A simple universal routing strategy for reducing the connectivity requirements of quantum LDPC codes

Technical Summary: A Simple Universal Routing Strategy for Reducing the Connectivity Requirements of Quantum LDPC Codes

Problem Statement
Quantum Low-Density Parity-Check (qLDPC) codes offer reduced overhead for quantum error correction compared to surface codes but typically require dense, long-range connectivity between data and ancilla qubits. This requirement poses a significant challenge for hardware implementation, particularly in superconducting processors where each long-range gate necessitates a physical coupler, increases crosstalk, degrades fidelity, and complicates calibration. While "morphing circuits" and "mid-cycle" approaches have been proposed to reduce connectivity for specific codes, there is a need for a universal strategy that systematically reduces long-range connections without fundamentally altering the code structure or sacrificing circuit-level distance.

Methodology
The authors propose a routing-based syndrome measurement strategy that leverages the inherent topological properties of quantum codes. The core observation is that in many quantum codes (including surface codes and Bivariate Bicycle (BB) codes), X and Z ancilla qubits form short loops (specifically length-4 loops) with data qubits.

The methodology involves the following steps:

1. Sequential Measurement: The syndrome extraction circuit is structured to measure X and Z stabilizers sequentially (pattern $C_X C_Z C_X C_Z \dots$) rather than simultaneously.
2. Ancilla Routing: When a direct connection between a data qubit and its target ancilla is removed, the information is routed through an adjacent ancilla qubit of the opposite type. For example, to measure a Z stabilizer where a direct link to a Z ancilla is missing, the parity information is transferred through a nearby X ancilla.
3. Circuit Construction: This routing is implemented using additional layers of CNOT gates. Specifically, the authors demonstrate that removing a connection requires adding a "detour" through the remaining three edges of the loop, effectively replacing one direct gate with a sequence of gates that transfers and then decouples the information.
4. Application to BB Codes: For Bivariate Bicycle codes, which feature translation-invariant layouts with weight-6 stabilizers, the authors identify specific length-4 loops formed by pairs of X and Z ancillae and two data qubits. They systematically remove one long-range connection per loop. Two configurations are explored:
   • Three-quarters LR Circuit: Removes one long-range coupler from Z stabilizers only (retaining 75% of long-range connections).
   • Half LR Circuit: Removes one long-range coupler from both X and Z stabilizers (retaining 50% of long-range connections).

Key Contributions

• Universal Routing Strategy: The paper introduces a general method applicable to diverse quantum codes that reduces connectivity requirements by exploiting short loops between X and Z ancillae.
• Connectivity Reduction in BB Codes: For BB codes, the strategy successfully removes up to 50% of long-range connections. In the "half LR" configuration, the connectivity requirement is reduced to four local connections and one long-range connection per qubit.
• Surface Code Adaptation: The method is applied to rotated surface codes, achieving a degree-3 connectivity requirement (hexagonal lattice) comparable to the "mid-cycle" approach by McEwen et al., but derived through a distinct routing mechanism.
• Preservation of Circuit-Level Distance: Through numerical simulations, the authors demonstrate that despite the increased circuit depth, the circuit-level distance remains largely preserved. The slopes of the logical error rates indicate that the circuit-level distance matches the code distance.

Results

• Surface Codes: The routing approach achieves degree-3 connectivity. While the logical error rates are slightly higher than the conventional 4-layer circuit and the mid-cycle approach (due to the increased depth of 10 CNOT layers vs. 4 or 8), the code distance is preserved.
• BB Codes:
  • Simulations using the SI1000 error model show that the "half LR" circuit (16 CNOT layers) and "three-quarters LR" circuit (14 CNOT layers) maintain circuit-level distances comparable to the original IBM implementation (7 CNOT layers).
  • At a physical error rate of $10^{-3}$, the logical error rates of the routing circuits are approximately ten times higher than the original IBM implementation. The authors attribute this to the increased circuit depth (14–16 layers vs. 7 layers).
  • However, the trade-off is the elimination of up to 50% of long-range couplers.
• Decoder Performance: The authors note that using the X ancilla measurement results as additional detectors (flag qubits) did not improve performance with the Minimum Weight Perfect Matching (MWPM) decoder and slightly degraded it, leading them to exclude this data in their primary results.

Significance and Claims
The paper claims that this routing strategy offers a practical pathway for implementing qLDPC codes on hardware with strict connectivity constraints, such as superconducting processors. The primary significance lies in the trade-off: the method exchanges increased circuit depth for reduced hardware connectivity requirements.

The authors argue that if implementing dense long-range connections proves more challenging than managing the physical error rates associated with fewer connections (due to crosstalk and calibration complexity), their scheme is preferable. They suggest that as qubit coherence times improve, the penalty of increased circuit depth may become less significant, making reduced-connectivity architectures increasingly advantageous. The work is presented as a step toward enabling qLDPC demonstrations on current and near-term hardware by relaxing the stringent connectivity requirements that currently hinder their implementation.