Implementation of Clifford gates in the Ising-anyon topological quantum computer

2008/12/12 by André Ahlbrecht, Andre Ahlbrecht, Lachezar S. Georgiev +1 · 1 voice · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebra over a field #Anyon #Braid group #Clifford algebra #Combinatorics #Group (periodic table) #Ising model #Mathematics #Physics #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum and electron transport phenomena #Quantum computer #Quantum gate #Quantum many-body systems #Quantum mechanics #Qubit #Topological quantum computer #Topology (electrical circuits) #cond-mat.mes-hall #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1103/physreva.79.032311

published as Phys. Rev. A 79, 032311 (2009) · 17 pages, 10 figures, RevTeX

arxiv created 2008/12/12 · arxiv published 2008/12/12 · arxiv updated 2008/12/12 · openalex publication_date 2009/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We give a general proof for the existence and realizability of Clifford gates in the Ising topological quantum computer. We show that all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates. We find that the braiding gates for two qubits exhaust the entire two-qubit Clifford group. Analyzing the structure of the Clifford group for n\ensuremath≥3 qubits we prove that the image of the braid group is a nontrivial subgroup of the Clifford group so that not all Clifford gates could be implemented by braiding in the Ising topological quantum computation scheme. We also point out which Clifford gates cannot in general be realized by braiding.

Citations

• The Finite Simple Groups
• Non-Abelian braid statistics versus projective permutation statistics
• 2n-quasihole states realize 2n−1-dimensional spinor braiding statistics in paired quantum Hall states
• Improved simulation of stabilizer circuits
• Measurement-based quantum computation with cluster states
• Fault-tolerant quantum computation by anyons
• Non-Abelian anyons and topological quantum computation
• Nonabelions in the fractional quantum hall effect
• Conformal Field Theory

Cited by

• Fibonacci Anyons Versus Majorana Fermions: A Monte Carlo Approach to the Compilation of Braid Circuits in SU(2)k Anyon Models
• Qutrit magic state distillation

Discussions

• I'm having trouble figuring out a good approach to exercise 3.3(d) in Steve Simon's book Topological Quantum involving representations of the braid group on four Ising anyons (described as Majorana fe [stackexchange, 0 points]

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