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.
Zd parafermions are exotic non-Abelian quasiparticles generalizing Majorana fermions, which correspond to the case
d=2. In contrast to Majorana fermions, braiding of parafermions with
d>2 allows one to perform an entangling gate. This has spurred interest in parafermions, and a variety of condensed matter systems have been proposed as potential hosts for them. In this work, we study the computational power of braiding parafermions more systematically. We make no assumptions on the underlying physical model but derive all our results from the algebraical relations that define parafermions. We find a family of
2d representations of the braid group that are compatible with these relations. The braiding operators derived this way reproduce those derived previously from physical grounds as special cases. We show that if a
d-level qudit is encoded in the fusion space of four parafermions, braiding of these four parafermions allows one to generate the entire single-qudit Clifford group (up to phases), for any
d. If
d is odd, then we show that in fact the entire many-qudit Clifford group can be generated.
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.
Parafermions are Zn generalisations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. We study the simplest case of the Z3 parafermion chain and investigate the form of the non-topological gate that arises through direct short-range interaction of the parafermion edge modes. We show that such an interaction gives rise to a dynamical phase gate on the encoded ground space, with the strongest order of the interaction generating a non-Clifford gate which can be tuned to belong to even levels of the Clifford hierarchy. We also illustrate the accessibility of highly non-contextual states using this dynamical gate. Finally, we propose an experiment that simulates the braiding and dynamical evolutions of the Z3 topological states with Rydberg atom technology.
Parafermions are
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 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 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 topological states with Rydberg atom technology.
Topological superconductors (TSCs) provide a promising route to fault-tolerant quantum information processing. However, the canonical Majorana platform based on
2e TSCs remains computationally constrained. In this work, we find a
4e TSC that overcomes these constraints by combining a charge-
4e condensate with an Abelian chiral
Z3 topological order in an intertwined fashion. Remarkably, this
4e TSC can be obtained by proliferating vortex-antivortex pairs in a stack of two
2e p+ip TSCs, or by melting a
ν=2/3 quantum Hall state. Specific to this TSC, the
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 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.
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.