New frontiers for quantum information processing: from topological invariants to the theory of formal languages Mario Rasetti ScuDo & DIFIS @ PoliTO & ISI Foundation Algorithmic complexity: the central open problem in computer science is the conjecture whether the two complexity classes P (polynomial) and NP (non-deterministic polynomial; i.e. those decision problems for which a conjectured solution can be verified in polynomial time) are distinct or not within the standard Turing model of computation: P NP ? 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. There is 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 its own or ii) any quantum system is by itself a computing machine whose internal evolution can reproduce the proper dynamics of a class of physical systems. 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). 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. 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 diffeomorphism invariant (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 field theory. 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-Abelian TQFT 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. 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 interpolation between A and B. 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 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. 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 nmanifolds. Renormalization (Wilson;1970): How does the Langrangian evolve when reexpressed 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 Chern-Simons Action has one derivative: A d A + 2/3 (A A A) 2 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. Knots : what are they ? Knots Knots are equivalence classes with respect to isotopies Central problem knot theory is classification of knots : given two knots decide whether or not they are topologically equivalent. Classification is made by invariants in the form of polynomials, whose coefficients encode the topological properties of a class of knots (Jones, Alexander, etc.) The JP for the trefoil knot Complexity To evaluate the Jones polynomial is a #P-hard problem from the computational point of view There exist no efficient classical algorithms for its evaluation 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 Manifolds : What are they ? Manifolds are spaces every point of which has a neighbourhood homeomorphic to a Euclidean space The most general property of 3-manifolds is the "prime decomposition" : every compact orientable 3-manifold M decomposes uniquely as a connected sum M = P1 # # Pn of 3-manifolds Pi which are prime in the sense that they can be decomposed as connected sums only in the trivial way Pi = Pi # S3 Prime Manifolds Standard topological invariants were created in 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 ] On the contrary, Quantum Invariants of knots and three-manifolds were instead discovered, yet their 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. Homeomorphism invariants of 3-manifolds are the isotopy invariants of Knots and Links (invariants of homology cobordism) (George K Francis) Formal languages: 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. 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. 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. 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. Spin Network Quantum Simulator The spin network quantum simulator model bridges circuit schemes of quantum computation with TQFT. Its key tool is the "fibered graph-space" structure underlying it, which exhibits combinatorial properties related to SU(2) [SU(2)q] state sum models. Spin Network Quantum Simulator Spin Network Quantum Simulator Hilbert spaces Alphabet and Words and Quantum Codes Gn (V, E) Spin Network Quantum Simulator Racah bracketing words relations Biedenharn Elliott The two most important properties of 6j-symbols are their tetrahedral symmetry and the ElliottBiedenharn or pentagon identity. The tetrahedral symmetry is an equivariance property under permutation of the six labels, summarized by the labeled Mercedes badge: 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 ) 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) Pachner’s moves 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 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 Asymptotics (semiclassical limit: very large spins) 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 -hedron τ 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 V its volume. (Wigner) For k → ∞ (for k ∈ Z) there is an asymptotic formula 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-Elliot identity for 6j symbols H13(V) SNQS the computational graph G 3 (V, E) G3 (V, E) Fiber space structure of the spin network simulator for 4 spins. Vertices and edges on the perimeter of the graph G3 (V, E) have to be identified through the antipodal map. The “blown up” vertex shows the local computational Hilbert space. SNQS State transformations Quantum amplitudes: (s-cl : Feynman path sum) From the Spin Network Quantum Simulator to the Spin Network Quantum Automaton Cobordims & pant decomposition pants SNQA is defined by the 5-tuple and therefore can be thought of as a quantum recognizer (Wiesner and Crutchfield ) A quantum recognizer is a particular type of finite-states quantum machine defined as a 5-tuple {Q, H, X, Y, T( Y|X )}, Q is a set of basis states, the internal states of the machine; H is a Hilbert space in which a particular (normalized) state, | H is singled out as ''start state'' expressed in the basis Q; X and Y { a, r, } (a accept , r reject , the null symbol) are finite alphabets for input and output symbols respectively; T( Y|X ) is the subset of transition matrices. These general axioms can be adapted to make the machine able to recognize a language L endowed with a word-probability distribution p(w) over the set of words {w} L . For any w=x y z L the recognizer one-step transition matrix elements are obtained by reading each individual symbol in w. The recognizer upgrades the start state to U (w) | U(z) U (y) U (x) |. The Spin Network Quantum Automaton as Quantum Recognizer The Spin Network Quantum Automaton (SNQA) is the quantum finite-state machine generated by deformation of the Spin Network Quantum Simulator structure algebra (su(2)q instead of su(2)). With this assumption SNQA recognizes the language of the Braid Group. From the Spin Network Quantum Simulator to the Spin Network Quantum Automaton The Braid Group elements composition identity inverse Generators & Relations The additive approximation: if a quantum circuit of dimension O(poly(n)) operates over n qubits, and if is a pure state of n qubits which can be prepared in O(poly(n)) time, then it is possible to construct a statistical ensemble in which, sampling for a O(poly(n)) time two random variables X, Y one has E[X+iY]=|U| Quantum Computation Tools The formal definition of the Jones polynomial J(q) is given in terms of: a trace of the braid group representation into the Temperley Lieb algebra J(q) = (– A) -3w(L) d n–1 Trr (s) [ Let R be a commutative ring and λ R. The Temperley-Lieb algebra Ln (λ) is the Hecke R-algebra generated by elements U1U2 … Un-1, subject to relations : Ui Ui = λ Ui for all 1 i n-1 UiUi + 1Ui = Ui for all 1 i n-2 UiUi − 1Ui = Ui for all 2 i n-1 UiUj = UjUi for all 1 i,j n-1 such that |i – j| 1 ] here: r is a representation of the braid group Bn in Ln -1 with coefficients in c [ q , q ] and parameter -1 2 -2 d = - q - q , such that s q U + q I , and w (L) is the writhe number f or link L , V.F.R. Jones, A polynomial invariant for links via von Neumann algebras, Bull. Amer. Math. Soc. 129 (1985), 103-112. Knot-braid connection A given link L L (coloured) can always be seen as the closure of a braid (Alexander theorem) The standard closure of a braid pattern inside a 3-manifold The plat-closure of a braid inside a 3-manifold Use CS-TQFT exact solution, through a unitary representation of the braid group: given a knot present it as a closure of a braid cut the braid with horizontal lines so that between two lines there is at most one crossing use the unitary representation of the braid group to evaluate the (conformal) topological invariant R. Kaul, Chern-Simons theory, colored-oriented braids and links invariants, Comm. In Math.Phys. 162(1994), 289 The circuit Complexity # gates n poly (k) # qubits n log (k+1) n = index of the braid group Bn Measuring auxiliary qubit entangled with the system one can obtain an approximate efficient evaluation of the Jones polynomial . Modular functions (theta functions coherent states) A typical fundamental domain for the action of the modular group on the upper half-plane (2 + 1) (2 + 6) [ mod 14] Modular Group The braid group B3 is the universal central extension of the modular group Application to formal language theory A metaphorical and exhaustive study-case for all above notions is the efficient solution of Dehn’s word problem for the Dehornoy group B (which includes braid and Thompson's groups). Its presentation extends the standard presentations of both, starting from a geometric approach according to which the elements of B can be seen as parenthesized braids. Every element of B generates a free subsystem with respect to the bracketing (self-distribuive) operation (x(yz)) = ((xy)(xz)) The trefoil knot : parenthesized quantum presentation Conclusions and perspectives Quantum symbolic manipulation Quantum (artificial) Intelligence Emergence of structures in languages what next ? Origin of Life: molecules as message passers With photosynthesis, cyanobacteria are able to transfer sunlight energy to molecular reaction centers for conversion into chemical energy with 100% efficiency. Speed is the key: transfer of solar energy (single photons) takes place almost instantaneously so little energy is wasted as heat. The green sulfur bacterium Quantum coherence influences energy transfer in photosynthesis; when it hits the bacterial protein, the light energizes a series of reactions that ultimately lead the protein to emit light of its own. Individual electrons coordinate their movements ("entanglement" ??) as they jostle energy back and forth: shifts to the left or right make electrons connect, while vertical shifts imply energy being passed or received. The two crucial questions in developmental biology are: how does one tell when there is communication in living systems? how does developmental control depend on the meaning of communication? A better comprehension is required of the processes whereby molecules can function symbolically, i.e., as records, codes (messages) and signals. One seeks to know how a molecule can and does become a message; answering the question: what is the simplest set of physical conditions that would allow matter to branch into two pathways – the living and lifeless – but under a single set of microscopic dynamical laws? And how large (complex) a system one must consider before biological function has a meaning? Know how to distinguish communication between molecules from physical interactions between molecules. Make such distinction at the simplest possible level, since the answer to the basic question about the origin of life cannot come from highly evolved organisms, in which communication processes are clear, distinct and complex. The crucial question is to know how messages originated. The alphabet: {A,C,G,T} A molecule does not become a message because of any particular shape or structure or behaviour of the molecule itself. A molecule becomes a message only in the context of a larger system of physical constraints, that can be thought of as a "language". Aim to a higher level than that of conventional quantum physics: not molecular structures only but the structure of language which they mutually communicate with. Spin Network Quantum Automaton can possibly provide an answer!! Hereditary processes are crucial: biology asserts that the mystery of heredity is solved at the molecular level by the structure of DNA and the laws of chemistry. Current molecular biological interpretation of hereditary transmission begins with DNA which replicates by a "template" process and then passes its hereditary information to RNA, which in turn codes the synthesis of the proteins. Proteins function primarily as (enzyme) catalysts. Thus the "central dogma" of biology asserts that hereditary information passes from the nucleic acids to the proteins, and never the other way around: for this reason the most primitive hereditary reactions at the origin of life plausibly occur in template replicating nucleic acid molecules. The crucial logical point is here that the hereditary propagation of a trait, involving a code as it does, implies a classification process and not simply the operation of the physical laws of motion on a set of initial conditions. Such dynamical laws depend only on the immediate past whereas only through the notion of a physical system able to manipulate information one can associate the concept of memory, description, code and classification. In this complex framework quantum correlations must be recognized as relevant.

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# Quantum information, topological invariants and formal