Topological Quantum Computation: The Art of Computing with Icosahedral Group

1. Topological Quantum Computation: The Art of Computing with Icosahedral Group Giuseppe Mussardo SISSA-Trieste

2. Integrable Systems and Loop Models Lattice gauge theories CFT, Fusion Rules and Commutative Algebras Knot theories and Topological Invariants Topological phases Non-abelian anyons Quantum Hall Effects, P+iP Superconductivity, Cold Atoms, Dimers Quantum Computations Interferometry

3. “ If I have seen a bit further it is by standing on the shoulders of giants” • Topological Quantum Computation S. Kitaev, M. Freedman, J. Preskill, … • Topological Phases of matter F. Wilczek, X. Wen, G. Moore, N. Read, E. Rezayi, E. Fradkin, P. Fendley, M. Fisher, C. Nayak, F. Bais, J. Stingerland, S. Das Sarma, A. Cappelli, A. Stern… • braidings and fusions E. Witten, E. Verlinde, Z. Wang, L. Kauffman, S. Trebst, E. Ardonne, K. Schoutens, A. Ludwig, N. Bonesteel, L. Hormozi, S. Simons,…

4. “Topological Quantum Hashing With Icosahedral Group” M. Burrello, H. Xu, G.M. and X. Wan arXiv:0903.1497

5. Plan of the seminar • Topological Phases of Matter • Anyons • Fibonacci, Ising and Cardano anyons • The art of braiding • Quantum computation and universal gates • Icosahedron and topological hashing

6. L A Topological Phases • Topological entropy • Order parameters: Wilson loops • Gapped spectrum. Degeneracy of ground states which depends on topology • Anyonic excitations • Fractional (non-abelian) statistics

7. Braid group • The world-lines in 2+1 of N anyons form a N strand braid • These trajectories are robust with respect to local perturbations • States in this space can only be distinguished • by global measurements, i.e. a perfect place • to store information • Braids of N strands form an infinite group: this can be seen as the permutation group with memory of its history

8. Algebraic relations

9. Basic quantities of non-abelian anyons • Rules for fusing (and splitting) the excitations • carrying conserved charges. • Associativity of the fusion rules (F-matrices) • Rules for braiding the excitations (R-matrices) • Growing rates of their Hilbert space, alias • quantum dimensions.

10. Fusion algebra • a is a non-abelian anyon if • Associativity where are m x m matrices

11. Verlinde formula and classification of FR • There exists a real unitary matrix S that simultaneously diagonalize all matrices • S is the modular S-matrix • The classification of all FR consists of finding all possible • real unitary matrices of dimension m x m • The exaustive classification has been so far achieved up to m=4

12. Examples • (Fibonacci anyon) • (Ising anyons) • (Cardano anyons)

13. Pentagonal equations The F-matrices have to be found as solutions of a set of consistency equations Examples • Fibonacci anyons , • Ising anyons , • Cardano anyons …

14. Hexagonal equations The R-matrices have to be found as solutions of a set of consistency equations Examples • Fibonacci anyons , • Ising anyons , … • Cardano anyons , … ,

15. ….. a a a a a b c1 c2 ci Cn-1 Quantum dimension Alias, how fast the Hilbert space of n-anyons grows In the large n-limit, this is dominated by the largest eigenvalues of Na

16. , Mathematical and physical features The quantum dimensionsdaare, simultaneously, Perron-Frobenius eigenvalues and eigenvectors of the fusion algebra Examples Anyonic gas at equilibrium • Fibonacci anyons In the steady state, anyons of type a appear with probability • Ising anyons • Cardano anyons

17. Fibonacci anyons =1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…

18. Geometrical paradox

19. Enlightening Fibonacci

20. Counting the outcoming rays

21. Quantum Computation Quantum Circuits are unitary operators acting on a Hilbert space, generated by n-qubits, whose states encoded the information we want to process

22. “Divide et Impera” An Universal Quantum Computer is a device able to implement any unitary operator in SU(N) Every unitary operator in SU(N) can be decomposed in: (i) Single-qubit (element of U(2)) and (ii) Controlled NOT gates • Single-qubit rotation • C-NOT gate CNOT is the key element for creating Entanglement

23. Braid realization of quantum gates This will solve at once all problems of decoherence

24. Weaves Any quantum computation that can be done by braiding n identical quasi-particles can also be done by moving only a single particle around the n-1 other particles whose position remains fixed. Simon et al. (PRL 2006) p-generators -approximation This simplification may greatly reduce the technological difficulty for realizing topological quantum computation

25. 0 1 1 0 0 1 1 1 Qubit for Fibonacci anyons 1 x 1 = 0 + 1 Two Fibonacci span a 2-dimensional Hilbert space To have non-trivial operation we need however three Fibonacci anyons

26. Qubit and braidings of Fibonacci anyons 1 and 2 induce an ergodic motion on SU(2) group

27. Cardano anyons The same can be done with Cardano anyons • The braiding matrices 1 and 2 of the field  can span the whole SU(2) • Those associated to the field  can span instead the whole SU(3)

28. Ising anyons In this case the braiding matrices 1 and 2 of the field  cannot span the whole SU(2) In fact, they just generate the finite sub-group of SU(2) given by the cube

29. Fibonacci anyons Cardano anyons Universal computation Single qubit rotation In the following we focus our attention only on the single qubit rotation gate. Ising anyons

30. Brute force approach (Hormozi et al.) Using the ergodicity properties, one can pin down an arbitrary SU(2) gate by a brute force search algorithm All roads lead to Rome algorithm Even though this search can be improved thanks to Solovey-Kitaev algorithm, it becomes nevertheless unfeasible for long braids Can we do something better?

31. Luca Pacioli De Divina Proportione

32. Basic Geometric Data • One of the 5 regular Platonic solids • It hasF=20 triangular faces, • E=30 edges, V=12 vertices

33. Symmetries • 15 axes of 2nd order • 10 axes of 3rd order • 6 axes of 5th order

34. Icosahedron vs Dodecahedron Finite group of 60 elements (isomorphic to A5)

35. Festival of Golden Ratio With the center placed at the originand side a=2, the coordinates of the vertices are Radius of middle sphere (it touches the middle of all edges) Radius of circumscribed sphere (it touches all the vertices) Radius of inscribed sphere (it touches all the faces)

36. Golden Rectangles and Borromean Rings

37. How to pin down a gate by a finite number of moves: HASHING STRATEGY Find by brute force, once for all, the 60 generators of the icosahedral group, in terms of braids of a given length (say l=8 or L=24) 2. Use the pseudo-group structure so created to set up a dense set of points in the Bloch sphere. 3. Two-step process: finite and infinitesimal rotations.

38. Icosahedral Pseudo-Group • The pseudo-group is not a group and it is characterized by errors, • depending on the chosen length in the braid representation • Our algorithm definitely exploits these errors to create an efficient • sampling of SU(2) ! Group Pseudo-Group Closure Î is not closed Identity can be obtained in many ways We can span the vicinity of Identity in many ways

39. Pre-processor: L = 8 Thanks to the errors of the approximation, with the product of 3 elements of Î(8), we can span all SU(2) with 0=0.03 60 points 216.000 points The pre-processor approximates the target with

40. Hashing strategy. 1. Pre-processor error Target SU(2) gate Pseudo-group icosahedral approximation This requires only a finite number of searches! After a rotation nearby the target gate, the only thing left is to reduce the error near the identity

41. Hashing strategy. 2. Main processor We can sample with high precision the vicinity of the Identity. How can we do it ? • For any n-plet of rotations in I, we can find g n+1 such that Mapping it in the approximated pseudo-group Î(24) with braid length L=24, we obtain a fine rotation R where Hn is, essentially, an hermitian random matrix

42. Random matrices The distribution of the eigenvalue spacings (alias of errors) satisfy the Wigner-Dyson form of Gaussian Unitary Ensemble

43. Conclusions and open questions • Hashing algorithm is a very efficient procedure to realize • quantum gates by braids • Role of anyon systems with higher number of excitations • Topological phases of matter at criticality and SLE • Deep connection between topological phases of matter and • integrable models (Golden chain and generalization)