In a paper published in the journal Nature Communications, researchers introduced a family of three-dimensional topological codes designed for optimal quantum information storage. These codes, constructed from layers of surface code joined at one-dimensional junctions, featured a maximum stabilizer check weight of six.
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Study: Layer codes. Image Credit: Have a nice day Photo/Shutterstock.com
By transforming stabilizer codes into three-dimensional structures, they achieved optimal scaling code parameters and a polynomial energy barrier. The work revealed strongly correlated quantum states with enhanced protection from errors in three dimensions.
Background
Past work has shown that quantum error correction, essential for scalable quantum computing, relies heavily on topological quantum codes due to their simple stabilizer checks and high error thresholds. The surface code, a prominent example, has been widely studied but needs to improve optimal code parameter scaling compared to good low-density parity-check codes. While recent breakthroughs have led to low-density parity-check (LDPC) codes with optimal scaling, topological codes on regular lattices in flat space still face constraints like the Bravyi-Poulin-Terhal bound.
Topological Code Design
The layer codes were constructed by concatenating an efficient Calderbank-Shor-Steane (CSS) code with a surface code, resulting in optimal scaling parameters. To address high-weight and nonlocal stabilizers, a variant of lattice surgery was employed, producing a three-dimensional layer code.
The construction uses a topological defect network formalism to link the surface code layers and implement the desired stabilizers. In a cuboid, surface code layers are stacked as qubits or checks. The xz-planes represent qubit layers, while the yz- and xy-planes represent X-check and Z-check layers.
Topological defect lines connect these layers to enforce the X and Z checks, using specified anyons to establish nontrivial junctions and maintain stability across the network. This configuration ensures that excitations are appropriately managed and nonlocal interactions are avoided.
The resulting code parameters are optimal for three dimensions. The number of physical qubits, N, is dictated by the arrangement of surface code layers in the cuboid. The number of logical qubits, K, matches the input CSS code, preserving commutation relations by mapping logical operators to string operators in the surface code layers. Despite complications arising from stabilizer violations at junctions, the code maintains equivalence in logical operator commutation and ensures no redundant logical pairs.
Finally, the distance, D, of the code relates to the weight of logical operators, with a lower bound determined by the properties of the input code. In cases where the input code is an LDPC code, the distance retains favorable scaling, ensuring robustness.
The energy barrier of the output code is also maintained despite possible increases in excitation penalties. A method is described to translate Pauli operators in the layer code to corresponding operators in the input code, ensuring locality and optimal energy scaling, even for logical transformations.
Layer Code Construction Overview
This work presents a construction that transforms an arbitrary CSS stabilizer code into a topological CSS code known as a layer code, which is local in three-dimensional space. The construction relates the code parameters of the input and output codes, with the maximum check weight of the output code being 6.
The stabilizer checks of the layer codes are defined through a topological defect network, where surface code layers are stacked on different planes in a cubic lattice, corresponding to physical qubits, X checks, and Z checks. These layers are connected at nontrivial junctions determined by the Tanner graph of the input code, ensuring the stability of the topological structure.
The resulting layer codes achieve optimal scaling of code parameters in three dimensions. For example, applying the construction to a family of good CSS LDPC codes results in topological CSS codes that reach the bounded-power-time (BPT) bound, illustrating that these codes are optimal local codes in three dimensions.
The construction also preserves the scaling of the energy barrier, which relates to the minimum energy required to implement a nontrivial logical operator. The energy barrier of the output codes is proportional to the size of the input codes, ensuring that the scaling properties are retained.
The main result shows that topological CSS stabilizer codes in three dimensions achieve optimal parameters and energy barrier scaling as Θ(L) by applying layer code construction to good LDPC codes with the weight of six or fewer stabilizer checks. Finally, two examples illustrate the construction with input codes: the 3-qubit repetition code and Shor’s code.
The layer code derived from the repetition code consists of 5 surface code layers, and the one derived from Shor’s code consists of 17 surface code layers. Both constructions feature topological defects at the junctions of the layers, demonstrating the practical implementation of the layer code concept and its potential for constructing efficient, scalable topological codes.
Conclusion
To summarize, a family of three-dimensional topological codes with optimal scaling parameters and polynomial energy barriers was introduced. These codes were derived from a construction that transformed stabilizer codes into topological defect networks composed of surface code layers.
When applied to quantum low-density parity-check codes, the output codes achieved optimal scaling, offering strong error protection. The results highlighted the potential of these codes for storing quantum information with maximum error resilience in three dimensions.
Journal Reference
Williamson, D. J., & Baspin, N. (2024). Layer codes. Nature Communications, 15:1, 1-7. DOI: 10.1038/s41467-024-53881-3, https://www.nature.com/articles/s41467-024-53881-3
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