Session J26: Focus Session: Topological Quantum Computing

Topological Quantum Computation and Measurement

Topologically ordered phases of matter possess ``anyons'' - quasiparticle excitations with exotic exchange statistics. Anyonic systems can potentially be exploited to provide an intrinsically fault-tolerant medium for quantum information processing. I will explain how this topological approach to quantum computation works and examine the role of measurements in this scheme.   

Implementing arbitrary phase gates with non-Abelian anyons in the 5/2 fractional quantum Hall state

Recent experiments~[1] suggest that the elementary excitations in the 5/2 fractional quantum Hall (FQH) state are non-Abelian quasiparticles. It therefore may be possible to store and manipulate quantum information in this system in a non-local (and therefore fault-tolerant) fashion~[2]. Unfortunately, the single-qubit phase gate necessary for universal quantum computation is not an available protected operation. Nonetheless, a noisy phase gate could suffice, as it may be used with protected operations to produce a gate of higher accuracy~[3]. However, previous suggestions for the implementation of the phase gate are problematic due to their necessity for fine control over the motion of the quasiparticles. We propose a device that would implement a single-qubit phase gate by running a current of anyons through interfering paths around the computational anyons. While the resulting operation is not topologically protected, it is predicted to fall within the error correction threshold for physically realistic parameters. Supported in part by DARPA-QuEST and NSF grant DMR 0748925. \\ ~[1] R.~L. Willett et al., arXiv:0807.0221v3 \\ ~[2] S. Das Sarma et al., Phys. Rev. Lett. 94, 166802 (2005). \\ ~[3] S. Bravyi, Phys. Rev. A 73, 042313 (2006)   

Unification of Universal and Non-Universal Topological Quantum Computation

In topological quantum computation, information is encoded non-locally in exotic quasiparticles called anyons, and quantum gates are carried out by braiding the anyons in (2+1)-dimensional space-time. Universal topological quantum computation can be carried out by a universal set of quantum gates composed of single-qubit gates and controlled-phase gate and more efficiently together with controlled-controlled-phase gate. In this talk we show a unified scheme of encoding and computing for both universal and non-universal topological quantum computation with SU(2)$_k$ anyon models. And we give explicit construction of quantum gates, especially the controlled-controlled-phase gate, for universal topological quantum computation. The unified scheme can also be used to generalize the non-universal topological quantum computation to arbitrarily-many-qubit system. Besides, we will discuss error detection and correction in topological quantum computation.   

Exact results on the entanglement entropy of Abelian and non-Abelian topological states

Quantum entanglement is essential for the purpose of quantum computing. The Kitaev model defined on trivalent lattices is one of the most important prototype models for topological quantum computation, which supports quasi-particle excitations with non-Abelian statistics. In this paper, we obtain an exact formula for the entanglement entropy of the ground state and all excited states with or without vison excitations of the Kitaev model. Interestingly, the entanglement entropy can be expressed in a simple separable form S=S$_G$+S$_F$, with S$_F$ the entanglement entropy of a free Majorana fermion (``matter field'') system and S$_G$ that of Z$_2$ gauge fields. The Z$_2$ gauge field part contributes the universal ``topological entanglement entropy'' of the ground state while the fermion part is responsible for the non-local entanglement carried by the quasi-particles in the non-Abelian phase. We also studied the Reyni entropy of this system which distinguishes the Abelian and nonAbelian states. Our approach to the entanglement entropy can be generalized to a broad class of many-body wave functions.   

Topological Quantum Computing with p-Wave Superfluid

It is shown that Majorana fermions trapped in three p-wave superfluid vortices form a qubit in a topological quantum computing (TQC). Several similar ideas have already been proposed, in which a qubit operation is performed by braiding the world lines of these two or four Majorana fermions. Naturally the set of quantum gates thus obtained is a discrete subset of the relevant unitary group. We propose a new scheme, where three Majorana fermions form a qubit. We show that continuous qubit operations are made possible by braiding the Majorana fermions complemented with dynamical phase factors. Furthermore, it is possible to introduce entanglement between two such qubits by geometrical manipulation of some vortices involved.   

A generic new platform for topological quantum computation using semiconductor heterostructures

We show that a film of a semiconductor in which $s$-wave superconductivity and a Zeeman splitting are induced by proximity effect, supports zero-energy Majorana fermion modes in the ordinary vortex excitations. Since time reversal symmetry is explicitly broken, the edge of the film constitutes a chiral Majorana wire. The heterostructure we propose -- a semiconducting thin film sandwiched between an $s$-wave superconductor and a magnetic insulator -- is a generic system which can be used as the platform for topological quantum computation by virtue of the existence of non-Abelian Majorana fermions.   

A theorem for the existence of Majorana fermion modes in spin-orbit-coupled semiconductors

We prove a theorem for the existence of Majorana zero modes in a semiconducting thin film with a sizable spin-orbit coupling when it is adjacent to a $s$-wave superconductor. The theorem, which is analogous to the Jackiw-Rebbi index theorem for the zero modes in mass domain walls in one-dimensional Dirac theory, applies to vortices with odd flux quantum in a semiconducting film in which $s$-wave superconductivity and a Zeeman splitting are induced by proximity effect. Thus the theorem proves the existence of non-degenerate zero-energy Majorana excitations and the resultant non-Abelian topological order in the proposed semiconductor heterostructure.   

Stroboscopic Generation of Topological Protection

Trapped neutral atoms offer a powerful route to robust simulation of complex quantum systems. We present here a stroboscopic scheme for realization of a Hamiltonian with $n$-body interactions on a set of neutral atoms trapped in an addressable optical lattice, using only 1- and 2-body physical operations together with a dissipative mechanism that allows thermalization to finite temperature or cooling to the ground state. We demonstrate this scheme with application to the toric code Hamiltonian, ground states of which can be used to robustly store quantum information when coupled to a low temperature reservoir.   

Quantum noise and entanglement generated by a local quantum quench

We examine the growth of entanglement under a quantum quench at point contacts of simple fractional quantum Hall fluids and its relation with the measurement of local observables. Recently Klich and Levitov proposed that the noise generated from a local quantum quench provides a measure of the entanglement entropy. Their methods were specific to non-interacting electrons and the generalization to interacting systems was left as an open question. In this work, we generalize their result to the Laughlin states. We investigate the noise generated in the current along the edge of a fractional quantum Hall state at filling factors $\nu=1/m$, when a quantum point contact, initially closed, is fully opened at some initial time $t_0=0$. We find that local quenching in these systems gives time dependent correlation functions that have universal behavior on sufficiently long time and length scales. We calculate the noise and full counting statistics for $\nu=1/m$ and find that in general, the entanglement entropy and noise generated are unrelated quantities. We also discuss a generalization of this problem to the critical quantum Ising spin chain.   

A Realization of Quantum Non-Local Persistent Current: Coupled Metallic Rings through a P-Wave Chiral Superconducting Wire

The excitations in the p-wave superconductors $Sr_{2}Ru O_{4}$, $^{3}He-A$ and the $\nu=\frac{5}{2}$ quantum Hall effect are characterized by half vortices, which are zero mode energy Majorana fermions. We consider a p-wave superconducting wire for which the pairing order parameter vanishes at the edges and two zero modes appear at $x=0$ and $x=L$. Due to the charge conjugation of the Bogoliubov spectrum these zero modes are Majorana Fermions. The ground state is a $Z_{2}$ doubly degenerate state and the single particle excitations are non-local. We couple the p-wave wire to two rings, which are pierced by external fluxes, and compute persistent current. The non-locality of the persistent current is manifested in the following way: (a) For a single ring with flux coupled to the p-wave wire at $x=0$ and grounded at $x=L$, due to the Andreev reflection the current in one ring will decay faster than if we decouple the wire. (b) For two rings with equal fluxes attached to the p-wave wire at $x=0$ and the second at $x=L$ the current will be equal and independent of the length of the wire. When the flux is $\pi$ we will have a Berry phase of $\pi$. (c) For dufferent fluxes the currents in the two rings will vanish. This situation is very different if instead of a p-wave wire we attach a metallic wire for which the current will be uncorrelated in the two rings.   

Topological qubits in graphene-like systems

The fermion-doubling problem can be an obstacle to getting half-a-qubit in two-dimensional fermionic tight-binding models in the form of Majorana zero modes bound to the core of superconducting vortices. We argue that the number of such Majorana zero modes is determined by a $Z_{2} \times Z_{2}$ topological charge for a family of two-dimensional fermionic tight-binding models ranging from noncentrosymmetric materials to graphene. This charge depends on the dimension of the representation (i.e., the number of species of Dirac fermions -- where the doubling problem enters) and the parity of the Chern number induced by breaking time-reversal symmetry. We show that in graphene there are as many as 10 order parameters that can be used in groups of 4 to change the topological number from even to odd.   

Non-Abelian anyons: when Ising meets Fibonacci

We analyze the edge lying on the interface of two non-Abelian quantum Hall states: the Moore-Read spin-polarized state at filling factor 1/2, supporting Ising anyons, and the non-Abelian spin-singlet state at filling factor 4/7, supporting Fibonacci anyons. We find that the neutral sector of the edge is described by a minimal model with central charge 7/10. We explore the role of the edge as a mediator between regions of different quantum statistics.   

Non-Abelian Quantum Hall States and their Quasiparticles: from the Pattern of Zeros to Vertex Algebra

In the pattern-of-zeros approach to quantum Hall states, a set of data obtained from fusing electrons is introduced to characterize a quantum Hall wavefunction, i.e. to classify different topological orders. In this work we combine (associative) vertex algebra with the pattern of zeros to classify different quantum Hall states. We find sufficient conditions on the pattern of zeros so that the data correspond to a valid wavefunction. We also show that an expanded set of data, i.e. the pattern of zeros together with the central charge of the vertex algebra, provides a more complete characterization of quantum Hall states and corresponding topological orders. This more complete characterization allows us to construct quantum Hall wavefunctions, and to obtain more topological properties of those states, such as the scaling dimensions and the statistics of quasiparticle excitations.