Constant-overhead quantum error correction with thin planar connectivity (2109.14609v1)

Abstract: Quantum LDPC codes may provide a path to build low-overhead fault-tolerant quantum computers. However, as general LDPC codes lack geometric constraints, na\"ive layouts couple many distant qubits with crossing connections which could be hard to build in hardware and could result in performance-degrading crosstalk. We propose a 2D layout for quantum LDPC codes by decomposing their Tanner graphs into a small number of planar layers. Each layer contains long-range connections which do not cross. For any CSS code with a degree-$\delta$ Tanner graph, we design stabilizer measurement circuits with depth at most $(2\delta +2)$ using at most $\lceil \delta/2 \rceil$ layers. We observe a circuit-noise threshold of 0.28\% for a positive-rate code family using 49 physical qubits per logical qubit. For a physical error rate of $10^{-4}$, this family reaches a logical error rate of $10^{-15}$ using fourteen times fewer physical qubits than the surface code.

• The paper introduces a crossing-free planar layout that decomposes Tanner graphs into non-crossing layers for efficient quantum LDPC code implementation.
• It designs stabilizer measurement circuits with constant depth (at most 2δ+2) to balance connectivity simplicity and error correction performance.
• Numerical simulations demonstrate a circuit-noise threshold of 0.28% and a logical error rate of 10⁻¹⁵ with significantly fewer physical qubits than surface codes.

Constant-Overhead Quantum Error Correction with Thin Planar Connectivity

The paper "Constant-overhead Quantum Error Correction with Thin Planar Connectivity" presents a novel approach to implementing quantum low-density parity-check (LDPC) codes in a manner that is more hardware-friendly and less error-prone. Quantum LDPC codes are attractive for quantum error correction (QEC) because they offer the potential for low overhead in terms of the number of physical qubits required per logical qubit, yet they pose practical challenges due to their non-local stabilizer generators. The authors address these challenges by proposing a crossing-free planar layout of quantum LDPC codes, aiming to overcome hardware difficulties and reduce performance-degrading crosstalk.

Key Results and Contributions

1. Planar Layout Using Multiple Layers: The paper introduces a 2D qubit layout for quantum LDPC codes by decomposing Tanner graphs into a small number of non-crossing planar layers. Each layer contains long-range connections that do not cross each other, allowing physical implementation with minimal interference among qubits.
2. Stabilizer Measurement Circuits: For any CSS code with a degree-$\delta$ Tanner graph, the authors design stabilizer measurement circuits with depth at most $(2\delta + 2)$ using at most $\lceil \delta/2 \rceil$ layers. This approach achieves a balance between connectivity simplicity and error correction efficacy, facilitating implementations with constant-depth circuits.
3. Numerical Simulation and Performance Analysis: The authors provide numerical evidence of a circuit-noise threshold of 0.28% for a positive-rate code family using 49 physical qubits per logical qubit. Compared to the surface code, this code family achieves a logical error rate of $10^{-15}$ using fourteen times fewer physical qubits.
4. Cardinal Circuit and Balanced Ordering: The paper details the construction of the cardinal circuit for hypergraph product codes, emphasizing its reduced circuit depth. For certain graph configurations, balanced ordering enables more efficient direction assignments for connectivity graphs, lowering the gates' operation depth significantly.

Implications and Future Directions

The paper addresses a crucial aspect of quantum computing architecture—efficient error correction with minimal hardware overhead. By tackling the non-local nature of stabilizer generators in positive-rate LDPC codes, it opens avenues for advancements in fault-tolerant quantum computers.

The paper suggests future exploration of quantum hardware design that accommodates long-range connections within layers. Challenges include not only building such connectivity but also minimizing crosstalk, which often arises from proximity or overlapping physical layers. The approach may also inspire future research on optimizing planar graph algorithms or developing improved decoders that further lower logical error rates.

In conclusion, this research contributes significantly to the practical implementation of quantum LDPC codes, potentially reducing the gap between theoretical error correction architectures and realizable quantum computers. As the field progresses, these advancements will be instrumental in achieving scalable quantum computing technologies.