Non-Abelian braiding of lattice bosons.

We report on a numerical experiment in which we use time-dependent potentials to braid non-Abelian quasiparticles. We consider lattice bosons in a uniform magnetic field within the fractional quantum Hall regime, where ν, the ratio of particles to flux quanta, is near 1/2, 1, or 3/2. We introduce time-dependent potentials which move quasiparticle excitations around one another, explicitly simulating a braiding operation which could implement part of a gate in a quantum computation. We find that different braids do not commute for ν near 1 and 3/2, with Berry matrices, respectively, consistent with Ising and Fibonacci anyons. Near ν=1/2, the braids commute.

[1]  S. Simon,et al.  Monte Carlo evaluation of non-Abelian statistics. , 2003, Physical review letters.

[2]  E. Fradkin,et al.  Landau-Ginzburg Theories for Non-Abelian Quantum Hall States , 1998, cond-mat/9811005.

[3]  J. Dalibard,et al.  Many-Body Physics with Ultracold Gases , 2007, 0704.3011.

[4]  R. A. Williams,et al.  Observation of vortex nucleation in a rotating two-dimensional lattice of Bose-Einstein condensates. , 2010, Physical review letters.

[5]  R. Joynt.  The Quantum Hall Effect , 1988 .

[6]  Haldane,et al.  Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect. , 1985, Physical review. B, Condensed matter.

[7]  R. Thomale,et al.  Spin hamiltonian for which the chiral spin liquid is the exact ground state. , 2007, Physical review letters.

[8]  Parsa Bonderson,et al.  Detecting non-Abelian statistics in the nu = 5/2 fractional quantum hall state. , 2006, Physical review letters.

[9]  P. Schmitteckert,et al.  Trapping Abelian anyons in fractional quantum Hall droplets , 2007, 0712.1461.

[10]  R. Thomale,et al.  Non-Abelian statistics in a quantum antiferromagnet. , 2009, Physical Review Letters.

[11]  Quantum phase transitions and vortex dynamics in superconducting networks , 2000, cond-mat/0011152.

[12]  M. Baraban,et al.  Numerical analysis of quasiholes of the moore-read wave function. , 2009, Physical review letters.

[13]  Alternation and interchange of e/4 and e/2 period interference oscillations as evidence for filling factor 5/2 non-Abelian quasiparticles , 2009, 0911.0345.

[14]  Gregory W. Moore,et al.  Nonabelions in the fractional quantum Hall effect , 1991 .

[15]  E. Kapit,et al.  Exact parent Hamiltonian for the quantum Hall states in a lattice. , 2010, Physical review letters.

[16]  A. Kitaev.  Fault tolerant quantum computation by anyons , 1997, quant-ph/9707021.

[17]  R. Laughlin.  Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations , 1983 .

[18]  Alexei Kitaev,et al.  Anyons in an exactly solved model and beyond , 2005, cond-mat/0506438.

[19]  F. Haldane,et al.  Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States , 1983 .

[20]  N. Cooper.  Rapidly rotating atomic gases , 2008, 0810.4398.

[21]  L. B. Ioffe,et al.  Topologically protected quantum bits from Josephson junction arrays , 2022 .

[22]  D. Hofstadter.  Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields , 1976 .

[23]  Berry phases for composite fermions: Effective magnetic field and fractional statistics , 2004, cond-mat/0407137.

[24]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[25]  M. Freedman,et al.  Topologically protected qubits from a possible non-Abelian fractional quantum Hall state. , 2004, Physical review letters.

[26]  M. Freedman,et al.  Towards universal topological quantum computation in the ν = 5 2 fractional quantum Hall state , 2005, cond-mat/0512066.

[27]  I. Spielman.  Raman processes and effective gauge potentials , 2009, 0905.2436.

[28]  F. Wilczek,et al.  Paired Hall states , 1992 .

[29]  E. Prodan,et al.  Mapping the braiding properties of the Moore-Read state , 2009, 1001.1930.

[30]  Sergey Bravyi.  Universal quantum computation with the v=5/2 fractional quantum Hall state , 2006 .

[31]  L. Fu,et al.  Superconducting proximity effect and majorana fermions at the surface of a topological insulator. , 2007, Physical review letters.

[32]  Steven H. Simon,et al.  Construction of a paired wave function for spinless electrons at filling fractionν=2∕5 , 2007 .

[33]  Dexter Kozen,et al.  New , 2020, MFPS.

[34]  Jiannis K. Pachos,et al.  Non-Abelian statistics as a Berry phase in exactly solvable models , 2009, 0901.3674.

[35]  H. V. D. van der Zant,et al.  Quantum phase transitions in two dimensions: Experiments in Josephson-junction arrays. , 1996, Physical review. B, Condensed matter.

[36]  Read,et al.  Quasiholes and fermionic zero modes of paired fractional quantum Hall states: The mechanism for non-Abelian statistics. , 1996, Physical review. B, Condensed matter.

[37]  N R Cooper,et al.  Quantum phases of vortices in rotating Bose-Einstein condensates. , 2001, Physical review letters.

[38]  Parsa Bonderson,et al.  Plasma analogy and non-Abelian statistics for Ising-type quantum Hall states , 2010, 1008.5194.

[39]  N. Read,et al.  Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level , 1998, cond-mat/9809384.

[40]  Immanuel Bloch,et al.  Single-spin addressing in an atomic Mott insulator , 2011, Nature.

[41]  Frank Wilczek,et al.  2n-quasihole states realize 2n−1-dimensional spinor braiding statistics in paired quantum Hall states , 1996 .

[42]  Wen,et al.  Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces. , 1990, Physical review. B, Condensed matter.

[43]  I. B. Spielman,et al.  Synthetic magnetic fields for ultracold neutral atoms , 2009, Nature.

[44]  D. Ivanov.  Non-Abelian statistics of half-quantum vortices in p-wave superconductors. , 2000, Physical review letters.

[45]  M. Troyer,et al.  Topologically protected quantum bits using Josephson junction arrays , 2001, Nature.

[46]  S. Simon,et al.  Topological quantum compiling , 2006, quant-ph/0610111.

[47]  S. Simon,et al.  Non-Abelian Anyons and Topological Quantum Computation , 2007, 0707.1889.

[48]  S. Pancharatnam,et al.  Generalized theory of interference, and its applications , 1956 .

[49]  S. Tewari,et al.  Topological degeneracy of non-Abelian states for dummies , 2006, cond-mat/0607743.

[50]  R. Thomale,et al.  Non-Abelian statistics and a hierarchy of fractional spin liquids in spin-S antiferromagnets , 2011 .

[51]  Xiong-Jun Liu,et al.  Quantum anomalous Hall effect with cold atoms trapped in a square lattice , 2010, 1003.2736.

[52]  Xiao-Gang Wen,et al.  String-net condensation: A physical mechanism for topological phases , 2004, cond-mat/0404617.

[53]  R. Thomale,et al.  Parent Hamiltonian for the chiral spin liquid , 2009, 0905.3257.

[54]  L. Georgiev.  Topologically protected gates for quantum computation with non-Abelian anyons in the Pfaffian quantum Hall state , 2006, cond-mat/0607125.