Download Quantum computing the Jones polynomial

Topological reach of
field-theoretical topological
quantum computation
Mario Rasetti
Politecnico di Torino
&
ISI Foundation
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
Scheme


Part I
 knot theory;
 The Jones polynomial;
 Computational complexity;
 Quantum computation;
Part II
 The Jones polynomial in QFT;
 Unitary representation of the braid group;
 The quantum circuit;
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?
Part I: the problem
Part 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
q q q
4
Part I: the problem
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
The braid group
The braid group on n strands
is a group with an
intuitive geometrical realization
Presentation
:  1 ,...,  n 1 |  i j   j i ; i i 1 i   i 1 i i 1
for
j  i 1
The Automaton based on the
Spin Network Quantum Simulator
accepts the Braid language
Part I: the problem
Part I: the problem
Part I: the problem
Part I: the problem
The Temperley-Lieb algebra
e1,..., en1
en2   en
emen  en em m  n  1
en1en en1  en1
en en1en  en
....
....
Part I: the problem
Knot-braid connection
A given link L
L
L (coloured)
can always be seen as the closure
of a braid (Alexander theorem)
Part I: the problem
Defining a representation
with coefficients in
such that
and
 i  Aei  A11
The Jones polynomial is given by
  A
3 w L 
 n 1Tr   
V.F.R. Jones, A polynomial invariant for links via von Neumann algebras,
Bull. Amer. Math. Soc. 129 (1985), 103-112.
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:
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
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...
That much for exact solutions, but what about
approximate solutions?
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
Part II: the method
Additive approximation

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 
In between part I and II:
quantum computation
What is a quantum algorithm?
A computational procedure which can be
performed on a quantum system
Ingredients:


Superposition
Entanglement
Quantum Computation
Turing’s machine
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.
Spin Network Quantum Simulator
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
Spin Network Quantum Simulator
Hilbert spaces
and Quantum Codes
n (V, E)
Alphabet and Words
Spin Network Quantum Simulator
Explicitly:
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:
the Biedenharn-Elliot equalities
the Racah identities
and the orthogonality relations
Spin Network Quantum Simulator
Racah
bracketing
Biedenharn
Elliott
words
Spin Network Quantum Simulator
N.B. Mapping class group – Hatcher & Thurston
J3()
SNQS
the graph 3 (V, E)
3 (V, E)
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.
Spin Network
Quantum Simulator
cobordims
pant decomposition
pants
Spin Network Quantum Simulator
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


 f I   f I 
 3
Pr 

4
u I 




Bordewich, Freedman, Lovasz, Welsh, Approximate counting and quantum
Computation, Comb. Probab. Comput. 14(2005), 737-754
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
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
2


S
Tr
A

dA

A

A

A


4 M 
3

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
Part II: the method
Part II: the method
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
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.
Part II: the method
Quantum computing the Jones polynomial
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
Part II:
the method
The Kaul unitary representation
of the braid group
Part II: the method
The Kaul unitary representation of the braid group
 i  U  i 
The finite dimensional Hilbert space which we use to build the
representation is the space of conformal blocks of WZW-CFT
Part II: the method
# qubits
 n  log  k  1 
# gates
 n  poly  k 
n is the index of the braid group
Bn
Part II: the method
The unitary gate acting on the last register is blockdiagonal and its dimension is fixed by the coupling
constant k.
It can be efficiently compiled by elementary unitary
gates.
Part II: the method
Measuring an auxiliary qubit entangled with the system
we can obtain an approximate evaluation of the Jones
polynomial efficiently
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.
3 manifolds