We introduce tile codes, a simple yet powerful way of constructing quantum codes that are local on a planar 2D lattice. Tile codes generalize the usual surface code by allowing for a bit more flexibility in terms of locality and stabilizer weight. Our construction does not compromise on the fact that the codes are local on a lattice with open boundary conditions. Despite its simplicity, we use our construction to find codes with parameters [[288,8,12]] using weight-6 stabilizers and [[288,8,14]] using weight-8 stabilizers, outperforming all previously known constructions in this direction. Allowing for a slightly higher nonlocality, we find a [[512,18,19]] code using weight-8 stabilizers, which outperforms the rotated surface code by a factor of more than 12. Our approach provides a unified framework for understanding the structure of codes that are local on a 2D planar lattice and offers a systematic way to explore the space of possible code parameters. In particular, due to its simplicity, the construction naturally accommodates various types of boundary conditions and stabilizer configurations, making it a versatile tool for quantum error-correction code design.
We introduce tile codes, a simple yet powerful way of constructing quantum codes that are local on a planar 2D lattice. Tile codes generalize the usual surface code by allowing for a bit more flexibility in terms of locality and stabilizer weight. Our construction does not compromise on the fact that the codes are local on a lattice with open boundary conditions. Despite its simplicity, we use our construction to find codes with parameters [[288,8,12]] using weight-6 stabilizers and [[288,8,14]] using weight-8 stabilizers, outperforming all previously known constructions in this direction. Allowing for a slightly higher nonlocality, we find a [[512,18,19]] code using weight-8 stabilizers, which outperforms the rotated surface code by a factor of more than 12. Our approach provides a unified framework for understanding the structure of codes that are local on a 2D planar lattice and offers a systematic way to explore the space of possible code parameters. In particular, due to its simplicity, the construction naturally accommodates various types of boundary conditions and stabilizer configurations, making it a versatile tool for quantum error-correction code design.
The recently introduced tile codes are a promising alternative to surface codes, combining two-dimensional locality with higher encoding efficiency. While surface codes are well understood in terms of their logical operators and boundary behavior, much less is known about tile codes. In this work, we establish a natural and precise description of their logical operator space. We prove that, under mild assumptions, any tile code admits a canonical symplectic basis of logical operators supported along lattice boundaries, which can be generated efficiently by a simple cellular automaton with the number of update rules only depending on the non-locality of the tile code. Further, we develop algebraic and algebro-geometric frameworks for tile codes, by resolving them by translationally invariant Pauli stabilizer models and showing that they arise as derived sections of a Koszul complex on P1×P1. Finally, we introduce the concept of derived automorphisms for quantum codes. These are automorphism-like operations that can exist even for codes that do not have symmetries. We explain how derived automorphisms can be implemented for tile codes in a low-overhead and fault-tolerant manner by extending the lattice on one side and shrinking it on the other. While this operation is trivial for the surface code, it induces a product of logical CNOT gates on the encoded information. Our results provide new structural insights into tile codes and lay the groundwork for tile codes as building blocks for fault-tolerant quantum computation.
Abstract In recent years, surface codes have become a leading method for quantum error correction in theoretical large-scale computational and communications architecture designs. Their comparatively high fault-tolerant thresholds and their natural two-dimensional nearest-neighbour (2DNN) structure make them an obvious choice for large scale designs in experimentally realistic systems. While fundamentally based on the toric code of Kitaev, there are many variants, two of which are the planar- and defect-based codes. Planar codes require fewer qubits to implement (for the same strength of error correction), but are restricted to encoding a single qubit of information. Interactions between encoded qubits are achieved via transversal operations, thus destroying the inherent 2DNN nature of the code. In this paper we introduce a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer. Our lattice surgery technique comprises splitting and merging planar code surfaces, and enables us to perform universal quantum computation (including magic state injection) while removing the need for braided logic in a strictly 2DNN design, and hence reduces the overall qubit resources for logic operations. Those resources are further reduced by the use of a rotated lattice for the planar encoding. We show how lattice surgery allows us to distribute encoded GHZ states in a more direct (and overhead friendly) manner, and how a demonstration of an encoded CNOT between two distance-3 logical states is possible with 53 physical qubits, half of that required in any other known construction in 2D.
Tile codes are a family of planar quantum low-density parity-check (qLDPC) codes with weight-6 stabilizers and open boundary conditions, offering an encoding efficiency kd2/n of up to four times that of the surface code. In this work, we develop an exhaustive search algorithm for finding SWAP-based routing schemes that implement syndrome extraction for four tile-code families using only nearest-neighbor interactions on a two-dimensional square lattice, matching the connectivity of the surface code. Using explicitly constructed routed syndrome-extraction circuits decoded with BP+OSD, we estimate the circuit-level thresholds of these code families. For the SI1000 noise model, the threshold without such a connectivity constraint is obtained in a range 0.23%-0.31%, while it decreases to 0.11%-0.13% with routing, representing a reduction factor of around two to three. Despite this threshold penalty, our resource-footprint analysis shows that routed tile codes require fewer physical qubits per logical qubit than the surface code at sufficiently low physical error rates: Under the SI1000 noise model, we find a crossover near p∗≈0.08%, below which routed tile codes become more qubit-efficient, with an advantage that grows monotonically as the physical error rate decreases.
Abstract. In recent years, surface codes have become the preferred method for quantum error correction in large scale computational and communications architectures. Their comparatively high fault-tolerant thresholds and their natural 2-dimensional nearest neighbour (2DNN) structure make them an obvious choice for large scale designs in experimentally realistic systems. While fundamentally based on the toric code of Kitaev, there are many variants, two of which are the planar-and defect- based codes. Planar codes require fewer qubits to implement (for the same strength of error correction), but are restricted to encoding a single qubit of information. Interactions between encoded qubits are achieved via transversal operations, thus destroying the inherent 2DNN nature of the code. In this paper we introduce a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer. Our lattice surgery technique comprises splitting and merging planar code surfaces, and enables us to perform universal quantum computation (including magic state injection) while removing the need for braided logic in a strictly 2DNN design, and hence reduces the overall qubit resources for logic operations. We show how lattice surgery allows us to distribute encoded GHZ states in a more direct (and overhead friendly) manner, and how a demonstration of an encoded CNOT between two distance 3 logical states is possible with 53 physical qubits, half of that required in any other known construction in 2D. ar
High-performance quantum low-density parity-check codes promise substantial reductions in the overhead of fault-tolerant quantum computation, but most constructions require long-range connectivity or qubit shuttling, both of which are difficult to realise in superconducting architectures. Here we introduce a family of quantum low-density parity-check codes that, for the first time, combines planar open-boundary layouts, finite-size advantages over surface codes, and syndrome extraction using only nearest-neighbour gates on a planar grid of qubits with either square or hexagonal connectivity. The resulting circuits achieve optimal constant-depth stabiliser measurement, independent of code size, and naturally remove leakage from the system by exchanging the roles of check and data qubits at each syndrome extraction round. We find finite-size instances such as a [[323,14,15]] code, whose code-efficiency ratio is nearly an order of magnitude larger than that of rotated surface-code patches. At around 40 circuit qubits per logical qubit, the best directional tile-code layouts reduce the per-logical per-round logical error rate by a factor of 100 relative to rotated surface-code memories. These results show that the advantages of quantum low-density parity-check codes can survive compilation into strictly planar nearest-neighbour circuits, bringing low-overhead fault-tolerant memories closer to near-term hardware.
Although high-threshold and low-overhead quantum low-density parity-check (qLDPC) codes, such as bivariate bicycle (BB) codes, can reduce the physical-qubit cost by an order of magnitude compared to the Kitaev toric code, their torus layout remains difficult for physical implementation. In this work, we introduce the first systematic procedure to convert BB codes into fully planar, open-boundary qLDPC codes, preserving their performance. We present planar code families with logical dimensions 6 ≤ k ≤ 13 , e.g., ⟦ 78 , 6 , 6 ⟧ , ⟦ 107 , 7 , 7 ⟧ , ⟦ 268 , 8 , 12 ⟧ , ⟦ 405 , 9 , 15 ⟧ , ⟦ 348 , 10 , 13 ⟧ , ⟦ 450 , 11 , 15 ⟧ , ⟦ 386 , 12 , 12 ⟧ , and ⟦ 362 , 13 , 11 ⟧ , all with geometrically local weight-6 stabilizers. Allowing weight-8 stabilizers produces a ⟦ 282 , 12 , 14 ⟧ code, exhibiting an efficiency metric ( k d 2 / n ) an order of magnitude higher than the surface code. The construction combines boundary-anyon condensation with the “lattice-grafting” optimization, yielding high-performance qLDPC codes natively compatible with planar hardware architectures. It also uncovers Sierpinski-type fractal logical operators the distance of which scales with the fractal area on finite lattices. These planar qLDPC codes provide an implementable route to resource-efficient, high-threshold fault tolerance and a flexible framework for future code design on realistic two-dimensional hardware.