[Paper Review] Physical Architecture for a Universal Topological Quantum Computer based on a Network of Majorana Nanowires

Maissam Barkeshli, Jay D. Sau|arXiv (Cornell University)|Sep 23, 2015

Neural Networks and Applications20 citations

TL;DR

This paper proposes a physical architecture for a universal topological quantum computer using a network of semiconductor-superconductor nanowires hosting Majorana zero modes. By engineering a two-dimensional Ising×Ising topological state via coupled nanowires, the system realizes genons—non-Abelian defects that enable topologically protected braiding operations, including the missing π/8 phase gate, thus achieving universal quantum computation with fault-tolerant operations within experimentally accessible energy scales.

ABSTRACT

The idea of topological quantum computation (TQC) is to store and manipulate quantum information in an intrinsically fault-tolerant manner by utilizing the physics of topologically ordered phases of matter. Currently, one of the most promising platforms for a topological qubit is in terms of Majorana fermion zero modes (MZMs) in spin-orbit coupled superconducting nanowires. However, the topologically robust operations that are possible with MZMs can be efficiently simulated on a classical computer and are therefore not sufficient for realizing a universal gate set for TQC. Here, we show that an array of coupled semiconductor-superconductor nanowires with MZM edge states can be used to realize a more sophisticated type of non-Abelian defect: a genon in an Ising $ imes$ Ising topological state. This leads to a possible implementation of the missing topologically protected $π/8$ phase gate and thus universal TQC based on semiconductor-superconductor nanowire technology. We provide detailed numerical estimates of the relevant energy scales, which we show to lie within accessible ranges.

Motivation & Objective

• To address the gap in topological quantum computation: MZM-based systems can only perform Clifford group operations, which are classically simulatable and insufficient for universality.
• To demonstrate that a network of coupled Majorana nanowires can realize a two-dimensional Ising×Ising topological order with intrinsic topological order.
• To show that genons—non-Abelian defects in the Ising×Ising state—can be engineered in this network to enable topologically protected braiding operations.
• To realize the missing topologically protected π/8 phase gate via genon braiding, completing the universal gate set for topological quantum computation.
• To provide detailed numerical estimates of energy scales, confirming feasibility within current experimental parameters.

Proposed method

• Engineer an effective Kitaev honeycomb spin model in a network of coupled Majorana nanowires, where each spin corresponds to a pair of Majorana zero modes.
• Use controlled tunneling and tuning of hopping amplitudes to realize a topological phase with Ising×Ising order, supporting non-Abelian genons.
• Implement measurement-based braiding of genons via sequential projections of topological charge through loops in the system.
• Control the fusion channel of genons by tuning tunneling amplitudes tσ and tψ, ensuring only desired fusion outcomes (e.g., (σ,σ) or (ψ,ψ)) are accessible.
• Utilize gapped boundaries in an Ising×Ising or Ising×Īsing system to emulate higher-genus topological surfaces, enabling Dehn twist-like operations.
• Adjust local couplings to suppress the energy gap along specific paths, enabling controlled quasiparticle tunneling and effective anyon braiding.

Experimental results

Research questions

• RQ1 Can a network of coupled Majorana nanowires realize a two-dimensional topological phase with Ising×Ising order and support non-Abelian genons?
• RQ2 Can the braiding of genons in this system implement the topologically protected π/8 phase gate required for universality in topological quantum computation?
• RQ3 What are the experimentally accessible energy scales for the relevant tunneling and pairing terms in the proposed architecture?
• RQ4 How can measurement-based braiding protocols be adapted to realize universal gate operations in a network of nanowires?
• RQ5 Can the Ising×Īsing topological state with gapped boundaries serve as an equivalent platform for universal TQC, preserving topological protection?

Key findings

• The network of coupled Majorana nanowires realizes a two-dimensional Ising×Ising topological order, supporting non-Abelian genons as emergent defects.
• Genon braiding operations can be mapped onto the dynamical topology changes required for universal topological quantum computation, as proposed by Bravyi and Kitaev (2000).
• The π/8 phase gate—previously missing in MZM-based systems—can be implemented via topologically protected braiding of genons, completing the universal gate set.
• Numerical estimates show that the relevant energy scales (e.g., tunneling amplitudes tσ, tψ and pairing terms) lie within experimentally accessible ranges, supporting feasibility.
• The system can be equivalently described as an Ising×Īsing state with gapped boundaries, enabling topologically robust operations via measurement-based protocols on genus-g surfaces.
• By tuning local couplings and controlling the gap along specific paths, the fusion channel of genons can be selectively projected, enabling controlled topological operations.