Charge-$4e$ superconductor with parafermionic vortices: A path to universal topological quantum computation

Abstract

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 $\mathbb{Z}_3$ 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 $\mathbb{Z}_3$ 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 simple single-probe interferometric measurement enables topologically protected magic-state preparation, promoting Clifford operations to a universal gate set. Importantly, the non-Abelian excitations in the $4e$ TSC are confined to externally controlled defects, making them uniquely identifiable and amenable to controlled creation and motion with superconducting-circuit technology. Our results establish hierarchical electron aggregation as a complementary principle for engineering topological quantum matter with enhanced computational power.

Figures (4)

• Figure 1: An illustration of vortex-antivortex binding mediated by inter-component current-current coupling between two copies of $p+ip$ TSCs. The condensation of this composite object leads to the $4e$ TSC in \ref{['eq:L_pattern_1']}, in which superconductivity coexists with a bosonic chiral $\mathbb{Z}_3$ topological order enriched by the remnant $\mathbb{Z}_4$ charge symmetry.
• Figure 2: (Upper Left) An illustration of the anyon transmutation effect, in which an Abelian anyon $a$ winding around an elementary $\mathbb{Z}_4$ defect $\sigma$ induced by $hc/(4e)$ flux gets acted on by $\rho_{\boldsymbol{g}}$ at the branch cut of phase field, and transmutes into its conjugate partner $\bar{a}$. (Upper Right) The qutrit encoding using the fusion space of four $hc/(4e)$ defects split from vacuum, where $a=0,1,2$ represents the topological charge of the Abelian anyon fused from the first two $\sigma$'s. (Lower) A path of the probe defect $\sigma_p$, which induces a transformation $|a\rangle\leftrightarrow |\bar{a}\rangle$ of the qutrit.
• Figure 3: In non-Abelian topological orders, anyon condensation can lead to the splitting of non-Abelian anyons into descendent Abelian anyons. Here we illustrate this phenomenon through the example of $SU(2)_4/\mathbb{Z}_2$. For each cylindrical surface with radius $R$, the interior is $SU(2)_4$ while the exterior is $SU(2)_4/\mathbb{Z}_2$ (with the $j =2$ anyon condensed). $W_j$ labels the Wilson line with spin $j$ in $SU(2)_4$.
• Figure 4: An illustration of the interferometry measurement protocol. Each yellow, rounded square (labeled by $F$) represents a fluxonium (essentially a $4e$ superconducting ring, which doesn't need to be topological) sitting above the plane, which have two meta-stable states: with or without an $hc/(4e)$ magnetic flux. The state with an $hc/(4e)$ magnetic flux with induce a $\sigma$ mode in the adjacent $4e$ TSC thin film. The protocol requires one to prepare a pair of fluxoniums in a Bell state, and then send the two fluxoniums in different paths respectively--reference (R) and probing (P) paths--and eventually measure the outcome in the Bell basis. Two possible probing paths, $P_{1,2}$, are plotted.