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 C 2, a “qubit”.  The standard model for quantum computing: Local gates on C 2, 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 There is an equivalent model for quantum computation [FLW 1,FKW 2 ] 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. 1. The two-eigenvalue problem and density of Jones representation of braid ground. Comm. Math. Phys.228 (2002),no.1,177 – Simulation of topological field theories by quantum computers. Comm. Math.Phys.227 (2002), no.3, The Topological Model

5 time birth braiding Particle - antiparticle pairs are created out of the vacuum. 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 = set of curves on a surface S. [S] = set of isotopy class of curves on S S is a surface: An example of a “multi loop” d on S,, etc…

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

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

11 Consider: V d is the associated TQFT V d (S) with a rich and known structure 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/ , it is argued that V d as likely to collopse to V d. -

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

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 The colors encode differing chemical potentials. Tunneling amplitudes t ab also vary with colors. c Hubbard Model

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 ”.“ game

17 HamiltonianGround State Manifold H =H 1/6 = { all particle positions } ( U 0 large) { one particle per bond } D ={ dimer cover T } Now small terms: j

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

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=1b=d-1a=1b=d-1 However is most general. mob rule: a = d 1/4 b = 1

24 C dimerizations C multiloops 1 to prevent process 3 2 5

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