabilizer “tiles” across a finite square-grid region and truncating those tiles at the boundary so that commutation is preserved. In the recent literature they are presented as a planar, hardware-friendly counterpart to more connectivity-intensive periodic bivariate bicycle constructions: they retain true $\mathcal{O}(1)$-locality on a 2D planar lattice, but typically trade some logical efficiency for markedly simpler physical realization (Mathews et al., 30 Jul 2025). At the same time, they generalize the surface code by relaxing the rigid nearest-neighbor, weight-4 stabilizer pattern while preserving locality on a lattice with open boundary conditions, and they have become a central example of planar qLDPC code design (Steffan et al., 12 Apr 2025).

1. Definition and conceptual position

Open-boundary tile codes are introduced as a broad constructive class of planar CSS stabilizer codes with open boundaries. The basic ingredients are a pair of local patterns, an $X$-tile and a $Z$-tile, together with a planar layout specifying where physical qubits and tile anchors are placed. In the square-grid formulation, physical qubits sit on edges of a finite planar subgraph, while stabilizers are anchored at vertices; a tile is a subset of edges in a $B\times B$ box, and placing that tile at an anchor defines the support of one stabilizer generator (Steffan et al., 12 Apr 2025).

This framework is explicitly positioned between the surface code and periodic bivariate bicycle constructions. Relative to the surface code, tile codes preserve planar locality with open boundaries while allowing more flexible stabilizer support patterns and higher check weights. Relative to bivariate bicycle codes on a torus, they remove periodic identifications and the associated wraparound couplers that are especially costly in planar hardware (Mathews et al., 30 Jul 2025). This places them in the class of planar qLDPC codes whose main design objective is to maintain bounded-range 2D locality without reverting to the low-rate surface-code regime.

A recurrent theme in the literature is that open-boundary tile codes should not be viewed as variants of radial codes. Rather, they are treated as closely related to bivariate bicycle and generalized toric constructions: the tileable, translational flavor is preserved, but periodicity is replaced by open boundaries and boundary truncation (Mathews et al., 30 Jul 2025).

2. Local tile construction and commutation structure

The construction is governed by two local conditions. First, both $X$- and $Z$-tiles have bounded support inside a $B\times B$ box, with no support on the top horizontal edges or rightmost vertical edges of that box. Second, the $X$- and $Z$-tiles satisfy a reflection or duality rule ensuring even overlap for every relative displacement. In support language, the induced CSS commutation condition is

$$|\operatorname{supp}(s_X)\cap \operatorname{supp}(s_Z)| \equiv 0 \pmod 2.$$

Equivalently, in binary notation,

$X$0

The paper introducing tile codes emphasizes that this even-overlap condition is enforced geometrically by the tile construction itself (Steffan et al., 12 Apr 2025).

The geometry is local but not necessarily nearest-neighbor in the surface-code sense. “Local” means bounded support diameter independent of system size. The principal locality ranges studied are $X$1 and $X$2, corresponding to stabilizers supported in $X$3 and $X$4 boxes, respectively. The support can be an irregular pattern of horizontal and vertical edges rather than a single plaquette or star (Steffan et al., 12 Apr 2025).

The open-boundary tile-code literature also makes clear that the construction does not, in general, enforce a weight-4 qubit tier with nearest-neighbor connectivity or a fixed check-qubit location inside each tile. In the hardware-placement study, this residual geometric freedom is resolved by placing check qubits at positions minimizing Euclidean distance to the data qubits in their support (Mathews et al., 30 Jul 2025). A plausible implication is that tile codes combine algebraic regularity with nontrivial code-to-layout degrees of freedom that are absent in the ordinary square-lattice surface code.

3. Open boundaries, truncation, and pruning

Open boundaries are implemented by truncating stabilizers whose nominal support extends beyond the finite planar patch. The defining rule is explicit: if the available qubit set does not contain the whole support of a stabilizer, then the support of that stabilizer is truncated to the available edges or qubits. Boundary layouts are chosen so that when an $X$5-boundary tile and a $X$6-boundary tile overlap, their overlap lies entirely inside the actual qubit set, preserving commutation after truncation (Steffan et al., 12 Apr 2025).

The standard boundary pattern uses a rectangular bulk array of anchors, then adds $X$7 layers of $X$8-only boundary anchors at top and bottom and $X$9 layers of $Z$0-only boundary anchors at left and right. This is the direct analogue of having opposite $Z$1-type and $Z$2-type boundaries. A final pruning step removes qubits not touched by any $Z$3-type stabilizer or not touched by any $Z$4-type stabilizer, and removes any stabilizer whose support becomes empty (Steffan et al., 12 Apr 2025).

In the hardware-oriented comparison study, the same open-boundary modification is described more operationally: bounded stabilizer tiles are repeated in the plane; tiles reaching beyond the supported lattice are truncated; then data qubits and stabilizers are pruned appropriately to ensure commutativity and distance preservation. The effect is to eliminate toroidal wraparound couplers and reduce edge density near the perimeter (Mathews et al., 30 Jul 2025).

A more algebraic extension of this picture appears in the theory of tile-code logical operators. Under mild assumptions, omitted boundary stabilizers are not merely absent checks but the building blocks of the logical sector: canonical logical operators can be written as products of omitted boundary stabilizers, and boundary translation acts nontrivially on the logical space (Breuckmann et al., 18 Nov 2025). This suggests that open boundaries in tile codes are structurally generative rather than merely truncative.

4. Parameters, families, and efficiency tradeoffs

The search-based tile-code literature reports several benchmark families with substantially better $Z$5 than standard planar surface-code variants, while retaining true planar locality. Representative examples include

$Z$6

using weight-6 stabilizers in $Z$7 boxes,

$Z$8

using weight-8 stabilizers in $Z$9 boxes, and

$B\times B$0

using weight-8 stabilizers in $B\times B$1 boxes (Steffan et al., 12 Apr 2025).

The same paper states that the $B\times B$2 code has

$B\times B$3

that the $B\times B$4 code has

$B\times B$5

and that the $B\times B$6 code has

$B\times B$7

These are presented as substantial improvements over the rotated surface code in the same metric (Steffan et al., 12 Apr 2025).

The hardware-cost study refines this parameter picture by organizing open-boundary tile codes into bands corresponding to different tile or check-weight families. For the visible $B\times B$8 family, examples range from $B\times B$9 through $X$0, with logical-efficiency values $X$1 rising from $X$2 to $X$3 while the hardware-cost metric remains essentially flat at $X$4–$X$5. Higher-weight families reach larger $X$6, for example $X$7 with $X$8 and $X$9, or $Z$0 with $Z$1 and $Z$2 (Mathews et al., 30 Jul 2025).

The tradeoff is explicit across the literature. Open boundaries substantially reduce hardware cost but generally lower logical efficiency compared with periodic bivariate bicycle realizations. To recover high $Z$3, one typically needs higher stabilizer weight and larger qubit count (Mathews et al., 30 Jul 2025). This tradeoff is central rather than incidental: open boundaries simplify physical realization by removing the hardest couplers, but they also weaken the toric-like encoding efficiency that periodicity can support.

5. Boundary-supported logical operators and derived automorphisms

A major structural advance is the proof that, under total topological order and a minimal-support condition, any tile code admits a canonical symplectic basis of logical operators supported near boundaries. More precisely, there is one pair of logical $Z$4 operators for each physical qubit in a $Z$5 box in the bottom-left corner; the $Z$6 operators can be chosen in a strip of width $Z$7 along the left boundary, and the $Z$8 operators in a strip of height $Z$9 along the bottom boundary. Within those strips, their representative is unique and can be generated by a cellular automaton with $B\times B$0 local update rules (Breuckmann et al., 18 Nov 2025).

The same work identifies the logical space with

$B\times B$1

using a Koszul-complex resolution of the translationally invariant bulk model. The finite open-boundary tile code is realized as a derived section of a Koszul complex on $B\times B$2, and the logical dimension is

$B\times B$3

under the generic assumptions of the construction (Breuckmann et al., 18 Nov 2025).

This algebraic control makes it possible to define derived automorphisms: boundary-manipulation operations that act like symmetries on the logical sector even when the code has no literal lattice automorphism. Extending the lattice on one side and shrinking it on the other induces logical maps $B\times B$4 and $B\times B$5, which act algebraically by multiplication by $B\times B$6 and $B\times B$7 in $B\times B$8. For the surface code this operation is trivial, but for tile codes it induces a product of logical CNOT gates (Breuckmann et al., 18 Nov 2025). This gives open-boundary tile codes a boundary-driven logical gate mechanism absent in the ordinary planar surface-code setting.

6. Hardware realization, planar cost, and related open-boundary families

Open-boundary tile codes are repeatedly identified as especially favorable for planar superconducting hardware because their locality is genuine in the plane and because truncation removes wraparound couplers. In the HAL placement-and-routing study, the representative $B\times B$9 tile code requires 3 routing tiers, average edge length $X$0, bumps $X$1, and TSVs $X$2, compared with the periodic BB example $X$3 requiring 5 tiers, length $X$4, bumps $X$5, and TSVs $X$6. The caption explicitly notes “about a fourfold reduction in the average edge length” (Mathews et al., 30 Jul 2025).

The same study attributes this advantage to true $X$7-locality, regular tiling motifs, reduced edge density near truncated boundaries, and shorter couplers than BB codes. The result is that tile-code points form nearly horizontal bands in hardware-cost plots: enlarging the patch mostly repeats a fixed local motif rather than introducing increasingly burdensome long links (Mathews et al., 30 Jul 2025).

More recent open-boundary planar qLDPC work extends the design space beyond the original tile-code formulation while preserving the same basic objectives. “Planar quantum low-density parity-check codes with open boundaries” constructs planar bivariate-bicycle-style codes using boundary anyon condensation and “lattice grafting,” reporting examples such as $X$8 and $X$9 with local stabilizers of weight 6 or lower (Liang et al., 11 Apr 2025). “Vine Codes” introduces another planar-square-grid framework with open boundaries, routing qubits, and generalized boundaries beyond the familiar $Z$0 boundaries of the surface and tile codes (Nixon et al., 18 Jun 2026). These constructions suggest that open-boundary tile codes sit inside a broader program of planarizing high-rate translationally structured qLDPC codes.

A complementary direction studies noise tailoring rather than boundary construction itself. Clifford-deformed variants of open-boundary tile codes preserve locality and LDPC structure while improving performance under biased noise; the open-boundary family with $Z$1 and $Z$2 is used as the main planar testbed, with bulk weight-6 checks and truncated lower-weight boundary checks (Das et al., 14 May 2026). This indicates that open-boundary tile codes are not only geometrically implementable but also flexible under basis deformation and circuit design.

7. Limitations, caveats, and open problems

The literature also identifies several limitations. The original tile-code paper does not provide a universal closed-form expression for $Z$3 in terms of lattice dimensions and tile parameters, and much of the evidence remains benchmark- and search-based rather than formulaic (Steffan et al., 12 Apr 2025). The hardware-placement study is heuristic: HAL optimizes placements and routings but does not prove global optimality, and its conclusions remain tied to a forward-looking multilayer superconducting architecture (Mathews et al., 30 Jul 2025).

Boundary theory, while now substantially clarified, is still incomplete. The logical-operator paper develops a canonical boundary basis and derived automorphisms, but its strongest results require total topological order and a minimal-support condition, and it explicitly leaves nongeneric cases for future work (Breuckmann et al., 18 Nov 2025). Likewise, open-boundary planarizations of general bivariate bicycle codes remain only partially understood: pruning is fully systematic for certain hypergraph-product subclasses, while genuinely bivariate examples still rely on explicit constructions rather than a general theorem (Eberhardt et al., 2024).

A further subtlety is that open boundaries do not improve every hardware submetric monotonically. The HAL robustness analysis observes that as some tile-code families grow, the average number of TSVs per edge can increase slightly even as bump congestion decreases, due to changing bulk-to-boundary ratios (Mathews et al., 30 Jul 2025). Thus “open boundaries help” is accurate only at the level of the total hardware balance, not as a universal monotonic statement about every submetric.

The main unresolved design question is whether one can find more logically efficient open-boundary planar qLDPC constructions without sacrificing the locality benefits that make tile codes hardware-cheap. The existing papers repeatedly frame this as an open problem: high-efficiency tile codes appear comparatively rare and are mostly obtained with higher weights and larger qubit counts (Mathews et al., 30 Jul 2025). In that sense, open-boundary tile codes presently define not a finished endpoint but a well-structured frontier between planar hardware feasibility and qLDPC coding efficiency.