Mario RasettiMario RasettiScuDo & DIFIS @ PoliTOScuDo & DIFIS @ PoliTO

&&ISI Foundation ISI Foundation

New frontiers for quantum New frontiers for quantum information processinginformation processing: :

from topological invariants to the from topological invariants to the theory of formal languagestheory of formal languages

Algorithmic complexityAlgorithmic complexity::

the central open problem in the central open problem in computer science is the conjecture computer science is the conjecture whether the two complexity classes whether the two complexity classes PP (polynomial) and(polynomial) and NPNP (non-(non-deterministic polynomial; i.e. those deterministic polynomial; i.e. those decision problems for which a decision problems for which a conjectured solution can be verified conjectured solution can be verified in polynomial time) are distinct or in polynomial time) are distinct or not within the standard Turing not within the standard Turing model of computation: model of computation:

P P NP NP ??

It is by now generally assumed that It is by now generally assumed that each physical theory supports each physical theory supports computation modelscomputation models whose power whose power is limited by the physical theory is limited by the physical theory itself. itself.

Classical physics,Classical physics, quantum quantum mechanics and topological mechanics and topological quantum field theoryquantum field theory (TQFT) (TQFT) are are believed to support a multitude of believed to support a multitude of different implementations of the different implementations of the Turing machine (or equivalent: Turing machine (or equivalent: boolean circuits, automata) model boolean circuits, automata) model of computation. of computation.

There is a conceptual dilemma There is a conceptual dilemma here: whetherhere: whether

i) an abstract universal model of i) an abstract universal model of computation, able to simulate any computation, able to simulate any discrete quantum system, including discrete quantum system, including solvable topological field theories, solvable topological field theories, exists exists on its ownon its own

ororii) any quantum system is by itself ii) any quantum system is by itself a computing machine whose a computing machine whose internal evolution can reproduce internal evolution can reproduce the proper dynamics of a class of the proper dynamics of a class of physical systems.physical systems.

The capability of quantum The capability of quantum information theory of efficiently information theory of efficiently computing topological or geometric computing topological or geometric quantities was first conjectured by quantities was first conjectured by Michael Freedman and co-workers. Michael Freedman and co-workers.

Their 'topological quantum Their 'topological quantum computation' setting, is designed computation' setting, is designed to comply with the behavior of to comply with the behavior of 'modular functors' of Chern-'modular functors' of Chern-Simons-Witten 3-D non-abelian Simons-Witten 3-D non-abelian topological quantum field theory, topological quantum field theory, with gauge group SU(2). with gauge group SU(2).

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

The term "The term "topological quantum field topological quantum field theorytheory" is used to refer to two " is used to refer to two distinct but related concepts: i) any distinct but related concepts: i) any quantum field theory in which the quantum field theory in which the actionaction is is diffeomorphism invariantdiffeomorphism invariant (the best known example is Chern-(the best known example is Chern-Simons theory); ii) any structure Simons theory); ii) any structure satisfying the Atiyah axioms. satisfying the Atiyah axioms.

The two concepts are not unrelated. The two concepts are not unrelated. The matrix elements of the linear The matrix elements of the linear transformation corresponding to a transformation corresponding to a cobordism are analogous to the cobordism are analogous to the transition amplitudes that one would transition amplitudes that one would compute by path integral in a compute by path integral in a conventional formulations of conventional formulations of quantum quantum field theory.field theory.

A universal model of computation, A universal model of computation, capable of solving (in the additive capable of solving (in the additive approximation) approximation) # # PP problems in problems in polynomial time stems out of a polynomial time stems out of a discrete, finite version of a non-discrete, finite version of a non-Abelian TQFT with Chern-Simons Abelian TQFT with Chern-Simons action action

It can be thought of as an analog It can be thought of as an analog computer able to solve a variety of computer able to solve a variety of hard problems in hard problems in TopologyTopology (knots (knots and manifolds invariants), in and manifolds invariants), in Formal Language TheoryFormal Language Theory and and perhaps in perhaps in Life ScienceLife Science..

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

For example, the circle SFor example, the circle S11 is a 1- is a 1-manifold, and a tube Smanifold, and a tube S11 × [0, 1] is a × [0, 1] is a cobordism between two circles. A 2-cobordism between two circles. A 2-dimensional TQFT associates a dimensional TQFT associates a Hilbert space Hilbert space H H SS11 to S to S1 1 and a linear and a linear transformation Mtransformation M0 0 the tube the tube

A different cobordism between the A different cobordism between the same pair of boundaries may be same pair of boundaries may be mapped to a different linear mapped to a different linear transformation between the same transformation between the same pair of Hilbert spacespair of Hilbert spaces

If we compose two cobordisms, we If we compose two cobordisms, we compose the corresponding linear compose the corresponding linear transformations.transformations.

Mathematicians express this propert Mathematicians express this propert saying that a TQFT is a "saying that a TQFT is a "functorfunctor". ". The linear transformation The linear transformation associated to a cobordism by a TQFT associated to a cobordism by a TQFT depends only on the topology of the depends only on the topology of the cobordism, not the geometric cobordism, not the geometric details.details.

For example, MFor example, M00 ◦ M ◦ M00 = M = M00. To the . To the empty boundary empty boundary we associate the we associate the Hilbert space Hilbert space CC. We can think of a . We can think of a closed manifold as a cobordism closed manifold as a cobordism between between and and . Therefore an n-. Therefore an n-dimensional TQFT associates to any dimensional TQFT associates to any closed n-manifold a map from closed n-manifold a map from CC to to CC, that is, a complex number. Such , that is, a complex number. Such map is a map is a CC-valued -valued topological topological invariantinvariant of closed n-manifolds. of closed n-manifolds.

RenormalizationRenormalization ( (Wilson;1970): Wilson;1970):

How does the Langrangian evolve How does the Langrangian evolve when re-expressed using longer and when re-expressed using longer and longer length scales, i.e., lower longer length scales, i.e., lower frequencies, colder temperatures ? frequencies, colder temperatures ? The terms with the fewest The terms with the fewest derivatives dominate because in derivatives dominate because in momentum space, differentiation momentum space, differentiation becomes multiplication by becomes multiplication by kk and and: k k k k

>> > > 22

Chern-Simons Action has one Chern-Simons Action has one derivative: derivative: A d A + 2/3 (A A d A + 2/3 (A A A A) A)

while kinetic energy p /2m is while kinetic energy p /2m is written with written with two derivatives (p = - i h d/dx) two derivatives (p = - i h d/dx) Thus, in condensed matter at Thus, in condensed matter at low enough low enough temperatures, temperatures, we may expect we may expect to see systems to see systems in which the in which the topological topological effects dominate effects dominate and geometric detail becomes and geometric detail becomes irrelevant. irrelevant.

22

//

KnotsKnots : what are they ? : what are they ?

KnotsKnots

Central problem knot theory is Central problem knot theory is classification of knots : given two classification of knots : given two knots decide whether or not they knots decide whether or not they are topologically equivalent. are topologically equivalent. Classification is made by invariants Classification is made by invariants in the form of polynomials, whose in the form of polynomials, whose coefficients encode the topological coefficients encode the topological properties of a class of knotsproperties of a class of knots (Jones, Alexander, etc.) (Jones, Alexander, etc.)

Knots are equivalence Knots are equivalence classes classes

with respect to isotopieswith respect to isotopies

The JP for The JP for the trefoil the trefoil

knotknot

To evaluate the Jones polynomial To evaluate the Jones polynomial is a is a #P-hard #P-hard problem from the problem from the computational point of viewcomputational point of view

There exist no efficient classical There exist no efficient classical algorithms for its evaluationalgorithms for its evaluation

ComplexityComplexity

Jaeger, Vertigan and Welsh, Jaeger, Vertigan and Welsh, On the computational complexity of the Jones and On the computational complexity of the Jones and Tutte PolynomialsTutte Polynomials, , Mathematical Proceedings of the Cambridge Phil. Soc. Mathematical Proceedings of the Cambridge Phil. Soc. 108(1990), 35-53108(1990), 35-53

ManifoldsManifolds : : What are they ?What are they ?

Manifolds are spaces every Manifolds are spaces every point of which has a point of which has a neighbourhood homeomorphic neighbourhood homeomorphic to a Euclidean space to a Euclidean space The most general property of The most general property of 3-manifolds is the "prime 3-manifolds is the "prime decomposition" : decomposition" :

every compact orientable 3-every compact orientable 3-manifold manifold MM decomposes decomposes uniquely as a connected sum uniquely as a connected sum M = PM = P11 # # # P # Pnn of 3-of 3-manifolds manifolds PPii which are prime which are prime in the sense that they can be in the sense that they can be decomposed as connected decomposed as connected sums only in the trivial waysums only in the trivial way PPii = P= Pii # S # S33

Prime ManifoldsPrime Manifolds

Standard topological invariants Standard topological invariants were were createdcreated in order to distinguish in order to distinguish between things: it is their intrinsic between things: it is their intrinsic definition that makes clear what definition that makes clear what kind of properties they reflect, e.g., kind of properties they reflect, e.g., the Euler number the Euler number χχ of a smooth, of a smooth, closed, oriented surface closed, oriented surface SS defined defined asas

χ(S) = 2 − 2gχ(S) = 2 − 2g, ,

where genus where genus gg is the number of is the number of handles of handles of S, S, fully determines its fully determines its topological type. topological type.

[ [ χ χ can be evaluated upon can be evaluated upon tessellation by Euler’s formula tessellation by Euler’s formula χ(S) χ(S) = V + F – E= V + F – E ; ; VV= # Vertices ; = # Vertices ; FF = # = # Faces ; Faces ; EE = # Edges ] = # Edges ]

On the contrary, Quantum On the contrary, Quantum Invariants of knots and three-Invariants of knots and three-manifolds were instead manifolds were instead discovereddiscovered, , yetyet their indirect their indirect construction, based as it is on construction, based as it is on quantum technology, provides quantum technology, provides information about the purely information about the purely topological properties we were topological properties we were unable to detect, even to hint.unable to detect, even to hint.

Beyond prime decomposition, 3-Beyond prime decomposition, 3-manifolds admit as well a manifolds admit as well a canonical decomposition along canonical decomposition along tori rather than spheres. tori rather than spheres.

Homeomorphism invariants of Homeomorphism invariants of 3-manifolds are the isotopy 3-manifolds are the isotopy invariants of Knots and Links invariants of Knots and Links (invariants of homology (invariants of homology cobordism)cobordism)

(George K Francis) (George K Francis)

Formal languagesFormal languages: what are they ?: what are they ?The basic ingredient of a languageThe basic ingredient of a languageis its alphabet is its alphabet AA. An alphabet is a . An alphabet is a finite set of symbols. finite set of symbols.

A language A language LL is a sequence of finite is a sequence of finite sequences of symbols over the sequences of symbols over the alphabet alphabet AA ( (wordswords). ).

All sequences in a language are All sequences in a language are finite, yet the language itself can be finite, yet the language itself can be infinite. Any non-empty set of infinite. Any non-empty set of languages over finite alphabets languages over finite alphabets defines a family of languages. defines a family of languages.

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

Any formal language can be Any formal language can be reconducted to a machine reconducted to a machine which recognizes it: which recognizes it:

regular setsregular sets are recognized by are recognized by finite state finite state automataautomata,, context-free languagescontext-free languages are are recognized by recognized by (non- deterministic) pushdown (non- deterministic) pushdown automataautomata, , recursively enumerable recursively enumerable and and recursive sets recursive sets respectively, are recognized by respectively, are recognized by Turing Turing machinesmachines and and halting Turing halting Turing machinesmachines. .

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

Spin Network Spin Network Quantum SimulatorQuantum Simulator

The spin network quantum The spin network quantum simulator model bridges circuit simulator model bridges circuit schemes of quantum schemes of quantum computation with TQFT. computation with TQFT.

Its key tool is the "fibered graph-Its key tool is the "fibered graph-space" structure underlying it, space" structure underlying it, which exhibits combinatorial which exhibits combinatorial properties related to SU(2) properties related to SU(2) [SU(2)[SU(2)qq] state sum models. ] state sum models.

Spin Network Spin Network Quantum Quantum SimulatorSimulator

Hilbert Hilbert spacesspaces

and Quantum and Quantum CodesCodes

Alphabet and Alphabet and WordsWords

GGnn (V, E) (V, E)

Spin Network Quantum Spin Network Quantum SimulatorSimulator

RacRacahah

BiedenharBiedenharnn

ElliottElliott

bracketingbracketing

wordwordss

Spin Network Quantum SimulatorSpin Network Quantum Simulator

relationsrelations

The two most important properties The two most important properties of 6j-symbols are their tetrahedral of 6j-symbols are their tetrahedral symmetry and the Elliott-symmetry and the Elliott-Biedenharn or pentagon identity. Biedenharn or pentagon identity.

The tetrahedral symmetry is an The tetrahedral symmetry is an equivariance property under equivariance property under permutation of the six labels, permutation of the six labels, summarized by the labeled summarized by the labeled Mercedes badgeMercedes badge:

( N.B. Cfr. Mapping Class Group – Hatcher ( N.B. Cfr. Mapping Class Group – Hatcher & Thurston )& Thurston )

Spin Network Quantum Spin Network Quantum SimulatorSimulator

The Elliott-Biedenharn identity expresses the The Elliott-Biedenharn identity expresses the fact that the composition of five successive fact that the composition of five successive change-of-basis operators inside a space of change-of-basis operators inside a space of 5-linear invariants is the 5-linear invariants is the identity.identity.

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

(the 6j symbol is invariant under the 24 (the 6j symbol is invariant under the 24 symmetries of the tetrahedron)symmetries of the tetrahedron)

Pachner’s movesPachner’s moves

Pachner’s theoremPachner’s theorem

Two triangulations specify the same Two triangulations specify the same 3-manifold if and only if they are 3-manifold if and only if they are connected by a finite sequence of connected by a finite sequence of the 2-3 and 1-4 moves and their the 2-3 and 1-4 moves and their inversesinverses

In the Ponzano-Regge model, given In the Ponzano-Regge model, given a triangulated 3-manifold one a triangulated 3-manifold one associates one j-variable to each associates one j-variable to each edge of each tetrahedron. j-edge of each tetrahedron. j-variables represent quantum spins variables represent quantum spins and take integer and half-integer and take integer and half-integer values. To a closed manifold the values. To a closed manifold the Ponzano-Regge model associates Ponzano-Regge model associates the amplitude :the amplitude :

Notice that the 6j symbol is Notice that the 6j symbol is invariant under the 6! = 24 invariant under the 6! = 24 symmetries of the spin tetrahedronsymmetries of the spin tetrahedron

associate to the six labels associate to the six labels aa, , bb, . . . , . . . ff a metric tetra -hedron a metric tetra -hedron ττ with with these these as side lengths. as side lengths. conditions conditions guarantee that the individual faces guarantee that the individual faces may be realized in Euclidean 2-may be realized in Euclidean 2-space. space. ττ has an isometric has an isometric embedding into Euclidean or embedding into Euclidean or Minkowskian 3-space according to Minkowskian 3-space according to the sign of the Cayley determinant. the sign of the Cayley determinant. If If ττ is Euclidean, let is Euclidean, let θθaa, , θθbb, . . . , , . . . , θθff be its corresponding exterior be its corresponding exterior dihedral angles dihedral angles and and VV its volume. its volume.

Asymptotics (semiclassical limit: Asymptotics (semiclassical limit: very large spins)very large spins)

conditions: a ≤ b + c ; b ≤ c + a ; c ≤ a + b ; a + b + c = evenconditions: a ≤ b + c ; b ≤ c + a ; c ≤ a + b ; a + b + c = even

For k → ∞ (for k ∈ For k → ∞ (for k ∈ ZZ) there is an ) there is an asymptotic formulaasymptotic formula

(Wigner)(Wigner)

For a three-dimensional quantum For a three-dimensional quantum field theory on triangulated field theory on triangulated manifolds to be topological, it manifolds to be topological, it should be independent of should be independent of triangulation, that is, invariant triangulation, that is, invariant under the Pachner moves. under the Pachner moves.

N.B.: the Ponzano-Regge model N.B.: the Ponzano-Regge model fails to be fully topological in fails to be fully topological in general, as it is invariant only general, as it is invariant only under the 2-3 Pachner move as a under the 2-3 Pachner move as a consequence of the Beidenharn-consequence of the Beidenharn-Elliot identity for 6j symbols Elliot identity for 6j symbols

Fiber space structure of the spin network simulator for Fiber space structure of the spin network simulator for 44 spins. spins. Vertices and edges on the perimeter of the graphVertices and edges on the perimeter of the graph GG33 (V, E) have to (V, E) have to

be identified through the antipodal map. The “blown up” vertex be identified through the antipodal map. The “blown up” vertex shows the local computational Hilbert space.shows the local computational Hilbert space.

G3 (V, E)

H13(V)

SNQSSNQS

the computational the computational graph graph G (V, E) G (V, E)33

SNQSSNQS

State transformationsState transformations

Quantum amplitudes: Quantum amplitudes: (s-cl : Feynman path sum)(s-cl : Feynman path sum)

pantpantss

CobordimsCobordims & & pant pant decompositiondecomposition

From the Spin From the Spin Network Quantum Network Quantum SimulatorSimulator to the Spin to the Spin Network Quantum Network Quantum AutomatonAutomaton

SNQASNQA is defined by the is defined by the 5-tuple5-tuple

and therefore can be thought of as and therefore can be thought of as a quantum recognizer (a quantum recognizer (WiesnerWiesner and and CrutchfieldCrutchfield ) )

A quantum recognizer is a particular type of finite-states quantum A quantum recognizer is a particular type of finite-states quantum machine defined as a 5-tuple machine defined as a 5-tuple

{Q, {Q, HH, X, Y, T( Y|X )}, , X, Y, T( Y|X )},

Q is a set of basis states, the internal states of the machine; Q is a set of basis states, the internal states of the machine; HH is a Hilbert space in which a particular (normalized) state, is a Hilbert space in which a particular (normalized) state, || HH is singled out as ''start state'' expressed in the basis Q; is singled out as ''start state'' expressed in the basis Q; X and Y X and Y { a, r, { a, r, } (a } (a accept , r accept , r reject , reject , the null symbol) the null symbol) are finite alphabets for input and output symbols respectively; are finite alphabets for input and output symbols respectively; T( Y|X ) is the subset of transition matrices. T( Y|X ) is the subset of transition matrices.

These general axioms can be These general axioms can be adapted to make the machine able adapted to make the machine able to recognize a language to recognize a language LL endowed endowed with a word-probability distribution with a word-probability distribution p(w)p(w) over the set of words over the set of words {w} {w} L L . .

For any For any w=x y w=x y z z LL the the recognizer one-step transition recognizer one-step transition matrix elements are obtained by matrix elements are obtained by reading each individual symbol inreading each individual symbol in ww. .

The recognizer upgrades the start The recognizer upgrades the start state to state to

U (w) |U (w) | U(z) U(z) U (y) U (x) | U (y) U (x) |. .

The Spin Network Quantum The Spin Network Quantum Automaton Automaton

as Quantum Recognizer as Quantum Recognizer The Spin Network Quantum The Spin Network Quantum Automaton (Automaton (SNQASNQA) is the quantum ) is the quantum finite-state machine generated by finite-state machine generated by deformation of the Spin Network deformation of the Spin Network Quantum Simulator structure Quantum Simulator structure algebra (algebra (su(2)su(2)qq instead of instead of su(2)su(2)).).

With this assumptionWith this assumption SNQA SNQA recognizes the language of the recognizes the language of the Braid Group.Braid Group.

From the Spin Network Quantum From the Spin Network Quantum SimulatorSimulator

to the Spin Network Quantum to the Spin Network Quantum AutomatonAutomaton

The Braid GroupThe Braid Group

elementselements compositioncomposition

identityidentity

inverseinverse

Generators Generators &&

RelationsRelations

The additive The additive approximationapproximation::

if a quantum circuit of dimension if a quantum circuit of dimension O(poly(n)) O(poly(n)) operates overoperates over nn qubits, qubits, and ifand if is a pure state ofis a pure state of n n qubits which can be prepared in qubits which can be prepared in O(poly(n))O(poly(n)) time, then it is possible time, then it is possible to construct a statistical ensemble to construct a statistical ensemble in which, sampling for ain which, sampling for a O(poly(n)) O(poly(n)) time two random variablestime two random variables X X,, Y Y one has one has

E [ X + i Y ] = E [ X + i Y ] = | U | | U |

The formal definition of the Jones The formal definition of the Jones polynomial polynomial J(q)J(q) is given in terms of: is given in terms of: a trace of the braid group a trace of the braid group representation into the Temperley representation into the Temperley Lieb algebraLieb algebra

Quantum Computation ToolsQuantum Computation Tools

J(q) = (– A) J(q) = (– A) Tr Tr(())[ Let [ Let RR be a commutative ring and be a commutative ring and λλ R R. The Temperley-Lieb algebra L. The Temperley-Lieb algebra Ln n ((λλ) is ) is the Hecke the Hecke RR-algebra generated by elements -algebra generated by elements UU11UU2 2 … U … Un-1n-1, subject to relations :, subject to relations :

UUi i UUi i = = λλ UUi i for all for all 1 1 i i n-1 n-1 UUiiUUii + 1 + 1UUii = = UUii for all 1 for all 1 i i n-2 n-2 UUiiUUii − 1 − 1UUii = = UUii for all 2 for all 2 i i n-1 n-1UUiiUUjj = = UUjjUUii for all 1 for all 1 i,j i,j n-1 such that |i – j| n-1 such that |i – j| 1 ] 1 ]

-3w(L) n – 1-3w(L) n – 1

is a representation of the braid is a representation of the braid group Bgroup Bnn in in LLnn with coefficients in with coefficients in cc [ q , q ] [ q , q ] and parameter and parameter = - q - q = - q - q , , such that such that q U + q U + q q I , I , andandw (L) is the writhe number f or link w (L) is the writhe number f or link L , L , V.F.R. Jones, V.F.R. Jones, A polynomial invariant for A polynomial invariant for links via von Neumann algebraslinks via von Neumann algebras, Bull. , Bull. Amer. Math. Soc. 129 (1985), 103-112.Amer. Math. Soc. 129 (1985), 103-112.

-1-1

here: here:

-1-12 -22 -2

Knot-braid connectionKnot-braid connection

A given A given

link link LL LL (coloured)(coloured)

can always be seen as the can always be seen as the closure of a braid closure of a braid

((Alexander theoremAlexander theorem))

The plat-closure of a The plat-closure of a braid inside a 3-manifoldbraid inside a 3-manifold

The standard closure of a braid The standard closure of a braid pattern inside a 3-manifoldpattern inside a 3-manifold

Use CS-TQFT exact solution, through a unitary Use CS-TQFT exact solution, through a unitary representation of the braid group: representation of the braid group:

given a knot present it as a closure of given a knot present it as a closure of a braida braid cut the braid with horizontal lines so cut the braid with horizontal lines so that between that between two lines there is at most one two lines there is at most one crossingcrossing use the unitary representation of the use the unitary representation of the braid group to braid group to evaluate the (conformal) topological evaluate the (conformal) topological invariantinvariant

R. Kaul, R. Kaul, Chern-Simons theory, colored-oriented braids and links Chern-Simons theory, colored-oriented braids and links invariantsinvariants, Comm. In Math.Phys. 162(1994), 289, Comm. In Math.Phys. 162(1994), 289

The circuitThe circuit

ComplexityComplexity

n = index of the braid n = index of the braid groupgroup

BBnn

# gates # gates n n poly (k) poly (k)# qubits # qubits n n log (k+1)log (k+1)

Measuring Measuring auxiliary qubit auxiliary qubit entangled with entangled with the system one the system one can obtain an can obtain an approximate approximate efficient efficient evaluation of evaluation of the Jones the Jones polynomial .polynomial .

Modular functionsModular functions((theta functions theta functions coherent states) coherent states)

A typical fundamental domain for A typical fundamental domain for the action of the modular group the action of the modular group

on the upper half-planeon the upper half-plane

(2 (2 + 1) + 1) (2 (2 + 6) + 6)[ mod 14][ mod 14]

The braid group The braid group BB33 is the universal is the universal central extension of the modular groupcentral extension of the modular group

Modular GroupModular Group

A metaphorical and exhaustive study-A metaphorical and exhaustive study-case for all above notions is the case for all above notions is the efficient solution of Dehn’s word efficient solution of Dehn’s word problem for the Dehornoy group problem for the Dehornoy group BB (which includes braid and Thompson's (which includes braid and Thompson's groups). Its presentation extends the groups). Its presentation extends the standard presentations of both, standard presentations of both, starting from a geometric approach starting from a geometric approach according to which the elements of according to which the elements of BB can be seen as can be seen as parenthesized braidsparenthesized braids. . Every element of Every element of BB generates a free generates a free subsystem with respect to the subsystem with respect to the bracketing (self-distribuive) operation bracketing (self-distribuive) operation

(x(yz)) = ((xy)(xz))(x(yz)) = ((xy)(xz))

Application to formal language theoryApplication to formal language theory

The trefoil The trefoil knot : knot :

parenthesizeparenthesized d quantum quantum

presentation presentation

Conclusions and perspectivesConclusions and perspectives Quantum symbolic Quantum symbolic manipulation manipulation Quantum (artificial) Quantum (artificial) Intelligence Intelligence Emergence of structures in Emergence of structures in languageslanguages

what next ? what next ? Origin of Life: molecules as Origin of Life: molecules as message passersmessage passers

With With photosynthesisphotosynthesis, , cyanobacteria cyanobacteria are able to are able to transfer sunlight energy to transfer sunlight energy to molecular reaction centers for molecular reaction centers for conversion into chemical energy conversion into chemical energy with with 100% efficiency. Speed is 100% efficiency. Speed is the key: transfer of solar energy the key: transfer of solar energy (single photons) takes place (single photons) takes place almost instantaneously so little almost instantaneously so little energy is wasted as heat. energy is wasted as heat.

The green sulfur bacteriumThe green sulfur bacterium

Quantum coherence influences Quantum coherence influences energy transfer in photosynthesis; energy transfer in photosynthesis; when it hits the bacterial protein, when it hits the bacterial protein, the light energizes a series of the light energizes a series of reactions that ultimately lead the reactions that ultimately lead the protein to emit light of its own. protein to emit light of its own. Individual electrons Individual electrons coordinatecoordinate their movements their movements ("("entanglement" ?entanglement" ??) as they jostle ?) as they jostle energy back and forth: shifts to energy back and forth: shifts to the left or right make electrons the left or right make electrons connect, while vertical shifts imply connect, while vertical shifts imply energy being passed or received.energy being passed or received.

The two crucial questions in The two crucial questions in developmental biology are: developmental biology are: how does one tell when there how does one tell when there is is communication in living communication in living systems? systems? how does developmental how does developmental control depend control depend on the meaning of on the meaning of communication?communication?

A better comprehension is A better comprehension is required of the processes required of the processes whereby molecules can function whereby molecules can function symbolically, symbolically, i.e.i.e., as records, , as records, codes (messages) and signals.codes (messages) and signals.

One seeks to know how a One seeks to know how a molecule can and does become a molecule can and does become a message; answering the question: message; answering the question: what is the simplest set of what is the simplest set of physical conditions that would physical conditions that would allow matter to branch into two allow matter to branch into two pathways – the living and lifeless – pathways – the living and lifeless – but under a single set of but under a single set of microscopic dynamical laws? microscopic dynamical laws?

And how large (complex) a system And how large (complex) a system one must consider before one must consider before biological biological function has a meaning? function has a meaning?

Know how to distinguish Know how to distinguish communication between molecules communication between molecules from physical interactions between from physical interactions between molecules. molecules. Make such distinction at the Make such distinction at the simplest possible level, since the simplest possible level, since the answer to the basic question about answer to the basic question about the origin of life cannot come from the origin of life cannot come from highly evolved organisms, in which highly evolved organisms, in which communication processes are communication processes are clear, distinct and complex. clear, distinct and complex.

The crucial question is to The crucial question is to know how messages know how messages originated. originated.

The alphabet: The alphabet:

{ A , C , G , T } { A , C , G , T }

A molecule does not become a A molecule does not become a message because of any particular message because of any particular shape or structure or behaviour of shape or structure or behaviour of the molecule itself. A molecule the molecule itself. A molecule becomes a message only in the becomes a message only in the context of a larger system of context of a larger system of physical constraints, that can be physical constraints, that can be thought of as a "language". thought of as a "language".

Aim to a higher level than that of Aim to a higher level than that of conventional quantum physics: not conventional quantum physics: not molecular structures only but the molecular structures only but the structure of language which they structure of language which they mutually communicate with. mutually communicate with.

Spin Network Quantum Automaton Spin Network Quantum Automaton can possibly provide an answer!!can possibly provide an answer!!

Hereditary processes are crucial: Hereditary processes are crucial: biology asserts that the mystery of biology asserts that the mystery of heredity is solved at the molecular heredity is solved at the molecular level by the structure of DNA and level by the structure of DNA and the laws of chemistry. the laws of chemistry.

Current molecular biological Current molecular biological interpretation of hereditary interpretation of hereditary transmission begins with DNA transmission begins with DNA which replicates by a "template" which replicates by a "template" process and then passes its process and then passes its hereditary information to RNA, hereditary information to RNA, which in turn codes the synthesis of which in turn codes the synthesis of the proteins. Proteins function the proteins. Proteins function primarily as (enzyme) catalysts. primarily as (enzyme) catalysts.

Thus the "central dogma" of biology Thus the "central dogma" of biology asserts that hereditary information asserts that hereditary information passes from the nucleic acids to the passes from the nucleic acids to the proteins, and never the other way proteins, and never the other way around: for this reason around: for this reason the most primitive the most primitive hereditary reactions hereditary reactions at the origin of life at the origin of life plausibly occur in plausibly occur in template replicating template replicating nucleic acid molecules. nucleic acid molecules.

The crucial logical point is here The crucial logical point is here that the hereditary propagation of that the hereditary propagation of a trait, involving a code as it does, a trait, involving a code as it does, implies a classification process and implies a classification process and not simply the operation of the not simply the operation of the physical laws of motion on a set of physical laws of motion on a set of initial conditions. Such dynamical initial conditions. Such dynamical laws depend only on the immediate laws depend only on the immediate past whereas only through the past whereas only through the notion of a physical system able to notion of a physical system able to manipulate information one can manipulate information one can associate the concept of memory, associate the concept of memory, description, code and classification.description, code and classification. In this complex framework In this complex framework quantum correlations must be quantum correlations must be recognized as relevant.recognized as relevant.