In topological quantum computation, quantum information is stored in states which are intrinsically protected from decoherence, and quantum gates are carried out by dragging particlelike excitations (quasiparticles) around one another in two space dimensions. The resulting quasiparticle trajectories define world lines in three-dimensional space-time, and the corresponding quantum gates depend only on the topology of the braids formed by these world lines. We show how to find braids that yield a universal set of quantum gates for qubits encoded using a specific kind of quasiparticle which is particularly promising for experimental realization.

Quantitative investigations of the fractional quantum Hall effect ͑FQHE͒ have been limited in the past to systems containing typically fewer than 10-12 particles, except for the 1/(2pϩ1) Laughlin states. We develop a method, using the framework of the composite-fermion theory, that enables a treatment of much bigger systems and makes it possible to obtain accurate quantitative information for other incompressible states as well. After establishing the validity of this method by comparison with few-particle exact-diagonalization results, we compute the ground-state energies and transport gaps for a number of FQHE states.

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A method for compiling quantum algorithms into specific braiding patterns for nonabelian quasiparticles described by the so-called Fibonacci anyon model is developed. The method is based on the observation that a universal set of quantum gates acting on qubits encoded using triplets of these quasiparticles can be built entirely out of three-stranded braids (three-braids). These three-braids can then be efficiently compiled and improved to any required accuracy using the Solovay-Kitaev algorithm.

We demonstrate numerically that non-Abelian quasihole (qh) excitations of the ν = 5/2 fractional quantum Hall state have some of the key properties necessary to support quantum computation. We find that as the qh spacing is increased, the unitary transformation which describes winding two qh's around each other converges exponentially to its asymptotic limit and that the two orthogonal wavefunctions describing a system with four qh's become exponentially degenerate. We calculate the length scales for these two decays to be ξU ≈ 2.7 0 and ξE ≈ 2.3 0 respectively. Additionally we determine which fusion channel is lower in energy when two qh's are brought close together.The proposal to use quantum Hall states as a platform for quantum computation has spurred a great deal of interest 1,2,3 . These quantum Hall systems are believed to have natural "topological" immunity to decoherence and therefore hold particular promise for quantum computation. In so-called non-Abelian quantum Hall systems, the ground state is highly degenerate in the presence of quasiparticles (qp's), and this degenerate space can be used to store quantum information. Operations on this space are then performed by adiabatically dragging qp's around each other, thus "braiding" their world-lines in 2+1 dimensions.Although there is currently no definitive experimental evidence that non-Abelian quantum Hall states even exist, the community now strongly suspects 1 that the quantum Hall plateau observed at Landau level (LL) filling fraction ν = 5/2 is the non-Abelian Moore-Read (MR) phase 4 (or its closely related particle-hole conjugate 5 ). While the MR phase is, strictly speaking, not capable of universal topological quantum computation (computation by braiding qp's around each other at large distances), a scheme has been devised 6 that in principle allows error free quantum computation by supplementing these topological processes with nontopological processes where qp's are moved together and allowed to interact. Furthermore, the MR phase is frequently viewed as the simplest paradigm of a non-Abelian state of matter, and is therefore a logical starting point for detailed analysis 1 . In order for topological (or partially topological) schemes for quantum computation to be scalable (i.e., to allow large scale quantum computation), a number of crucial conditions must hold 1 . Condition (1) As all of the qp's are moved apart from one another, the splitting of the energy levels of the putatively degenerate ground state space must converge to zero at least as fast as e −R/ξ E where R is the minimum distance between qp's. In the literature, there has been numerical work suggesting that condition (1) may not be true 7 for the MR state. One of the goals of our work is to perform more precise numerical calculations to determine whether this numerical conclusion holds up to more careful scrutiny. Condition (2) As qp's are moved apart from each other, the unitary transformation that results from adiabatically dragging one qp around another must converge to its...

One dimensional chains of non-Abelian quasiparticles described by SU (2) k Chern-Simons-Witten theory can enter random singlet phases analogous to that of a random chain of ordinary spin-1/2 particles (corresponding to k → ∞). For k = 2 this phase provides a random singlet description of the infinite randomness fixed point of the critical transverse field Ising model. The entanglement entropy of a region of size L in these phases scales as SL ≃ ln d 3 log 2 L for large L, where d is the quantum dimension of the particles.A particularly exotic form of quantum order is possible in two space dimensions -so-called non-Abelian order [1]. In states with non-Abelian order, when certain localized quasiparticle excitations are present there is a lowenergy Hilbert space whose dimensionality grows exponentially with the number of these quasiparticles. When these quasiparticles are well separated, this low-energy space becomes degenerate, and its states are characterized by purely topological quantum numbers, meaning they cannot be distinguished by local measurements. If these quasiparticles are then adiabatically moved around one another, unitary transformations corresponding to non-Abelian representations of the braid group are carried out on this degenerate space. Aside from their intrinsic scientific interest, recent attention has focused on the possibility of one day using non-Abelian states to perform fault-tolerant quantum computation [2,3].Recently Feiguin et al. [4] have studied models of interacting non-Abelian quasiparticles, specifically uniform chains in which neighboring quasiparticles are close enough together to lift the degeneracy of the topological Hilbert space. In this Letter we study a related class of random interacting chains of non-Abelian quasiparticles. We are motivated both by [4] and by recent work of Refael and Moore [5,6] showing that the entanglement entropy of certain random one-dimensional models scales logarithmically with a universal coefficient. We find the same is true here for an infinite class of models.Exact diagonalization studies [7,8,9] provide compelling evidence that the experimentally observed ν = 5/2 fractional quantum Hall (FQH) state is a nonAbelian state described by the Moore-Read "Pfaffian" state [1]. This state belongs to a wider class of nonAbelian FQH states introduced by Read and Rezayi [10], labeled by index k. In this class, the k = 1 state is an ordinary (Abelian) Laughlin state, the k = 2 state is the Moore-Read state, and all subsequent integer k values describe new non-Abelian states. There is some numerical evidence [10,11] that the k = 3 Read-Rezayi state describes the experimentally observed ν = 12/5 FQH state [12].The quasiparticle excitations of the Read-Rezayi states with index k can be viewed (up to Abelian phases) as par- ticle excitations in SU (2) k Chern-Simons-Witten theory [13]. These particles are characterized by their topological charge, a quantum number which can be viewed as a "q-deformed" spin [14]. At level k, topological charge can take the v...

We find that for the pure Coulomb repulsion the composite Fermi sea at ν = 1/2 is on the verge of an instability to triplet pairing of composite fermions.It is argued that a transition into the paired state, described by a Pfaffian wave function, may be induced if the short-range part of the interaction is softened by increasing the thickness of the two-dimensional electron system.

We show how to eliminate the first-order effects of the spin-orbit interaction in the performance of a two-qubit quantum gate. Our procedure involves tailoring the time dependence of the coupling between neighboring spins. We derive an effective Hamiltonian which permits a systematic analysis of this tailoring. Time-symmetric pulsing of the coupling automatically eliminates several undesirable terms in this Hamiltonian. Well chosen pulse shapes can produce an effectively isotropic exchange gate, which can be used in universal quantum computation with appropriate coding. PACS: 03.67.Lx, 71.70.Ej, 85.35.Be The exchange interaction between spins is a promising physical resource for constructing two-qubit quantum gates in quantum computers [1][2][3][4][5]. In the idealized case of vanishing spin-orbit coupling, this interaction is isotropic, and any Hamiltonian describing time-dependent exchange between two spin-1/2 qubits, H 0 (t) = J(t)S 1 · S 2 , commutes with itself at different times. Thus, the resulting quantum gate depends on J(t) only through its time integral -a convenient simplification, particularly because, when carrying out quantum gates, the exchange interaction should be pulsed adiabatically on time scales longer thanh/∆E, where ∆E is a typical level spacing associated with the internal degrees of freedom of the qubits [3]. In addition, isotropic exchange alone has been shown to be sufficient for universal quantum computation, provided the logical qubits of the computer are properly encoded [6,7].Given the potential advantages of isotropic exchange for quantum gates, it is important to understand the effect of the inevitable anisotropic corrections due to spinorbit coupling. When these corrections are included, the Hamiltonian describing time-dependent exchange iswhereHere β(t) is the Dzyaloshinski-Moriya vector, which is first order in spin-orbit coupling, and IΓ(t) is a symmetric tensor which is second order in spin-orbit coupling [8]. Although these corrections may be small, they will, in general, not be zero unless forbidden by symmetry. For example, Kavokin has recently estimated that β(t) can be as large as 0.01 for coupled quantum dots in GaAs [9]. In this Letter we construct the quantum gates produced by pulsing H(t). This is nontrivial because H(t) typically does not commute with itself at different times. We represent the resulting gates using an effective Hamiltonian H(t), which we derive perturbatively in powers of the spin-orbit coupling. H(t) is simple to work with because it does commute with itself at different times. As an application of this effective Hamiltonian, we use it to tailor pulse forms that effectively eliminate any firstorder anisotropic corrections.The quantum gate obtained by pulsing a particular H(t) is found by solving the time-dependent Schrödinger equation i d dt |Ψ(t) = H(t)|Ψ(t) where |Ψ(t) is the state vector describing the two spin-1/2 qubits (here, and in what follows,h = 1). In general this problem cannot be solved analytically. However, since we expect spin...