1. Topological reach of field-theoretical topological quantum computation Mario Rasetti Politecnico di Torino & ISI Foundation

2. Preliminaries Preliminaries The object: Construction of new efficient quantum algorithms for topological invariants The general context: Quantum Information Theory ( Turing machine, circuit model, Lambda calculus, Post system and all that in quantum version … ) in its Quantum Field Theory version The results: Efficient quantum algorithms for any observable of Chern-Simons topological quantum field theory, in particular the Jones polynomial for knots in  3 and the invariants of 3-manifolds

3. Scheme Part I Part I knot theory; The Jones polynomial; Computational complexity; Quantum computation; Part II Part II The Jones polynomial in QFT; Unitary representation of the braid group; The quantum circuit;

4. Part I: the problem Knot theory is the branch of topology concerning with the properties of knots. The most important problem in knot theory is the classification of knots: given two knots can we determine whether they are topologically equivalent or not?

5. Part I: the problem Part I: the problem

6. A knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode for some of the topological properties of a class of knots. The Jones polynomial is one of the most important such knot invariants Part I: the problem

7. Definition of the Jones polynomial by braid representation The original definition of the Jones polynomial results from: a trace of the braid group representation into the Temperley Lieb algebra Part I: the problem

8. The braid group The braid group on n strandsis a group with an intuitive geometrical realization Presentation for The Automaton based on the Spin Network Quantum Simulator accepts the Braid language :

9. Part I: the problem

12. The Temperley-Lieb algebra....

13. Part I: the problem Knot-braid connection A given link L LL (coloured) can always be seen as the closure of a braid (Alexander theorem)

14. Part I: the problem The Jones polynomial is given by V.F.R. Jones, A polynomial invariant for links via von Neumann algebras, Bull. Amer. Math. Soc. 129 (1985), 103-112. Defining a representation with coefficients inand such that

15. Part I: the problem  How hard is to evaluate the Jones polynomial from a computational point of view?  We know that there are no efficient classical algorithms for its evaluation: P the Jones polynomial is a #P-hard problem  Can we provide an efficient quantum algorithm? Jaeger, Vertigan and Welsh, On the computational complexity of the Jones and Tutte Polynomials, Mathematical Proceedings of the Cambridge Phil. Soc. 108(1990), 35-53

16. Part I: Computational Complexity  The Jones polynomial is #P-hard: hard means that all the problems in #P can be polynomially reduced to it.  From this it follows that, efficiently solving #P-hard problems we could even solve NP-complete problems, and so we could prove P=NP......too good to be true......too good to be true... That much for exact solutions, but what about approximate solutions?

17. Part I: Computational Complexity  Some #P-hard problems admit an efficient approximate solution  We showed that evaluation of the Jones polynomial can be done efficiently with a quantum computer if we search for an approximate solution  In fact the approximate evaluation of the Jones polynomial is the first known BQP-complete problem

18. Part II: the method Let the quantum circuit constructed be of length O(poly(n)) acting on n qubits, and let be a pure state of n qubits which can be prepared in time O(poly(n)). It is then possible to sample in O(poly(n)) time from random variables X, Y in such a way that Additive approximation

19. In between part I and II: quantum computation Superposition Superposition Entanglement Entanglement Ingredients: quantum algorithm What is a quantum algorithm? A computational procedure which can be performed on a quantum system

20. Turing’s machine Quantum Computation

21. Spin Network Quantum Simulator The spin network simulator (SNQS) models bridge circuit schemes for standard quantum computation and notions from TQFTs. Its key tool is provided by the fiber space structure underlying the model, which exhibits combinatorial properties closely related to SU(2) state sum models. It can be thought of as non-Boolean version of the quantum circuit model, with unitary gates expressed in terms of: i) recoupling coefficients ( 3nj symbols) between inequivalent binary coupling schemes of N  (n+1) SU(2)-angular momenta; ii) Wigner rotations in the eigenspace of the total angular momentum.

22. i)the combinatorial structure – induced by the SU(2) coalgebra – allows representing any computation process as a path over a graph, as in the classical case. The graph is the base space of a fiber bundle which sustains the simulator dynamics as well as information coding. ii)the extension to the quantum deformed algebra su(2) q maps it to a quantum automaton structure; v)The 3n-strand braid group acts on the functor: it is this action that defines the evolution of the initial state. Spin Network Quantum Simulator

23. Spin Network Quantum Simulator

24. Hilbert spaces and Quantum Codes Alphabet and Words  n (V, E) Spin Network Quantum Simulator

25. where there appears the Racah -Wigner 6j symbol of SU(2) and f plays the role of the total angular momentum quantum number. 6j symbols satisfy consistency conditions given by: Explicitly: the Racah identities and the orthogonality relations the Biedenharn-Elliot equalities Spin Network Quantum Simulator

26. Racah Biedenharn Elliott bracketing words Spin Network Quantum Simulator

27. N.B. Mapping class group – Hatcher & Thurston Spin Network Quantum Simulator

28. The fiber space structure of the spin network simulator for (n+1) = 4 spins.Vertices and edges on the perimeter of the graph  3 (V, E) have to be identified through the antipodal map. The “blown up” vertex shows the local computational Hilbert space.  3 (V, E) J3()J3() SNQS the graph  3 (V, E)

29. cobordims pants pant decomposition Spin Network Quantum Simulator

30. Spin Network Quantum Simulator

31. In between part I and II: quantum computation Approximate evaluation of the Jones polynomial is BQP-c  BQP=Bounded error Quantum Polynomial time: it is the class of decision problems solvable by a quantum computer in polynomial time with an error probability < ¼  These are the problems which a quantum computer can “reasonably” solve  A BQP-complete problem is important to compare quantum computers and classical computers Bordewich, Freedman, Lovasz, Welsh, Approximate counting and quantum Computation, Comb. Probab. Comput. 14(2005), 737-754

32. Part II: the method We use the realization of the Jones polynomial in quantum field theory, i.e. as the expectation value of observables in Chern-Simons Topological Quantum Field Theory (CS-TQFT) In CS-TQFT the Jones polynomial is the expectation value of Wilson loop operators

33. Part II: the method Chern-Simons TQFT Is a 3-dimensional topological quantum field theory In TQFT the correlation functions do not depend on the metric of space-time and can be used to derive topological invariants k is a (integer) coupling parameter A is a connection one-form, valued in the Lie algebra of the group G (=SU(2)), the gauge group of the theory M is a 3-dimensional manifold E. Witten, Quantum field theory and the Jones polynomial, Comm. In Math. Phys. 121(1989), 351-399

34. Part II: the method

35. Chern-Simons TQFT To solve the theory it is important to use the connection between CS-TQFT and WZW-CFT WZW is constructed on a finite dimensional Hilbert space which is the space of conformal blocks

36. Part II: the method Chern-Simons TQFT The observables are called Wilson loop operators: ρ is an irreps of the gauge group G and C is a knot; T are the generators of SU(2) in representation ρ; A is a connection on the principal fibre bundle P(M,G) The expectation value of Wilson loop operators is a topological invariant of manifold M. In particular if G=SU(2) we have the Jones polynomial.

37. Part II: the method Quantum computing the Jones polynomial CS-TQFT We use CS-TQFT exact solution, through a unitary representation of the braid group, to provide a quantum algorithm for the evaluation of the Jones polynomial  given a knot present it as a closure of a braid  cut the braid with horizontal lines in such a way that between two lines there is at most one crossing  use the unitary representation of the braid group to explicitly evaluate the topological invariant R. Kaul, Chern-Simons theory, colored-oriented braids and links invariants, Comm. In Math.Phys. 162(1994), 289

38. Part II: the method The Kaul unitary representation of the braid group

39. Part II: the method The finite dimensional Hilbert space which we use to build the WZW-CFT representation is the space of conformal blocks of WZW-CFT The Kaul unitary representation of the braid group

40. Part II: the method # qubits # gates n is the index of the braid group

41. Part II: the method The unitary gate acting on the last register is block- diagonal and its dimension is fixed by the coupling constant k. It can be efficiently compiled by elementary unitary gates.

42. Part II: the method Measuring an auxiliary qubit entangled with the system we can obtain an approximate evaluation of the Jones polynomial efficiently

43. Results and discussion Efficient quantum algorithm for the approximation of the Jones polynomial It can be generalized to colored Jones polynomials It can be used to evaluate 3-manifold invariants Links with the theory of quantum automata in the framework of the q-deformed spin network simulator.

44. 3 manifolds