The modern conception of phases of matter has undergone tremendous developments since the first observation of topologically ordered state in fractional quantum Hall systems in the 1980s. Topological orders are exotic states of matter characterized by patterns of long-range entanglement that lie beyond Landau's symmetry breaking paradigm. In 2+1 dimensions, these phases support signature anyonic excitations with fractional statistics or even non-Abelian braiding effects. One important theoretical question is how we should characterize these phases. In 2+1 dimensions, plenty of evidence suggests that the \emph{modular matrices}, S and T, suffice to characterize the order completely. It is thus of equal importance to find out if these quantities are experimental observables. Here we demonstrate how such measurements can indeed be achieved using only the given Hamiltonian, allowing one to identify the corresponding topological order using a nuclear magnetic resonance simulator. This is a first experimental realization of these fundamental signatures of a topological order; it opens up new future avenues toward identifying more generic topological orders based purely on experimental measurements.
The modern conception of phases of matter has undergone tremendous developments since the first observation of topologically ordered state in fractional quantum Hall systems in the 1980s. Topological orders are exotic states of matter characterized by patterns of long-range entanglement that lie beyond Landau's symmetry breaking paradigm. In 2+1 dimensions, these phases support signature anyonic excitations with fractional statistics or even non-Abelian braiding effects. One important theoretical question is how we should characterize these phases. In 2+1 dimensions, plenty of evidence suggests that the \emph{modular matrices}, S and T, suffice to characterize the order completely. It is thus of equal importance to find out if these quantities are experimental observables. Here we demonstrate how such measurements can indeed be achieved using only the given Hamiltonian, allowing one to identify the corresponding topological order using a nuclear magnetic resonance simulator. This is a first experimental realization of these fundamental signatures of a topological order; it opens up new future avenues toward identifying more generic topological orders based purely on experimental measurements.
ABSTRACT Topological order is a new quantum phase that is beyond Landau’s symmetry-breaking paradigm. Its defining features include robust degenerate ground states, long-range entanglement and anyons. It was known that R and F matrices, which characterize the fusion-braiding properties of anyons, can be used to uniquely identify topological order. In this article, we explore an essential question: how can the R and F matrices be experimentally measured? We show that the braidings, i.e. the R matrices, can be completely determined by the half braidings of boundary excitations due to the boundary-bulk duality and the anyon condensation. The F matrices can also be measured by comparing the quantum states involving the fusion of three anyons in two different orders. Thus we provide a model-independent experimental protocol to uniquely identify topological order. By using quantum simulations based on a toric code model with boundaries encoded in three- and four-qubit systems and state-of-the-art technology, we obtain the first experimental measurement of R and F matrices by means of an NMR quantum computer at room temperature.
Topological orders can be used as media for topological quantum computing-a promising quantum computation model due to its invulnerability against local errors. Conversely, a quantum simulator, often regarded as a quantum computing device for special purposes, also offers a way of characterizing topological orders. Here, we show how to identify distinct topological orders via measuring their modular S and T matrices. In particular, we employ a nuclear magnetic resonance quantum simulator to study the properties of three topologically ordered matter phases described by the string-net model with two string types, including the Z_{2} toric code, doubled semion, and doubled Fibonacci. The third one, non-Abelian Fibonacci order is notably expected to be the simplest candidate for universal topological quantum computing. Our experiment serves as the basic module, built on which one can simulate braiding of non-Abelian anyons and ultimately, topological quantum computation via the braiding, and thus provides a new approach of investigating topological orders using quantum computers.
A topological order is a new quantum phase that is beyond Landau's symmetry-breaking paradigm. Its defining features include robust degenerate ground states, long-range entanglement and anyons. It was known that R- and F-matrices, which characterize the fusion-braiding properties of anyons, can be used to uniquely identify topological order. In this article, we explore an essential question: how can the R- and F-matrices be experimentally measured? By using quantum simulations based on a toric code model with boundaries and state-of-the-art technology, we show that the braidings, i.e. the R-matrices, can be completely determined by the half braidings of boundary excitations due to the boundary-bulk duality and the anyon condensation. The F-matrices can also be measured in a scattering quantum circuit involving the fusion of three anyons in two different orders. Thus we provide an experimental protocol for measuring the unique identifiers of topological order.
After the discovery of fraction quantum Hall states in the 1980s,[1] it became more and more clear that Landau symmetry breaking theory does not describe all possible quantum phases of matter. The new quantum phases of matter were called topologically ordered phases[2,3] (for gapped cases) or quantum ordered phases[4] (for gapless cases), which correspond to patterns of many-body entanglement.[5–7] One may wonder: besides quantum Hall systems, are there other systems that realize the new topological/quantum order?
To perform topological quantum computation we first need to experimentally realise anyons in a topological system. These systems are characterised by intriguing non-local quantum correlations that give rise to the anyonic statistics. What are the diagnostic tools we have to identify if a given system is indeed topological or not? Different phases of matter are characterised by their symmetries. This information is captured by order parameters. Usually, order parameters are defined in terms of local operators that can be measured in the laboratory. For example, the magnetisation of a spin system is given as the expectation value of a single spin with respect to the ground state. Such local properties can describe fascinating physical phenomena efficiently, such as ferromagnetism, and can pinpoint quantum phase transitions. But what about topological systems? Experimentally, we usually identify the topological character of systems, such as the fractional quantum Hall liquids, by probing the anyonic properties of their excitations (Miller et al. , 2007). However, topological order should be a characteristic of the ground state (Thouless et al. , 1982; Wen, 1995). The natural question arises: is it possible to identify a property of the ground state of a system that implies anyonic excitations? The theoretical background that made it possible to answer this question came from entropic considerations of simple topological models. Hamma et al . (2005) studied the entanglement entropy of the toric code ground state and noticed an unusual behaviour.
The non-Abelian geometric phases of the robust degenerate ground states were proposed as physically measurable defining properties of topological order in 1990. In this paper we discuss in detail such a quantitative characterization of topological order, using generic Abelian fractional quantum Hall states as examples. We find a way to determine even the pure U(1) phase of the non-Abelian geometric phases. We show that the non-Abelian geometric phases not only contain information about the quasi-particle statistics, they also contain information about the chiral central charge Delta c mod 24 of the edge states, as well as the Hall viscosity. Thus, the non-Abelian geometric phases (both the Abelian part and the non-Abelian part) provide a quite complete way to characterize 2D topological order.