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The Topological approach to building a quantum computer.

The Topological approach to building a quantum computer. Michael H. Freedman Theory Group Microsoft Research. Classical computers work with bits: {0,1}. Quantum computers will store information in a superposition of and , i.e. a vector in C 2 , a “qubit”.

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The Topological approach to building a quantum computer.

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  1. The Topological approach to building a quantum computer. Michael H. Freedman Theory Group Microsoft Research

  2. Classical computers work with bits: {0,1}. • Quantum computers will store information in a superposition of and , i.e. a vector in C2, a “qubit”. • The standard model for quantum computing: • Local gates on C2, followed my measurement of the qubits.

  3. Successes: Shor's factoring algorithm Grover’s search algorithm great for simulating solid state physics theoretical fault tolerance But practical fault tolerance may require physical (not software) error correction inherent in topology.

  4. 1. The two-eigenvalue problem and density of Jones representation of braid ground. Comm. Math. Phys.228 (2002),no.1,177 –199.2. Simulation of topological field theories by quantum computers. Comm. Math.Phys.227 (2002), no.3, 587-603. The Topological Model There is an equivalent model for quantum computation [FLW1,FKW2] based on braiding the excitations of a 2-dimensional quantum media whose ground state space is the physical Hilbert space of a topological quantum field theory TQFT.

  5. Particle-antiparticle pairs are created out of the vacuum. birth braiding time death afterlife?

  6. But before we can implement this model in the real world, we must design and build a suitable 2-dimensional structure. • The design would be much easier if we already had a quantum computer!?! • So we use instead powerful mathematical ideas coming from algebras and the theory of Vaughan Jones.

  7. We will define a Hamiltonian with both large and small terms. The large terms will define “multi loops” on a surface and the small terms will be studied perturbatively. The small terms create an effective action which will be a sum of projectors. The projectors in define“d-isotopy” of curves: This is the (previously mentioned) rich mathematical theory derived from C*-algebras.

  8. S is a surface: , , etc… An example of a “multi loop” d on S S= set of curves on a surface S. [S]= set of isotopy class of curves on S

  9. For 2 strand relation ad -1 a = = = so a = d. In both cases: functions on Z -homology. 2

  10. It turns out that the only possible relation on 3- strands is: - d - d = 0 + + This gives something much more interesting than homology. The 4- strand relation is even more interesting: it yields a computationally universal theory.

  11. Consider: Vdis the associated TQFTVd (S) with a rich and known structure.1 1. In A magnetic model with a possible Chern-Simons phase. With an appendix by F. Goodman and H. Wenzl. Comm. Math. Phys. 234 (2003), no. 1, 129—183 and A Class of P, T-Invariant Topological Phases of Interacting Electrons, ArXiv:cond-mat/0307511, it is argued that Vd as likely to collopse to Vd. -

  12. Locating Topological Phases Inside Hubbard Type Models. Kirill Shtengel Chetan Nayak MichaelFreedman

  13. A two dimensional lattice of atoms, partially filled with a population of donated electrons can have it’s parameters tuned to become a (universal) quantum computer.

  14. In our model the sites (atoms) are arrayed on the Kagome lattice Hubbard Model The colors encode differing chemical potentials . Tunneling amplitudes tab also vary with colors. c

  15. We work with an equivalent triangular representation. • In this representation particles (e.g. electrons) live on edges. The important feature for us is that the triangular lattice is not bipartite.

  16. We discuss an “occupation model” at 1/6 fill. For example, imagine that each green atom has donated one electron which is now free to localize near any atom = site of Kagome (K).Let’s look at a “game”.

  17. Hamiltonian Ground State Manifold H=H1/6= {all particle positions} (U0 large) {one particle per bond} D ={dimer cover T} Now small terms: j

  18. Review - Perturbation Theory function of l don’t like: perturbed, but can recurse dynamic, off diag. terms of projectors . . diagonal terms of projectors balanced to keep

  19. To each “small” process there will be a contribution to an “effective Hamiltonian”:

  20. These matrix equations control all small processes:

  21. To make all processes projections, and thus obtain an exactly soluble point, we must impose:

  22. And if there is a Ring term:

  23. Some choice about how to treat : e.g. democracy: all loops = d a=db=d 3 aristocracy: a=1 b=d-1 mob rule: a=d1/4 b=1 However is most general.

  24. 2 1 3 5 CdimerizationsCmultiloops to prevent process

  25. CONCLUDING REMARKS • Ring terms vs. R-sublattice defects • Fermionic vs. Bosonic models

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