topological reach of field-theoretical topological quantum computation

Topological reach of Topological reach of field-theoretical field-theoretical

topological quantum topological quantum computationcomputation

Mario RasettiMario Rasetti Politecnico di Torino

&ISI Foundation

PreliminariesPreliminaries 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

SchemeScheme Part IPart I

knot theory; The Jones polynomial; Computational complexity; Quantum computation;

Part IIPart II The Jones polynomial in QFT; Unitary representation of the braid

group; The quantum circuit;

Part I: the problemPart I: the problemKnot 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?

Part I: the problemPart I: the problem

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

1 2 4q q q


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 a trace of the braid group representation into the the Temperley Lieb algebraTemperley Lieb algebra


Part I: the problemPart I: the problemThe braid groupThe braid group

The braid group on n strands is a group with anintuitive geometrical realization

Presentation 1 1 1 1 1,..., | ;n i j j i i i i i i i

for 1j i

The Automaton based on the Spin Network Quantum Simulator

accepts the Braid language

:

Part I: the problemPart I: the problemThe Temperley-Lieb algebra

2

1 1 1

1

1n n

m n n m

n n n n

n n n n

e ee e e e m ne e e ee e e e

.... ....

1 1,..., ne e

Part I: the problemPart I: the problemKnot-braid connection

A given link L

L L (coloured)

can always be seen as the closure of a braid (Alexander theorem)


11i iAe A

The Jones polynomial is given by 3 1w L nA Tr

V.F.R. Jones, A polynomial invariant for links via von Neumann algebras, Bull. Amer. Math. Soc. 129 (1985), 103-112.

Defining a representationwith coefficients in and

such that


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: the Jones polynomial is a #PP-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

Part I: Computational Part I: Computational ComplexityComplexity

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?

Part I: Computational Part I: Computational ComplexityComplexity

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

Part II: the methodPart 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

E X iY U

Additive approximation

In between part I and II:In between part I and II:quantum computationquantum computation

SuperpositionSuperposition EntanglementEntanglement

Ingredients:

What is a quantum algorithmquantum algorithm?A computational procedure which can be performed on a quantum system

Turing’s machineTuring’s machine

Quantum ComputationQuantum Computation

Spin Network Quantum SimulatorSpin 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.

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 Spin Network Quantum SimulatorQuantum Simulator

Hilbert spacesHilbert spaces

and Quantum and Quantum CodesCodes

Alphabet and Words

nn (V, E) (V, E)


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


Racah

Biedenharn Elliott

bracketing

words


N.B. Mapping class group – Hatcher & ThurstonN.B. Mapping class group – Hatcher & Thurston


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()

SNQSSNQS

the graphthe graph 3 (V, E)3 (V, E)

cobordims

pants

pant decomposition

Spin Network Spin Network Quantum SimulatorQuantum Simulator

In between part I and II:In between part I and II:quantum computationquantum computation

3Pr4

f I f Iu I

Approximate evaluation of the Jones polynomial is Approximate evaluation of the Jones polynomial is BQP-cBQP-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

Part II: the methodPart 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


24 3M

kS Tr A dA A A A

Chern-Simons TQFTIs a 3-dimensional topological quantum field theoryIn TQFT the correlation functions do not depend on the metric of space-time and can be used to derive topological invariantstopological invariants

k is a (integer) coupling parameterA is a connection one-form, valued in the Lie algebra of the group G (=SU(2)), the gauge group of the theoryM is a 3-dimensional manifold

E. Witten, Quantum field theory and the Jones polynomial, Comm. In Math. Phys. 121(1989), 351-399

Part II: the Part II: the methodmethod

Part II: the methodPart II: the method Chern-Simons TQFTTo 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

Part II: the methodPart II: the method Chern-Simons TQFTThe 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.

Part II: the methodPart II: the method Quantum computing the Jones polynomial

We use CS-TQFTCS-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 invariantR. Kaul, Chern-Simons theory, colored-oriented braids and links invariants, Comm. In Math.Phys. 162(1994), 289

Part II: Part II: the methodthe method

The Kaul unitary representation of the braid group


i iU

The finite dimensional Hilbert space which we use to build the representation is the space of conformal blocks of WZW-CFTWZW-CFT

The Kaul unitary representation of the braid group


log 1n k

nB

# qubits

# gates

n is the index of the braid group

n poly k


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.


Measuring an auxiliary qubit entangled with the system we can obtain an approximate evaluation of the Jones polynomial efficiently

Results and discussionResults 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.

3 3 manifoldsmanifolds

topological reach of field-theoretical topological quantum computation

Documents

problemthe jones polynomial

jones polynomial results

polynomial invariant

problema knot polynomial

quantum version

quantum computer

phard problems

braid language