Braiding by Majorana Tracking and Long-Range CNOT Gates with Color Codes

TL;DR

This paper proposes a software-based Majorana tracking method for implementing color codes in Majorana nanowire arrays, enabling fault-tolerant quantum computation with simplified hardware and efficient long-range CNOT gates.

Contribution

It introduces a Majorana tracking protocol that replaces physical braiding, simplifying hardware requirements and enabling long-range gates via lattice surgery in color code architectures.

Findings

Majorana tracking reduces hardware operations for Clifford gates.

Long-range CNOT gates scale logarithmically with distance.

The approach is compatible with various topological qubit systems.

Abstract

Color-code quantum computation seamlessly combines Majorana-based hardware with topological error correction. Specifically, as Clifford gates are transversal in two-dimensional color codes, they enable the use of the Majoranas' nonabelian statistics for gate operations at the code level. Here, we discuss the implementation of color codes in arrays of Majorana nanowires that avoid branched networks such as T-junctions, thereby simplifying their realization. We show that, in such implementations, nonabelian statistics can be exploited without ever performing physical braiding operations. Physical braiding operations are replaced by Majorana tracking, an entirely software-based protocol which appropriately updates the Majoranas involved in the color-code stabilizer measurements. This approach minimizes the required hardware operations for single-qubit Clifford gates. For Clifford completeness, we combine color codes with surface codes, and use color-to-surface-code lattice surgery for long-range multi-target CNOT gates which have a time overhead that grows only logarithmically with the physical distance separating control and target qubits. With the addition of magic state distillation, our architecture describes a fault-tolerant universal quantum computer in systems such as networks of tetrons, hexons, or Majorana box qubits, but can also be applied to non-topological qubit platforms.