Combining physics, mathematics and computer science, topological quantum computation is a rapidly expanding research area focused on the exploration of quantum evolutions that are immune to errors. In this book, the author presents a variety of different topics developed together for the first time, forming an excellent introduction to topological quantum computation. The makings of anyonic systems, their properties and their computational power are presented in a pedagogical way. Relevant calculations are fully explained, and numerous worked examples and exercises support and aid understanding. Special emphasis is given to the motivation and physical intuition behind every mathematical concept. Demystifying difficult topics by using accessible language, this book has broad appeal and is ideal for graduate students and researchers from various disciplines who want to get into this new and exciting research field.

A relation between geometric phases and criticality of spin chains is established. As a result, we show how geometric phases can be exploited as a tool to detect regions of criticality without having to undergo a quantum phase transition. We analytically evaluate the geometric phase that correspond to the ground and excited states of the anisotropic XY model in the presence of a transverse magnetic field when the direction of the anisotropy is adiabatically rotated. It is demonstrated that the resulting phase is resilient against the main sources of errors. A physical realization with ultra-cold atoms in optical lattices is presented.PACS numbers: 03.65. Vf, 05.30.Pr, 42.50.Vk Since the discovery by Berry [1], geometric phases in quantum mechanics have been the subject of a variety of theoretical and experimental investigations [2]. Possible applications range from optics and molecular physics to fundamental quantum mechanics and quantum computation [3]. In condensed matter physics a variety of phenomena have been understood as a manifestation of topological or geometric phases [4,5,6,7,8]. An interesting open question is whether the geometric phases can be used to investigate the physics and the behavior of condensed matter systems. Here we show how to exploit the geometric phase as an essential tool to reveal quantum critical phenomena in many-body quantum systems. Indeed, quantum phase transitions are accompanied by a qualitative change in the nature of classical correlations and their description is clearly one of the major interests in condensed matter physics [9,10]. Such drastic changes in the properties of ground states are often due to the presence of points of degeneracy and are reflected in the geometry of the Hilbert space of the system. The geometric phase, which is a measure of the curvature of the Hilbert space, is able to capture them, thereby revealing critical behavior. This provides the means to detect, not only theoretically, but also experimentally the presence of criticality without having to undergo a quantum phase transition.In this letter we analyze the XY spin chain model and the geometric phase that corresponds to the XX criticality. Since the XY model is exactly solvable and still presents a rich structure it offers a benchmark to test the properties of geometric phases in the proximity of a quantum phase transition. Indeed, we observe that, an excitation of the model obtains a non-trivial Berry phase if and only if it circulates a region of criticality. The generation of this phase can be traced down to the presence of a conical intersection of the energy levels located at the XX criticality. This geometric interpretation reveals a relation between the critical exponents of the model. The insights provided here shed light into the understanding of more general systems, where analytic solutions might not be available. A physical implementation is proposed with ultra-cold atoms superposed by optical lattices [11,12]. It utilizes Raman activated tunneling transitions as well as coher...

In this Letter, we provide a general methodology to directly measure topological order in cold atom systems. As an application, we propose the realization of a characteristic topological model, introduced by Haldane, using optical lattices loaded with fermionic atoms in two internal states. We demonstrate that time-of-flight measurements directly reveal the topological order of the system in the form of momentumspace Skyrmions. DOI: 10.1103/PhysRevLett.107.235301 PACS numbers: 67.85.Àd, 03.65.Vf Different phases of matter can be distinguished by their symmetries. This information is usually captured by locally measurable order parameters that summarize the essential properties of the phase. Topological insulators are materials with symmetries that depend on the topology of the energy eigenstates of the system [1]. These materials are of interest because they give rise to robust spin transport effects with potential applications ranging from sensitive detectors to quantum computation [2,3]. However, direct observation and measurement of topological order has been up to now impossible due to its nonlocal character. Instead, experiments have relied so far on indirect manifestations of this order, such as edge states and the quantization of conductivity.Ultracold atoms facilitate the implementation of artificial gauge fields [4]. Here, we distinguish proposals that generate continuous fields [5], such as the recent experiment by Lin et al. [6], from those that rely on optical lattices and engineering of hopping [7]. We will concentrate on the latter, introducing a method based on standard time-of-flight (TOF) measurements that can identify a topological character in the quantum state of the system. Our starting point is a possible implementation of Haldane's model using fermionic atoms in two internal states. The topological nature of its ground state is witnessed by the Chern number. This number counts the times the ground state, written as a spinor, wraps around the sphere, as a function of momentum. We demonstrate that TOF measurements reconstruct the Chern number in a way which is robust against the presence of external perturbations or state preparation. Our method can be adapted to other quantum simulations of topological order in optical lattices [8][9][10][11][12][13][14][15][16][17], as many already use internal degrees of freedom of the atoms to encode the order.One common mechanism for the appearance of topological order is based on the topology of the eigenstate manifolds. Consider a real-space lattice whose unit cell has d quantum degrees of freedom-position of the particle, spin, etc. In the case of the quantum Hall effect, the energy bands are separated from each other and the material becomes an insulator for appropriate Fermi energies, E F . In a real setup, with finite boundaries, the sample can have a quantized nonzero conductivity given by the topological invariant m xy ¼ e 2 =h P E m

In the holonomic approach to quantum computation information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and manipulated by the associated holonomic gates. These are realized in terms of the non-abelian Berry connection and are obtained by driving the control parameters along adiabatic loops. We show how it is possible, for a specific model, to explicitly determine the loops generating any desired logical gate, thus producing a universal set of unitary transformations. In a multi-partite system unitary transformations can be implemented efficiently by sequences of local holonomic gates. Moreover a conceptual scheme for obtaining the required Hamiltonian family, based on frequently repeated pulses, is discussed, together with a possible process whereby the initial state can be prepared and the final one can be measured.Comment: 5 pages, no figures, revtex, minor changes, version accepted by Phys. Rev A (rapid comm.

To use quantum systems for technological applications one first needs to preserve their coherence for macroscopic time scales, even at finite temperature. Quantum error correction has made it possible to actively correct errors that affect a quantum memory. An attractive scenario is the construction of passive storage of quantum information with minimal active support. Indeed, passive protection is the basis of robust and scalable classical technology, physically realized in the form of the transistor and the ferromagnetic hard disk. The discovery of an analogous quantum system is a challenging open problem, plagued with a variety of no-go theorems. Several approaches have been devised to overcome these theorems by taking advantage of their loopholes. The state-of-the-art developments in this field are reviewed in an informative and pedagogical way. The main principles of self-correcting quantum memories are given and several milestone examples from the literature of two-, three-and higher-dimensional quantum memories are analyzed.

We demonstrate that in a triangular configuration of an optical lattice of two atomic species a variety of novel spin-1/2 Hamiltonians can be generated. They include effective three-spin interactions resulting from the possibility of atoms tunneling along two different paths. This motivates the study of ground state properties of various three-spin Hamiltonians in terms of their two-point and n-point correlations as well as the localizable entanglement. We present a Hamiltonian with a finite energy gap above its unique ground state for which the localizable entanglement length diverges for a wide interval of applied external fields, while at the same time the classical correlation length remains finite.PACS numbers: 03.67.Mn, The combination of cold atom technology with optical lattices [1,2] gives rise to a variety of possibilities for constructing spin Hamiltonians [3,4]. This is particularly appealing as the high degree of isolation from the environment that can be achieved in these systems allows for the study of these Hamiltonians under idealised laboratory conditions. In parallel, techniques have been developed for minimising imperfections and impurities [5,6] in the implementation of the desired structures and for their subsequent probing and measurement [7]. These achievements permit the experimental investigation of Hamiltonians that are of interest in areas such as quantum information or condensed matter physics with the added advantage of a remarkable freedom in the choice of external parameters. Presently, attention both in condensed matter physics and in cold atom research is focusing on two-spin interactions as these are most readily accessible experimentally. However, the unique experimental capability provided by cold atom technology allows us to relax this restriction. Here we demonstrate that cold atom technology provides a laboratory to generate and study higher order effects such as three-spin interactions that give rise to unique entanglement properties.The present work serves two purposes. Firstly, it demonstrates that in a two species Bose-Hubbard model in a triangular configuration a wide range of Hamilton operators can be generated that include effective three-spin interactions. They result from the possibility of atomic tunneling through different paths from one vertex to the other. This can be extended to a one dimensional spin chain with three-spin interactions. Secondly, we take this novel experimental capability as a motivation to study unique ground state properties of Hamiltonians that include three-spin interactions. In this context one can study possible quantum phase transitions by considering both the classical correlation properties as well as the entanglement properties of these systems. Specifically we consider the so-called cluster Hamiltonian and its ground state, the cluster state which has previously been shown to play an important role as a resource in the context of quantum computation [20]. Subject to an additional Zeeman term the combined Hamiltonian possesses a fi...

Two-qubit logical gates are proposed on the basis of two atoms trapped in a cavity setup. Losses in the interaction by spontaneous transitions are efficiently suppressed by employing adiabatic transitions and the Zeno effect. Dynamical and geometrical conditional phase gates are suggested. This method provides fidelity and a success rate of its gates very close to unity. Hence, it is suitable for performing quantum computation.One of the main obstacles in realizing a quantum computer (QC) is decoherence resulting from the coupling of the system with the environment. There are theoretical proposals for models which avoid decoherence [1][2][3][4]. For this purpose decoherence-free subspaces (DFS) have been proposed in the literature for performing QC [5][6][7]. While they are easy to construct in the case of a single qubit, they are more complicated for the case of an externally controlled multipartite system. Their main decoherence channel is the "bus" that couples the different subsystems and is usually strongly perturbed by the environment. In the case of an ion trap the bus is the common vibrational mode which is subject to continuous heating. In the case of cavity QED the bus is a cavity mode which may leak to the environment. Additionally the cavity couples to an excited state of the atom that shows spontaneous emission. To avoid these phenomenon it is most convenient to transfer population by virtually populating the bus [3,4,[8][9][10]. Here we present a model with atoms in an optical cavity that bypasses the decoherence problem with, in principle, arbitrarily large fidelity and success rate.Atomic levels of the two atoms and laser and cavity couplings with their detunings. The qubits 1, 2 and the auxiliary states |σ are depicted.The system presented here consists of two four-level atoms fixed inside an optical cavity, Fig. 1. This can be achieved by, for example, having trapped ions in a cavity with its axis perpendicular to the ionic chain. It is assumed that the atoms have the lower states, |0 , |1 and |σ , which could be represented by different hyperfine levels or Zeeman levels, and an excited state |2 coupled individually to each ground state by laser radiation with different polarizations or frequencies. The atoms interact with each other via the common cavity radiation field.Our goal is to perform QC in such a way that coherent evolution can be performed even for high loss rates, κ, of the cavity and relatively large decay rates, Γ, of the excited atomic states. This is achieved by employing an adiabatic procedure that keeps the cavity empty and the excited state of the atoms depopulated. Information is transfered by virtual population of these decohering atomic and cavity states. The entangling adiabatic transfer of population between ground state levels occurs by slowly varying the Rabi frequencies of the lasers in a counterintuitive temporal sequence, similarly to the well-known STIRAP process for Λ systems, but now it is performed in the space spanned by the tensor product states of the two ato...

We demonstrate that Dirac fermions self-interacting or coupled to dynamic scalar fields can emerge in the low energy sector of designed bosonic and fermionic cold atom systems. We illustrate this with two examples defined in two spacetime dimensions. The first one is the self-interacting Thirring model. The second one is a model of Dirac fermions coupled to a dynamic scalar field that gives rise to the Gross-Neveu model. The proposed cold atom experiments can be used to probe spectral or correlation properties of interacting quantum field theories thereby presenting an alternative to lattice gauge theory simulations.