Mario Rasetti ScuDo & DIFIS @ PoliTO & ISI Foundation

1. New frontiers for quantum information processing: from topological invariants to the theory of formal languages • Mario Rasetti • ScuDo & DIFIS @ PoliTO • & • ISI Foundation

2. Algorithmiccomplexity: the central open problem in computer science is the conjecturewhether the twocomplexityclassesP(polynomial) andNP(non-deterministicpolynomial; i.e. thosedecisionproblemsforwhich a conjecturedsolution can beverified in polynomialtime) are distinct or notwithin the standard Turingmodelofcomputation: ? P NP

3. It is by now generally assumed that each physical theory supports computation models whose power is limited by the physical theory itself. Classical physics,quantum mechanics and topological quantum field theory(TQFT) are believed to support a multitude of different implementations of the Turing machine (or equivalent: boolean circuits, automata) model of computation.

4. Thereis a conceptual dilemma here: whether i) an abstract universal model of computation, able to simulate any discrete quantum system, including solvable topological field theories, exists on itsown or ii) any quantum system is by itself a computing machine whose internal evolution can reproduce the proper dynamics of a class of physicalsystems.

5. The capability of quantum information theory of efficiently computing topological or geometric quantities was first conjectured by Michael Freedman and co-workers. Their 'topological quantum computation' setting, is designed to comply with the behavior of 'modular functors' of Chern-Simons-Witten 3-D non-abelian topological quantum field theory, with gauge group SU(2).

6. In physicists’ language, functors are partition functions and correlators of the quantum theory; in view of gauge invariance and invariance under diffeomorphisms, which freeze out local degrees of freedom, they share a global, 'topological' character.

7. The term "topological quantum field theory" is used to refer to two distinct but related concepts: i) any quantum field theory in which the action is diffeomorphisminvariant (the best known example is Chern-Simons theory); ii) any structure satisfying the Atiyah axioms. The two concepts are not unrelated. The matrix elements of the linear transformation corresponding to a cobordism are analogous to the transition amplitudes that one would compute by path integral in a conventional formulations of quantum fieldtheory.

8. A universal model of computation, capable of solving (in the additive approximation) # P problems in polynomial time stems out of a discrete, finite version of a non-AbelianTQFT with Chern-Simons action It can be thought of as an analog computer able to solve a variety of hard problems in Topology (knots and manifolds invariants), in Formal Language Theory and perhaps in Life Science.

9. A n-dimensional axiomatic topological quantum field theory (TQFT) is a map that associates a Hilbert space to any (n-1)-manifold, and to any n-dimensional manifold "interpolating" between a pair of (n-1)-dimensional manifolds, and associates a linear transformation between the corresponding Hilbert spaces: cobordism, defined as a triple (M,A,B) where M is an n-manifold whose boundary is the disjoint union of (n-1)-manifolds A and B. This provides the notion of interpolationbetween A and B.

10. For example, the circle S1 is a 1-manifold, and a tube S1 × [0, 1] is a cobordism between two circles. A 2-dimensional TQFT associates a Hilbert space H S1 to S1 and a linear transformation M0 the tube A different cobordism between the same pair of boundaries may be mapped to a different linear transformation between the same pair of Hilbert spaces

11. If we compose two cobordisms, we compose the corresponding linear transformations. Mathematicians express this propert saying that a TQFT is a "functor". The linear transformation associated to a cobordism by a TQFT depends only on the topology of the cobordism, not the geometric details.

12. For example, M0 ◦ M0 = M0. To the empty boundary we associate the Hilbert space C. We can think of a closed manifold as a cobordism between and . Therefore an n-dimensional TQFT associates to any closed n-manifold a map from C to C, that is, a complex number. Such map is a C-valued topological invariant of closed n-manifolds.

13. Renormalization (Wilson;1970): How does the Langrangian evolve when re-expressed using longer and longer length scales, i.e., lower frequencies, colder temperatures ? The terms with the fewest derivatives dominate because in momentum space, differentiation becomes multiplication by k and: k k 2 > >

14. Chern-Simons Action has one derivative: • A d A + 2/3 (A  A  A) • while kinetic energy p /2m is written with • two derivatives (p = - i h d/dx) • Thus, in condensed matter at low enough • temperatures, • we may expect • to see systems • in which the • topological • effects dominate • and geometric detail becomes irrelevant. 2 /

15. Knots : what are they ? Knots

16. Knots are equivalenceclasses withrespecttoisotopies Centralproblemknottheoryisclassificationofknots : giventwoknots decide whether or notthey are topologicallyequivalent. Classificationismadebyinvariants in the formofpolynomials,whosecoefficientsencode the topologicalpropertiesof a classofknots(Jones, Alexander, etc.) The JP for the trefoilknot

17. Complexity Toevaluate the Jones polynomialis a #P-hard problemfrom the computationalpointofview Thereexist no efficientclassicalalgorithmsforitsevaluation Jaeger, Vertigan and Welsh, On the computationalcomplexityof the Jones and Tutte Polynomials, MathematicalProceedingsof the Cambridge Phil. Soc. 108(1990), 35-53

18. Manifolds: What are they ?

19. Manifolds are spaceseverypointofwhichhasa neighbourhoodhomeomorphicto a Euclideanspace The most general property of 3-manifolds is the "prime decomposition" : every compact orientable 3-manifold Mdecomposes uniquely as a connected sum M = P1 #  # Pnof 3-manifolds Piwhich are prime in the sense that they can be decomposed as connected sums only in the trivial wayPi = Pi # S3

20. Prime Manifolds

21. Standard topological invariants were createdin order to distinguish between things: it is their intrinsic definition that makes clear what kind of properties they reflect, e.g., the Euler number χ of a smooth, closed, oriented surface S defined as χ(S) = 2 − 2g, where genus g is the number of handles of S, fully determines its topological type. [ χ can be evaluated upon tessellation by Euler’s formula χ(S) = V + F – E ; V= # Vertices ; F = # Faces ; E = # Edges ]

22. On the contrary, Quantum Invariants of knots and three-manifolds were instead discovered, yettheir indirect construction, based as it is on quantum technology, provides information about the purely topological properties we were unable to detect, even to hint. Beyond prime decomposition, 3-manifolds admit as well a canonical decomposition along tori rather than spheres.

23. Homeomorphisminvariantsof 3-manifolds are the isotopyinvariantsofKnots and Links (invariantsofhomologycobordism) (George K Francis)

24. Formallanguages: what are they ? The basic ingredient of a language is its alphabet A. An alphabet is a finite set of symbols. A language L is a sequence of finite sequences of symbols over the alphabet A (words). All sequences in a language are finite, yet the language itself can be infinite. Any non-empty set of languages over finite alphabets defines a family of languages.

25. Many families of formal languages are known, including the four families of the Chomsky Hierarchy (regular sets, context-free languages, context sensitive languages and recursively enumerable sets), recursive sets, and indexed languages.  rigorous formal (group theoretical) setting of context-free languages and of formal language theory.

26. Any formal language can be • reconducted to a machine • which recognizes it: • regular sets are recognized by finite state • automata, • context-free languages are recognized by • (non- deterministic) pushdown automata, • recursively enumerable and recursive sets • respectively, are recognized by Turing • machines and halting Turing machines.

27. Alternatively, a formal language may be generated (i.e., defined) by the set of its grammatical rules, as it is the case for indexed languages, recognized by one way nested stack automata, and generated by grammars.

28. Spin Network Quantum Simulator The spin network quantum simulator modelbridgescircuitschemesof quantum computationwith TQFT. Its key toolis the "fiberedgraph-space" structureunderlyingit, whichexhibitscombinatorialpropertiesrelatedto SU(2) [SU(2)q] state sum models.

29. Spin Network Quantum Simulator

30. Spin Network Quantum Simulator Hilbertspaces Alphabet and Words and Quantum Codes Gn (V, E)

31. Spin Network Quantum Simulator Racah bracketing Biedenharn Elliott relations words

33. The two most important properties of 6j-symbols are their tetrahedral symmetry and the Elliott-Biedenharn or pentagon identity. The tetrahedral symmetry is an equivariance property under permutation of the six labels, summarized by the labeled Mercedes badge:

34. Spin Network Quantum Simulator The Elliott-Biedenharn identity expresses the fact that the composition of five successive change-of-basis operators inside a space of 5-linear invariants is the identity. ( N.B. Cfr. Mapping Class Group – Hatcher & Thurston )

36. Ponzano-Regge approximation associates linear transformations to 3-manifolds, thought of as cobordisms between 2-manifolds. Among the ways of describing 3-manifolds, the most intuitive is by triangulation: a prescription of tetrahedra and of which face is "glued" to which. For example, we could take two tetrahedra and glue their faces: (the 6j symbol is invariant under the 24 symmetries of the tetrahedron)

37. Pachner’smoves Pachner’s theorem Two triangulations specify the same 3-manifold if and only if they are connected by a finite sequence of the 2-3 and 1-4 moves and their inverses

38. In the Ponzano-Regge model, given a triangulated 3-manifold one associates one j-variable to each edge of each tetrahedron. j-variables represent quantum spins and take integer and half-integer values. To a closed manifold the Ponzano-Regge model associates the amplitude : Notice that the 6j symbol is invariant under the 6! = 24 symmetries of the spin tetrahedron

39. Asymptotics (semiclassicallimit: verylargespins)  conditions: a ≤ b + c ; b ≤ c + a ; c ≤ a + b ; a + b + c = even associate to the six labels a, b, . . . f a metric tetra -hedronτ with these as side lengths.  conditions guarantee that the individual faces may be realized in Euclidean 2-space. τ has an isometric embedding into Euclidean or Minkowskian 3-space according to the sign of the Cayley determinant. If τ is Euclidean, let θa, θb, . . . , θf be its corresponding exterior dihedral angles and Vits volume.

40. (Wigner) For k → ∞ (for k ∈ Z) there is an asymptotic formula

41. For a three-dimensional quantum field theory on triangulated manifolds to be topological, it should be independent of triangulation, that is, invariant under the Pachner moves. N.B.: the Ponzano-Regge model fails to be fully topological in general, as it is invariant only under the 2-3 Pachner move as a consequence of the Beidenharn-Elliotidentityfor 6j symbols

42. H13(V) SNQS the computational graphG (V, E) 3 G3 (V, E) Fiber space structure of the spin network simulator for4spins. Vertices and edges on the perimeter of the graphG3 (V, E)have to be identified through the antipodal map. The “blown up” vertex shows the local computational Hilbert space.

43. SNQS State transformations Quantum amplitudes: (s-cl : Feynmanpath sum)

44. From the Spin Network Quantum Simulatorto the Spin Network Quantum Automaton Cobordims & pant decomposition pants

45. SNQAisdefinedby the 5-tuple and therefore can bethoughtofas a quantum recognizer (Wiesner and Crutchfield ) • A quantum recognizeris a particulartypeoffinite-states quantum • machinedefinedas a 5-tuple • {Q, H, X, Y, T( Y|X )}, • Q is a set ofbasisstates, the internalstatesof the machine; • His a Hilbertspace in which a particular (normalized) state, • | Hissingled out as ''start state'' expressed in the basis Q; • X and Y  { a, r, } (a  accept , r  reject ,   the nullsymbol) • are finite alphabetsfor input and output symbolsrespectively; • T( Y|X ) is the subset oftransitionmatrices.

46. Thesegeneralaxioms can beadaptedtomake the machineabletorecognize a languageLendowedwith a word-probabilitydistributionp(w)over the set ofwords{w} L . Foranyw=x y  z Lthe recognizerone-steptransitionmatrixelements are obtainedbyreadingeachindividualsymbol inw. The recognizerupgrades the start state to U (w) | U(z)  U (y) U (x) |.

47. The Spin Network Quantum Automaton as Quantum Recognizer The Spin Network Quantum Automaton (SNQA) is the quantum finite-state machinegeneratedbydeformationof the Spin Network Quantum Simulator structure algebra (su(2)qinsteadofsu(2)). WiththisassumptionSNQArecognizes the languageof the Braid Group.

48. From the Spin Network Quantum Simulator to the Spin Network Quantum Automaton

49. The Braid Group elements composition

50. identity inverse