Hashing anyons

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Topological Quantum Computation The Art of
Computing with Icosahedral Group
Giuseppe Mussardo SISSA-Trieste
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Integrable Systems and Loop Models
Lattice gauge theories
CFT, Fusion Rules and Commutative Algebras
Topological phases
Knot theories and Topological Invariants
Non-abelian anyons
Quantum Hall Effects, PiP Superconductivity, Col
d Atoms, Dimers
Quantum Computations
Interferometry
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If I have seen a bit further it is by standing
on the shoulders of giants

• Topological Quantum Computation

S. Kitaev, M. Freedman, J. Preskill,

• Topological Phases of matter

F. Wilczek, X. Wen, G. Moore, N. Read, E. Rezayi,
E. Fradkin, P. Fendley, M. Fisher, C. Nayak, F.
Bais, J. Stingerland, S. Das Sarma, A. Cappelli,
A. Stern

• braidings and fusions

E. Witten, E. Verlinde, Z. Wang, L. Kauffman, S.
Trebst, E. Ardonne, K. Schoutens, A. Ludwig, N.
Bonesteel, L. Hormozi, S. Simons,

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Topological Quantum Hashing With Icosahedral
Group
M. Burrello, H. Xu, G.M. and X. Wan
arXiv0903.1497
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Plan of the seminar

• Topological Phases of Matter

• Anyons

• Fibonacci, Ising and Cardano anyons

• The art of braiding

• Quantum computation and universal gates

• Icosahedron and topological hashing

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Topological Phases

• Topological entropy

• Order parameters Wilson loops

• Gapped spectrum.

Degeneracy of ground states which depends on
topology

• Anyonic excitations

• Fractional (non-abelian) statistics

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Braid group

• The world-lines in 21 of N anyons form
• a N strand braid

• These trajectories are robust with respect
• to local perturbations

• States in this space can only be distinguished
• by global measurements, i.e. a perfect place
• to store information

• Braids of N strands form an infinite group
• this can be seen as the permutation group
• with memory of its history

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Algebraic relations
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Basic quantities of non-abelian anyons

• Rules for fusing (and splitting) the excitations
  
• carrying conserved charges.

• Associativity of the fusion rules (F-matrices)

• Rules for braiding the excitations (R-matrices)

• Growing rates of their Hilbert space, alias
• quantum dimensions.

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Fusion algebra

• a is a non-abelian anyon if

• Associativity

where
are m x m matrices
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Verlinde formula and classification of FR

• There exists a real unitary matrix S that
  simultaneously diagonalize all matrices

• S is the modular S-matrix

• The classification of all FR consists of finding
  all possible
• real unitary matrices of dimension m x m

• The exaustive classification has been so far
  achieved up to m4

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Examples

• (Fibonacci anyon)

• (Ising anyons)

• (Cardano anyons)

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Pentagonal equations
The F-matrices have to be found as solutions of a
set of consistency equations
Examples

• Fibonacci anyons

,

• Ising anyons

,

• Cardano anyons

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Hexagonal equations
The R-matrices have to be found as solutions of a
set of consistency equations
Examples

• Fibonacci anyons

,

• Ising anyons

,

• Cardano anyons

,

,
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Quantum dimension
Alias, how fast the Hilbert space of n-anyons
grows
In the large n-limit, this is dominated by the
largest eigenvalues of Na
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Mathematical and physical features
The quantum dimensions da are, simultaneously,
Perron-Frobenius eigenvalues and eigenvectors of
the fusion algebra
Examples
Anyonic gas at equilibrium

• Fibonacci anyons

In the steady state, anyons of type a appear
with probability

• Ising anyons

• Cardano anyons

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Fibonacci anyons
1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
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Geometrical paradox
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Enlightening Fibonacci
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Counting the outcoming rays
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Quantum Computation
Quantum Circuits are unitary operators acting on
a Hilbert space, generated by n-qubits, whose
states encoded the information we want to process
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Divide et Impera
An Universal Quantum Computer is a device able to
implement any unitary operator in SU(N)
Every unitary operator in SU(N) can be decomposed
in (i) Single-qubit (element of U(2)) and
(ii) Controlled NOT gates

• Single-qubit rotation

• C-NOT gate

CNOT is the key element for creating Entanglement
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Braid realization of quantum gates
This will solve at once all problems of
decoherence
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Weaves
Any quantum computation that can be done by
braiding n identical quasi-particles can also be
done by moving only a single particle around the
n-1 other particles whose position remains fixed.
Simon et al. (PRL 2006)
p-generators
?-approximation
This simplification may greatly reduce the
technological difficulty for realizing
topological quantum computation
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Qubit for Fibonacci anyons
1 x 1 0 1
Two Fibonacci span a 2-dimensional Hilbert space
To have non-trivial operation we need however
three Fibonacci anyons
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Qubit and braidings of Fibonacci anyons
?1 and ?2 induce an ergodic motion on SU(2) group
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Cardano anyons
The same can be done with Cardano anyons

• The braiding matrices ?1 and ?2 of the field ?
  can span the whole SU(2)

• Those associated to the field ? can span instead
  the whole SU(3)

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Ising anyons
In this case the braiding matrices ?1 and ?2 of
the field ? cannot span the whole SU(2)
In fact, they just generate the finite sub-group
of SU(2) given by the cube
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Single qubit rotation
In the following we focus our attention only on
the single qubit rotation gate.
Ising anyons
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Brute force approach (Hormozi et al.)
Using the ergodicity properties, one can pin down
an arbitrary SU(2) gate by a brute force search
algorithm
All roads lead to Rome algorithm
Even though this search can be improved thanks to
Solovey-Kitaev algorithm, it becomes
nevertheless unfeasible for long braids
Can we do something better?
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Luca Pacioli De Divina Proportione
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Basic Geometric Data

• One of the 5 regular Platonic solids

• It has F20 triangular faces,
• E30 edges, V12 vertices

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Symmetries

• 15 axes of 2nd order

• 10 axes of 3rd order

• 6 axes of 5th order

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Icosahedron vs Dodecahedron
Finite group of 60 elements (isomorphic to A5)
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Festival of Golden Ratio
With the center placed at the origin and side
a2, the coordinates of the vertices are
Radius of circumscribed sphere (it touches all
the vertices)
Radius of inscribed sphere (it touches all the
faces)
Radius of middle sphere (it touches the middle
of all edges)
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Golden Rectangles and Borromean Rings
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How to pin down a gate by a finite number of
moves
HASHING STRATEGY

• Find by brute force, once for all, the 60
  generators of
• the icosahedral group, in terms of braids of
  a given
• length (say l8 or L24)

2. Use the pseudo-group structure so created to
set up a dense set of points in the Bloch
sphere.
3. Two-step process finite and infinitesimal
rotations.
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Icosahedral Pseudo-Group

• The pseudo-group is not a group and it is
  characterized by errors,
• depending on the chosen length in the braid
  representation

• Our algorithm definitely exploits these errors
  to create an efficient
• sampling of SU(2) !

Group
Pseudo-Group
Closure
Î is not closed
Identity can be obtained in many ways
We can span the vicinity of Identity in many ways
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Pre-processor L 8
Thanks to the errors of the approximation, with
the product of 3 elements of Î(8), we can span
all SU(2) with ?00.03
60 points
216.000 points
The pre-processor approximates the target with
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Hashing strategy. 1. Pre-processor
error
Target SU(2) gate
Pseudo-group icosahedral approximation
This requires only a finite number of searches!
After a rotation nearby the target gate, the
only thing left is to reduce the error near the
identity
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Hashing strategy. 2. Main processor
We can sample with high precision the vicinity of
the Identity.
How can we do it ?

• For any n-plet of rotations in I, we can find g
  n1 such that

Mapping it in the approximated pseudo-group Î(24)
with braid length L24, we obtain a fine
rotation R
where Hn is, essentially, an hermitian random
matrix
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The Hermitian matrix Hn is related to the
accumulated deviations and tends to a random
matrix with Gaussian unitary ensemble.
The distribution of the eigenvalue spacings
(alias of errors) must then satisfy the
Wigner-Dyson form
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Random matrices
The distribution of the eigenvalue spacings
(alias of errors) satisfy the Wigner-Dyson form
of Gaussian Unitary Ensemble
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Conclusions and open questions

• Hashing algorithm is a very efficient procedure
  to realize
• quantum gates by braids

• Role of anyon systems with higher number of
  excitations

• Topological phases of matter at criticality and
  SLE

• Deep connection between topological phases of
  matter and
• integrable models (Golden chain and
  generalization)