Layer Codes

TL;DR

This work introduces layer codes, a construction that converts any CSS stabilizer code into a three-dimensional, local topological CSS code organized as networks of surface-code layers connected by defects. The output codes have maximum stabilizer weight 6 and satisfy a parameter mapping that, for input [[n,k,d]] with $n_X,n_Z$ checks and max weight $w$, yields [[Θ(n n_X n_Z), k, Ω((d/w) min(n_X,n_Z))]]; when the input is a good CSS LDPC code, the result is a 3D code with optimal scaling [[Θ(L^3), Θ(L), Θ(L^2)]] and a linear energy barrier Θ(L). The construction leverages lattice surgery and topological defects to sew together local surface-code patches, producing a non-translation-invariant defect-network code that saturates known bounds in 3D. The paper provides explicit lattice-check realizations with weight at most 6, detailed analyses of logical operators and error structures, and concrete examples (Repetition, [[4,2,2]], Shor, and Steane codes) to illustrate the defect-network architecture. Together, these results establish that layer codes achieve optimal 3D code parameters while preserving a scalable energy barrier when built from LDPC inputs, significantly advancing the landscape of 3D quantum error-correcting codes with practical locality and decoding potential.

Abstract

The surface code is a two-dimensional topological code with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here, we introduce a construction that takes as input a stabilizer code and produces as output a three-dimensional topological code with related code parameters. The output codes have the special structure of being topological defect networks formed by layers of surface code joined along one-dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good low-density parity-check codes, the output is a three-dimensional topological code with optimal scaling code parameters and a polynomial energy barrier.

Figures (38)

• Figure 1: An illustration of existing bounds on local codes in 3D, and constructions known to saturate them. The blue shaded area corresponds to the BPT bound bravyi2010tradeoffs${kd \leq O(n)}$, and the BT bound bravyi2009no$d \leq O(n^{2/3})$. The point labeled 'a' indicates the layer code construction, see Theorem \ref{['thm:1']}. The red shaded area corresponds to Bravyi's bounds for subsystem codes bravyi2011subsystem. The point 'b' indicates the construction in Ref. bacon2015sparse, see also Ref. WireCodes. The green shaded area corresponds to the bound on classical codes from Ref. bravyi2010tradeoffs. The point 'c' indicates the construction in Ref. baspin2023combinatorial. All points along the dashed line from 'a' to $(1,0)$ can be achieved by tiling smaller layer code blocks, see Corollary \ref{['corr:1']}, and similarly for points 'b' and 'c'.
• Figure 2: The layer code based on the 3 qubit repetition code. 3 grey $xz$-layers depict surface codes corresponding to input physical qubits. 2 red $yz$-layers depict surface codes corresponding to input $Z$ checks. Trijunctions between the red and grey layers correspond to nontrivial topological defects.
• Figure 3: The layer code based on the [[4,2,2]] code. 4 grey $xz$-layers depict surface codes corresponding to input physical qubits. The blue $xy$-layer depicts a surface code corresponding to an input $X$ check. The red $yz$-layer depicts a surface code corresponding to an input $Z$ check. Blue, red and green junctions where the layers meet correspond to nontrivial topological defects.
• Figure 4: The layer code based on Shor's code. 9 grey $xz$-layers depict surface codes corresponding to input physical qubits. 2 blue $xy$-layers depict surface codes corresponding to input $X$ checks. 6 red $yz$-layers depict surface codes corresponding to input $Z$ checks. Blue, red and green junction lines where the layers meet correspond to nontrivial topological defects.
• Figure 5: The layer code based on Steane's code. 7 grey $xz$-layers depict surface codes corresponding to input physical qubits. 3 blue $xy$-layers depict surface codes corresponding to input $X$ checks. 3 red $yz$-layers depict surface codes corresponding to input $Z$ checks. Blue, red and green junction lines where the layers meet correspond to nontrivial topological defects.

Theorems & Definitions (32)

• Theorem 1
• proof
• Corollary 1
• proof
• Lemma 1: Levin2013ProtectedBarkeshli2013Classification
• Definition 1: $m$-configuration
• Definition 2: Boundary equivalence
• Remark 1
• Remark 2
• Remark 3