Floquet Codes from Derived Semi-Regular Hyperbolic Tessellations on Orientable and Non-Orientable Surfaces

1. Problem Statement

Quantum computing requires fault-tolerant architectures to protect quantum information from noise and decoherence. While stabilizer codes (like surface codes) are well-established, Floquet codes offer a dynamic alternative where logical operators change over time, potentially offering better error correction and resource utilization.

Existing research on hyperbolic Floquet codes has primarily focused on regular tessellations (using a single type of polygon) on orientable surfaces. This limits the diversity of code parameters (length $n$, dimension $k$, and distance $d$) and the ability to optimize for specific hardware constraints. Furthermore, there is a lack of literature regarding Floquet codes constructed on non-orientable surfaces using semi-regular tessellations (tilings using multiple types of regular polygons).

The paper addresses these gaps by:

1. Constructing new Floquet codes using derived semi-regular tessellations rather than just regular ones.
2. Extending these constructions to both orientable and non-orientable compact surfaces with genus $g \ge 2$.
3. Analyzing the asymptotic behavior and performance metrics of these new code families.

2. Methodology

The authors employ a geometric approach rooted in hyperbolic geometry and topological quantum error correction.

A. Geometric Foundations

• Hyperbolic Surfaces: The codes are defined on compact surfaces of genus $g \ge 2$. These surfaces admit hyperbolic geometry, allowing for an infinite number of tessellations (unlike Euclidean geometry).
• Tessellation Types:
  • Regular Tessellations: Denoted as $\{p, q\}$, consisting of $p$-gons with $q$ meeting at each vertex.
  • Semi-Regular Tessellations: Denoted as $[m_1, m_2, \dots, m_k]$, consisting of multiple types of regular polygons arranged in a specific cyclic order around every vertex.
• Derivation Techniques: The core innovation involves generating new semi-regular tessellations from existing regular $\{p, q\}$ tessellations using two specific local operations:
  1. Clipping Derivation: "Cutting" the corners of a $p$-gon near its vertices to create a new $2p$-gon, resulting in a $[2p, 2p, q]$ tessellation.
  2. Incenter Derivation: Connecting the incenters of adjacent faces and decomposing the original faces into right triangles, resulting in a $[2p, 2q, 4]$ tessellation.
• Non-Orientable Surfaces: The authors establish that for a non-orientable surface of genus $g$, if the Euler characteristic $\chi(S_g)$ matches that of an orientable surface $S_h$ (where $g=2h$), the resulting Floquet codes share the same parameters. This allows the transfer of construction techniques to non-orientable manifolds.

B. Code Construction

• Physical Qubits: Assigned to the vertices of the tessellation.
• Check Operators: The tessellation must be trivalent (3 edges per vertex) and 3-colorable. Edges are colored (Red, Green, Blue) based on the faces they bound.
• Measurement Rounds: The code operates in a periodic sequence (Floquet cycle). In each round, specific two-qubit Pauli checks (XX, YY, or ZZ) are measured on edges of a specific color.
• Instantaneous Stabilizer Group (ISG): The group of stabilizers evolves over time. Logical operators are defined by homologically non-trivial cycles on the surface.

C. Parameter Calculation

The authors derive formulas to calculate the code parameters $[[n, k, d]]$:

• $n$ (Length): The number of vertices ($V$) in the tessellation.
• $k$ (Dimension): Determined by the genus $g$ and orientability ($k=2g$ for orientable, $k=g$ for non-orientable).
• $d$ (Distance): Calculated by estimating the length of the shortest non-trivial homological cycle (geodesic) on the hyperbolic surface. The authors use a geometric method involving the ratio of the geodesic length to the distance between incenters of adjacent faces to estimate the weight of logical operators.

3. Key Contributions

1. New Code Families via Derived Tessellations: The paper introduces a systematic method to generate Floquet codes from semi-regular tessellations ($[2p, 2p, q]$ and $[2p, 2q, 4]$) derived from regular $\{p, q\}$ bases. This significantly expands the design space beyond regular tilings.
2. Non-Orientable Surface Construction: The authors provide the first comprehensive construction of Floquet codes on non-orientable surfaces using semi-regular tessellations, proving that codes on non-orientable surfaces of odd genus offer unique parameter sets not found in orientable counterparts.
3. Extensive Parameter Tables: The paper provides exhaustive tables (Tables II–XIII) detailing code parameters for genera $g=2$ through $g=50$ (and specific non-orientable cases up to $g=51$).
4. Asymptotic Analysis: The study analyzes the behavior of the encoding rate ($k/n$) and the distance-weighted rate ($kd^2/n$) as the genus increases.

4. Results

• Performance Metrics:

  • Encoding Rate ($k/n$): Codes on non-orientable surfaces generally exhibit a higher encoding rate ($k/n$) compared to orientable surfaces of the same genus.
  • Distance-Weighted Rate ($kd^2/n$): Codes on orientable surfaces generally achieve a higher $kd^2/n$ ratio, which is a critical metric for fault tolerance thresholds.
  • Quasi-Euclidean Families: The authors identify "quasi-Euclidean" families (e.g., $[6, 6, p]$ for large even $p$) where the encoding rate approaches $1/3$ as $g, p \to \infty$.
  • Quasi-Regular Families: For families like $[2p, 2p, 2q]$, the encoding rate approaches $1$ as parameters grow, indicating high efficiency.

• Specific Findings:

  • The derived semi-regular tessellations allow for a trade-off between code length and distance that is not possible with regular tessellations.
  • For the $[6, 6, 8]$ tessellation, the code parameters improve significantly as genus increases, with the $kd^2/n$ ratio showing substantial growth (e.g., reaching values $>6$ for $g=50$).
  • Non-orientable codes of odd genus (e.g., $g=3, 5, 7$) provide unique parameter sets (e.g., $[[24, 3, 4]]$) that do not have direct analogs in orientable codes, offering new options for specific hardware topologies.

5. Significance

• Enhanced Flexibility: By moving beyond regular tessellations to semi-regular ones, the paper demonstrates that quantum error correction codes can be "tuned" more precisely to specific hardware constraints (e.g., connectivity limits or preferred check operator weights).
• Resource Efficiency: The higher encoding rates found in non-orientable constructions suggest that utilizing non-orientable topologies (or simulating them) could lead to more efficient use of physical qubits for a given logical capacity.
• Scalability: The asymptotic analysis confirms that these hyperbolic Floquet codes maintain robust performance as the system scales (increasing genus), making them strong candidates for large-scale fault-tolerant quantum computing.
• Theoretical Advancement: The work bridges the gap between geometric topology (specifically the theory of uniform tilings on non-orientable surfaces) and practical quantum error correction, providing a mathematical framework for generating new code families that were previously unexplored.

In conclusion, this paper establishes that derived semi-regular hyperbolic tessellations on both orientable and non-orientable surfaces constitute a rich and underutilized resource for constructing high-performance, scalable Floquet codes, offering superior flexibility in parameter selection compared to traditional regular tessellation approaches.