Zhenghan Wang

Simulation of topological field theories by quantum computers

Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold:1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”.2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.

The Two-Eigenvalue Problem and Density¶of Jones Representation of Braid Groups

Introduction 1. The two-eigenvalue problem 2. Hecke algebra representations of braid groups 3. Duality of Jones-Wenzl representations 4. Closed images of Jones-Wenzl sectors 5. Distribution of evaluations of Jones polynomials 6. Fibonacci representations

Identifying topological order by entanglement entropy

Topological entanglement entropy provides a robust measure for detecting the long-range entanglement that characterizes quantum ground states displaying topological order. A new method for calculating this entropy isolates minimally entangled states from the ground states of a topological phase—offering a reliable test for identifying topological spin liquids.

Implementation of universal quantum gates based on nonadiabatic geometric phases.

We propose an experimentally feasible scheme to achieve quantum computation based on nonadiabatic geometric phase shifts, in which a cyclic geometric phase is used to realize a set of universal quantum gates. Physical implementation of this set of gates is designed for Josephson junctions and for NMR systems. Interestingly, we find that the nonadiabatic phase shift may be independent of the operation time under appropriate controllable conditions. A remarkable feature of the present nonadiabatic geometric gates is that there is no intrinsic limitation on the operation time.

Unconventional geometric quantum computation.

We propose a new class of unconventional geometric gates involving nonzero dynamic phases, and elucidate that geometric quantum computation can be implemented by using these gates. Comparing with the conventional geometric gate operation, in which the dynamic phase shift must be removed or avoided, the gates proposed here may be operated more simply. We illustrate in detail that unconventional nontrivial two-qubit geometric gates with built-in fault-tolerant geometric features can be implemented in real physical systems.

A class of P ; T -invariant topological phases of interacting electrons

Abstract We describe a class of parity- and time-reversal-invariant topological states of matter which can arise in correlated electron systems in 2+1-dimensions. These states are characterized by particle-like excitations exhibiting exotic braiding statistics. P and T invariance are maintained by a ‘doubling’ of the low-energy degrees of freedom which occurs naturally without doubling the underlying microscopic degrees of freedom. The simplest examples have been the subject of considerable interest as proposed mechanisms for high-Tc superconductivity. One is the ‘doubled’ version of the chiral spin liquid. The chiral spin liquid gives rise to anyon superconductivity at finite doping and the corresponding field theory is U(1) Chern–Simons theory at coupling constant m=2. The ‘doubled’ theory is two copies of this theory, one with m=2 the other with m=−2. The second example corresponds to Z2 gauge theory, which describes a scenario for spin-charge separation. Our main concern, with an eye towards applications to quantum computation, are richer models which support non-Abelian statistics. All of these models, richer or poorer, lie in a tightly organized discrete family indexed by the Baraha numbers, 2cos(π/(k+2)), for positive integer k. The physical inference is that a material manifesting the Z2 gauge theory or a doubled chiral spin liquid might be easily altered to one capable of universal quantum computation. These phases of matter have a field-theoretic description in terms of gauge theories which, in their infrared limits, are topological field theories. We motivate these gauge theories using a parton model or slave-fermion construction and show how they can be solved exactly. The structure of the resulting Hilbert spaces can be understood in purely combinatorial terms. The highly constrained nature of this combinatorial construction, phrased in the language of the topology of curves on surfaces, lays the groundwork for a strategy for constructing microscopic lattice models which give rise to these phases.

EXTRASPECIAL 2-GROUPS AND IMAGES OF BRAID GROUP REPRESENTATIONS

We investigate a family of (reducible) representations of the braid groups Bn corresponding to a specific solution to the Yang‐Baxter equation. The images of Bn under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of Bn factoring over Temperley‐Lieb algebras and the corresponding link invariants.

Geometric Quantum Computation and Multiqubit Entanglement with Superconducting Qubits inside a Cavity

We analyze a new scheme for quantum information processing, with superconducting charge qubits coupled through a cavity mode, in which quantum manipulations are insensitive to the state of the cavity. We illustrate how to physically implement universal quantum computation as well as multiqubit entanglement based on unconventional geometric phase shifts in this scalable solid-state system. Some quantum error-correcting codes can also be easily constructed using the same technique. In view of the gate dependence on just global geometric features and the insensitivity to the state of cavity modes, the proposed quantum operations may result in high-fidelity quantum information processing.

Spin fluctuations, interband coupling and unconventional pairing in iron-based superconductors

Based on an effective two-band model and using the fluctuation-exchange (FLEX) approach, we explore spin fluctuations and unconventional superconducting pairing in Fe-based layer superconductors. It is elaborated that one type of interband antiferromagnetic (AF) spin fluctuation stems from the interband Coulomb repulsion, whereas the other type of intraband AF spin fluctuation originates from the intraband Coulomb repulsion. Due to the Fermi-surface topology, a spin-singlet extended s-wave superconducting state is more favorable than the nodal dXY-wave state if the interband AF spin fluctuation is more significant than the intraband one, otherwise vice versa. It is also revealed that the effective interband coupling plays an important role in the intraband pairings, which is a distinct feature of the present two-band system.

Sublattice entanglement and quantum phase transitions in antiferromagnetic spin chains

Entanglement of the ground states in the S = 1/2 XXZ chain, dimerized Heisenberg spin chain, two-leg spin ladders as well as S = 1 anisotropic Haldane chain is analysed using the entanglement entropy between a selected sublattice of spins and the rest of the system. In particular, we reveal that quantum phase transition points/boundaries may be identified based on the analysis on the local extreme of this sublattice entanglement entropy, which is illustrated to be superior over the concurrence scenario and may enable us to explore quantum phase transitions in many other systems including higher dimensional ones.Entanglement of the ground states in