C191 Final Presentation by Paolo Zoccante William Leibzon What is it New paradigm of fault tolerant quantum computing For efficient quantum computing error rate 10 4 still not achieved error correction algorithms Possible estimated error rate as low as 10 30 No need for error correction What studies Topology Properties that are not changed by smooth deformations Topological properties are robust against small perturbations The basic idea of TQC Create particles from vacuum initialization Thread their world line unitary operations This step is not based on interactions between the two particles Measure the result measure Physics Particles Fermions Particles of Matter Elementary Up and Down Quarks Electrons Muons Composite Protons Neutrons 3 quarks Obey Fermi Dirac Statistics Have half integer spins for electron No two can be in exactly same state Pauli Exclusion Principle Bosons Particles of Force for Elementary Elementary Photons Gluons Composite Mesons quark antiquark nucleus like He4 Obey Bose Einstein Statistics Have integer spins Two can be in same state Bosons fermions anyons If we exchange two fermions becomes Single particle properties unchanged but it interferes differently with other particles 3D only Bosons and fermions 2D we could have also anyons they can acquire any complex phase a ib exp i ANY ons depends on the kind of particle and it s fixed We may have instead of just a phase an unitary U Braiding in 2 1 dimensions Clockwise swapping counterclockwise swapping Classified by winding number a topological invariant Classical Hall Effect Fractional Quantum Hall Effect Electron gas at the interface in a GaAs heterojunction T 10 mK strong trasverse magnetic field J E Experimental confirmations FQH collective excitations are quasiparticles such that the ratio f between electrons and magnetic flux quanta is a fractional number If we circle a f 1 3 particle around a f 2 5 particle we can find a relative statistic of 1 15 So they behave like anyons Goldman 2006 Lattice of Abelian Anyons Physical system for complete quantum computation are abelians anyons on a lattice they could be quantum Hall effect excitations Lattice site j is occupied by anyon j is not occupied bj bj are creation and annihilation operators Aj bj bj applied on j gives 0 while j is the eigenvector with eigenvalue 1 Bjk bj bk bj bk used to swap the states in j and k Anyon circled around another anyon gets factor ei Qubits and 1 qubit Gates Qubit is a combination of an anyon occupied site and a unoccupied one on sites j and j 0 j jj 1 j jj This number operator generates rotation on z axis Swapping allows to do By applying the hamiltonian Bjj for a certain time we can generate the rotation exp i x 2 around the x axis Applying Aj will generate the rotations around the z axis The Aj and Bjkdo not involve interactions with other anyons Consequently with rotations about X and Z axis we have a full set of 1 qubit operations Universal Set of Gates Two qubit control gate for x j y k is done by repeated swaps we circle the content of the first site of j qubit around the first site of the k qubit The phase of 1 for state of two quibits is obtained if and only if the second sites of j and k both contain anyons corresponding to 1 giving CPHASE gate The size of the orbits used when circling is unimportant the phase is a topological effect not due to interactions We have a universal set of quantum gates 1 qubit operations local unprotected 2 qubit operations topologically protected Non Abelian Anyons For Abelian anyons 2 qubit operations are topologically protected from errors but single qubit operations are local and unprotected To have fully protected operations we need Non Abelian anyons They operate on fusion spaces interaction of 3 or more anyons and acquire a unitary instead of a phase The braid group is generated by moving up down every thread With opportune generators we can build a dense subset of SU 4 and SU 2 topologically protected single and two qubit operators Fibonacci Model A simplest of Non Abelian anyons models is Yang Lee Fibonacci model Two anyons can fuse in either of two ways i e 1 x 1 0 1 The resulting Hilbert space has dimensions that are Fibonacci numbers Qubits encoded in one anyon Fibonacci anyons CNOT accurate to 10 3 Future Perspectives Experimentalists think that the FQH excitations with f 5 2 easier to investigate and 12 5 harder to investigate are non abelians The 5 2 particles will not probably generate a dense subspace of SU 2 The 12 5 particle should work At the moment still trying to measure non abelian anyons In 2000 they proved that topological quantum computers and ordinary quantum computer are equivalent can simulate each other Possible Error Sources Errors occur if thermal fluctuations generate pair of anyons Errors are exponentially small for low temperature Probability of errors decreases exponentially with distance The rate of errors can be minimized to almost 0 with low temperature and keeping anyons sufficiently far apart References Collins Graham Computing with Quantum Knots Scientific American Magazine April 2006 http marcuslab harvard edu otherpapers SciamTQC pdf Day Charles Devices Based on the Fractional Quantum Hall Effect May Fulfill the Promise of Quantum Computing Physics Today October 2005 http nfo phys unm edu thedude topo phystodayTQC pdf Lloyd Seth Quantum computation with abelian anyons Journal of Quantum Information Processing 10 1023 April 2002 http arxiv org pdf quant ph 0004010v2 Preskill John Topological Quantum Computing fo Beginners Institute of Quantum Information Caltech June 2003 http pnm itp ucsb edu online exotic c04 preskill pdf Preskill pdf