Impact of non-Abelian anyons on criticality

James R. Wootton

Anyons are quasiparticles that may be realized in two dimensional systems. They come in two types, the simpler Abelian anyons and the more complex non-Abelian anyons. Both of these have been considered as a means for quantum computation, but non-Abelian anyons are usually assumed to be better suited to the task. Here we challenge this view, demonstrating that Abelian anyon models have as much potential as some simple non-Abelian models. First the means to perform quantum computation with Abelian anyon models is considered. These models, like many non-Abelian models, cannot realize universal quantum computation by braiding alone. Non-topological operations must be used in addition, whose complexity depends on the physical means by which the anyons are realized. Here we consider anyons based on spin lattice models, with single spin measurements playing the role of non-topological operations. The computational power achieved by various kinds of measurement is explored and the requirements for universality are determined. The possibility to simulate non-Abelian anyons using Abelian ones is then considered. Finally, a non-Abelian quantum memory is dissected in order to determine the means by which it provides fault-tolerant storage of information. This understanding is then employed to build equivalent quantum memories with Abelian anyon models. The methodology provides with the means to demonstrate that Abelian models have the capability to simulate non-Abelian anyons, and to realize the same computational power and fault-tolerance as non Abelian models. Apart from the intellectual interest in relating topological models with each other, and of understanding the properties of non-Abelian anyons in terms of the simpler Abelian ones, these results can also be applied in the lab. The simpler structure of Abelian anyons means that their physical realization is more straightforward. The demonstration of non-Abelian properties with Abelian models therefore allows features of non-Abelian anyons to be realized with present and near future technology. Based on this possibility, proposals are made here for proof of principle experiments.

Subir Sachdev A

Peter E. Finch, Holger Frahm, Jon Links

Tomer Alkalay, Jinhong Park, Minseong Oh +7 more

Unlike elementary particles, which obey either bosonic or fermionic exchange statistics, certain quasiparticles, known as anyons, are predicted to exhibit Abelian or non-Abelian braiding statistics. While braiding Abelian anyons modifies the wavefunction by a 'statistical phase', braiding non-Abelian anyons implements a unitary transformation of the state within a degenerate subspace of states. Experimental evidence of non-Abelian braiding has thus far remained elusive. Here, we report a 'time-domain braiding' signature of non-Abelian anyons in the $ν= 5/2$ fractional quantum Hall state, by extending our previously demonstrated approach with Abelian anyons at $ν= 1/3$. Our approach is based on measurements of the current fluctuations arising from weak partitioning of a highly dilute one-dimensional edge mode. We independently probe the partition noise of the downstream charged mode and also that of the upstream neutral mode. These independent measurements agree with our theoretical predictions for 'time-domain braiding' of the downstream Abelian and the upstream non-Abelian anyons, respectively, in the 'particle-hole Pfaffian' topological order. Together, these results provide evidence for the presence of non-Abelian anyons.

Zhaohui Zhi, Bo Han, Romain Vasseur +3 more

Non-Abelian anyon chains whose interactions favor a single fusion channel host quantum critical ground states described by minimal-model conformal field theories (CFTs) protected by a topological symmetry. We show that their measurement-only counterpart, where non-commuting fusion charge measurements act locally on the anyons of a chain of $L$ sites, provides a fundamental example of a non-unitary CFT controlling a non-Abelian learning problem. Mapping the monitored dynamics to a random tensor network, we numerically find $1/L$ scaling Lyapunov gaps, logarithmic entanglement scaling with the chord length, and an effective central charge in the Shannon entropy of the measurement record. The conserved topological charge sharpens with dynamical exponent $z= 1$ and is learnable by a decoder on an $O(L)$ time scale. As the topological symmetry is a non-invertible symmetry of the $SU(2)_k$ anyon theories, we establish the monitored golden chain $(k=3)$ as a non-Abelian generalization of the weakly (Kramers-Wannier) self-dual quantum critical point of the toric code $(k=2)$, and the first of a family for $k\geq 3$.

Nico Kirchner, Frank Pollmann, Technische Universität München