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arXiv:1803.05886 · Physical review. B./Physical review. B

Universal quantum computing with parafermions assisted by a half-fluxon

Arpit Dua, Boris A. Malomed, Meng Cheng, Liang Jiang

Published Oct 17, 2019• 12 citations
arXivHTMLPDF

Abstract

Braiding of anyons such as Majoranas or parafermions provides only Clifford gates which do not form a universal set of quantum gates. We propose a robust and resource-efficient scheme to perform non-Clifford gates on logical qudits encoded in parafermionic zero modes via the Aharonov-Casher effect. This gate can be implemented by moving a half-flux quantum around the pair of parafermionic zero modes. The parafermion modes can be realized in a two-dimensional setup via existing proposals, and a half-fluxon can be created as a part of half-fluxon/anti-half-fluxon pair in a spin-triplet Josephson junction. We provide evidence that the half-fluxon can be braided robustly around the parafermions and hence this is a reliable proposal for the implementation of the non-Clifford gate without magic state distillation. Supplementing this gate with the braiding of parafermions provides the avenue for universal quantum computing with parafermions.

DOI
10.1103/physrevb.100.144508

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Braiding of anyons such as Majoranas or parafermions provides only Clifford gates which do not form a universal set of quantum gates. We propose a robust and resource-efficient scheme to perform non-Clifford gates on logical qudits encoded in parafermionic zero modes via the Aharonov-Casher effect. This gate can be implemented by moving a half-flux quantum around the pair of parafermionic zero modes. The parafermion modes can be realized in a two-dimensional setup via existing proposals, and a half-fluxon can be created as a part of half-fluxon/anti-half-fluxon pair in a spin-triplet Josephson junction. We provide evidence that the half-fluxon can be braided robustly around the parafermions and hence this is a reliable proposal for the implementation of the non-Clifford gate without magic state distillation. Supplementing this gate with the braiding of parafermions provides the avenue for universal quantum computing with parafermions.
Abstract Majorana fermions and their generalizations to Zn parafermions are considered promising building blocks of fault-tolerant quantum computers for their ability to encode quantum information nonlocally. In such topological quantum computers, highly robust quantum gates are obtained by braiding pairs of these quasi-particles. However, it is well-known that braiding Majorana fermions or parafermions only leads to a Clifford gate, hindering quantum universality. This paper establishes an exact mapping between Majorana fermions to Z4 parafermions in systems under total parity nonconserving and total parity conserving setting. It is revealed that braiding of Majorana fermions may lead to non-Clifford quantum gates in the 4-dimensional qudit representation spanned by Z4 parafermions, whilst braiding of Z4 parafermions may similarly yield non-Clifford quantum gates in the qubit representation spanned by Majorana fermions. This finding suggests that topologically protected universal quantum computing may be possible with Majorana fermions (Z4 parafermions) by supplementing the usual braiding operations with the braiding of Z4 parafermions (Majorana fermions) that could be formed out of Majorana fermions (Z4 parafermions) via the mapping prescribed here. Finally, the paper discusses how braiding of Majorana fermions or Z4 parafermions could be obtained via a series of parity measurements.
Parafermions are \nZ \nn \n generalizations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. Here we investigate the generation of quantum gates by allowing \nZ \n3 \n parafermions to interact in order to achieve universality. In particular, we study the form of the nontopological gate that arises through direct short-range interaction of the parafermion edge modes in a \nZ \n3 \n parafermion chain. We show that such an interaction gives rise to a dynamical phase gate on the encoded ground space, generating a non-Clifford gate which can be tuned to belong to even levels of the Clifford hierarchy. We illustrate how to access highly noncontextual states using this dynamical gate. Finally, we propose an experiment that simulates the braiding and dynamical evolutions of the \nZ \n3 \n topological states with Rydberg atom technology.
Parafermions are Zn{\mathbb{Z}}_{n}Zn​ generalizations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. Here we investigate the generation of quantum gates by allowing Z3{\mathbb{Z}}_{3}Z3​ parafermions to interact in order to achieve universality. In particular, we study the form of the nontopological gate that arises through direct short-range interaction of the parafermion edge modes in a Z3{\mathbb{Z}}_{3}Z3​ parafermion chain. We show that such an interaction gives rise to a dynamical phase gate on the encoded ground space, generating a non-Clifford gate which can be tuned to belong to even levels of the Clifford hierarchy. We illustrate how to access highly noncontextual states using this dynamical gate. Finally, we propose an experiment that simulates the braiding and dynamical evolutions of the Z3{\mathbb{Z}}_{3}Z3​ topological states with Rydberg atom technology.
Charge-4e4e4e superconductor with parafermionic vortices: A path to universal topological quantum computation
Topological superconductors (TSCs) provide a promising route to fault-tolerant quantum information processing. However, the canonical Majorana platform based on 2e2e2e TSCs remains computationally constrained. In this work, we find a 4e4e4e TSC that overcomes these constraints by combining a charge-4e4e4e condensate with an Abelian chiral Z3\mathbb{Z}_3Z3​ topological order in an intertwined fashion. Remarkably, this 4e4e4e TSC can be obtained by proliferating vortex-antivortex pairs in a stack of two 2e2e2e p+ipp+ipp+ip TSCs, or by melting a ν=2/3ν=2/3ν=2/3 quantum Hall state. Specific to this TSC, the hc/(4e)hc/(4e)hc/(4e) fluxes act as charge-conjugation defects in the topological order, whose braiding with anyons transmutes anyons into their antiparticles. This symmetry enrichment leads to Z3\mathbb{Z}_3Z3​ parafermion zero modes trapped in the elementary vortex cores, which naturally encode qutrits. Braiding the parafermion defects alone generates the full many-qutrit Clifford group. We further show that a single-probe interferometric measurement enables topologically protected magic-state preparation, promoting Clifford operations to a universal gate set. Because the non-Abelian modes are bound to flux defects, they can, in principle, be externally controlled using superconducting circuit-based technology. More broadly, our results highlight hierarchical electron aggregation, the formation and condensation of higher-charge electron clusters, as a design principle for topological quantum matter with increased computational capability.
We propose and analyze a physical system capable of performing topological quantum computation with Majorana zero modes (MZMs) in a one-dimensional topological superconductor (1DTS). One of the leading methods to realize quantum gates in a 1DTS is to use T junctions, which allows one to maneuver MZMs in such a manner as to achieve braiding. In this paper, we propose a scheme for implementing quantum logical gates in a purely one-dimensional geometry without T junctions, instead replacing it with an auxiliary qubit. This has the additional benefit of introducing a non-Clifford gate, corresponding to one- and two-logical-qubit ZZZ rotations. We first design a topologically protected logical ZZZ gate based entirely on local interactions within the 1DTS. Using an auxiliary qubit coupled to the topological superconductors, we extend the ZZZ gate to non-Clifford single- and multiqubit arbitrary rotations with partial topological protection. Finally, to perform universal quantum computing, we introduce a scheme for performing arbitrary unitary rotations, albeit without topological protection. We develop a formalism based on unitary braids which creates transitions between different topological phases of the 1DTS system. The unitary formalism can be simply converted to an equivalent adiabatic scheme, which we numerically simulate, and we show that high-fidelity operations should be possible with reasonable parameters.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3S_3S3​, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3S_3S3​ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.