We present a constructive proof that anyonic magnetic charges with fluxes in a nonsolvable finite group can perform universal quantum computations. The gates are built out of the elementary operations of braiding, fusion, and vacuum pair creation, supplemented by a reservoir of ancillas of known flux. Procedures for building the ancilla reservoir and for correcting leakage are also described. Finally, a universal qudit gate set, which is ideally suited for anyons, is presented. The gate set consists of classical computation supplemented by measurements of the X operator.
We investigate the capability of multiple-braiding in
SU(2)k anyon models to realize universal quantum computation. The multiple elementary braiding matrices (MEBMs) are derived from the
q-deformed representation theory of
SU(2). Through a general analysis of the MEBMs, we find that multiple-braiding loses its universality only when the braiding multiplicity
m causes the MEBMs to collapse to a scalar (up to a global phase). Our numerical analysis of the MEBMs of
SU(2)k anyon models with
k=3,5,6,7, and
m∈[2,9], shows that the values of
m at which the braid group representation fails to be dense agree exactly with the theoretical prediction. For those
m that support universality, single-qubit gates are compiled to high precision by a genetic algorithm-enhanced Solovay-Kitaev algorithm (GA-enhanced SKA), and a genetic algorithm (GA) search with progressively increasing braid length yields an approximately the local equivalence class
[CNOT]. Notably, even-order braiding operations offer a physical advantage by reducing the number of non-Abelian anyons required in braiding-based topological quantum computation (TQC). Our analytical and numerical results together provide strong evidence for identifying which braiding multiplicities
m support universal quantum computation in
SU(2)k anyon models.
A class of anyonic models for universal quantum computation based on weakly-integral anyons has been recently proposed. While universal set of gates cannot be obtained in this context by anyon braiding alone, designing a certain type of sector charge measurement provides universality. In this paper we develop a compilation algorithm to approximate arbitrary
n-qutrit unitaries with asymptotically efficient circuits over the metaplectic anyon model. One flavor of our algorithm produces efficient circuits with upper complexity bound asymptotically in $O({3}^{2\phantom{\rule{0.16em}{0ex}}n}\phantom{\rule{0.16em}{0ex}}log1/\ensuremath{\varepsilon})$ and entanglement cost that is exponential in
n. Another flavor of the algorithm produces efficient circuits with upper complexity bound in $O(n\phantom{\rule{0.16em}{0ex}}{3}^{2\phantom{\rule{0.16em}{0ex}}n}\phantom{\rule{0.16em}{0ex}}log1/\ensuremath{\varepsilon})$ and no additional entanglement cost.
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,
S3, 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
S3 TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
We introduce and study a class of anyon models that are a natural generalization of Ising anyons and Majorana fermion zero modes. These models combine an Ising anyon sector with a sector associated with
SO(m)2 Chern-Simons theory. We show how they can arise in a simple scenario for electron fractionalization and give a complete account of their quasiparticle types, fusion rules, and braiding. We show that the image of the braid group is finite for a collection of
2n fundamental quasiparticles and is a proper subgroup of the metaplectic representation of $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})\ensuremath{\ltimes}H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$, where $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the symplectic group over the finite field
Fm and $H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the extra special group (also called the $(2n\ensuremath{-}1)$-dimensional Heisenberg group) over
Fm. Moreover, the braiding of fundamental quasiparticles combined with a restricted set of measurements can be efficiently simulated classically. However, computing the result of braiding a certain type of composite quasiparticle followed by fusion into the identity is $#P$-hard. It is not universal for quantum computation because it has a finite braid group image. This is a rare example of a topological phase that is not universal for quantum computation through braiding but nevertheless has $#P$-hard link invariants. We argue that our models are closely related to recent analyses finding non-Abelian anyonic properties for defects in quantum Hall systems, generalizing Majorana zero modes in quasi-1D systems.
We examine a class of operations for topological quantum computation based on fusing and measuring topological charges for systems with
SU(2)4 or
k=4 Jones-Kauffman anyons. We show that such operations augment the braiding operations, which, by themselves, are not computationally universal. This augmentation results in a computationally universal gate set through the generation of an exact, topologically protected irrational phase gate and an approximate, topologically protected controlled-
Z gate.