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Braid group representation on quantum computation
Ryan Kasyfil Aziz, and Intan Muchtadi-Alamsyah
Citation: AIP Conference Proceedings 1677, 030011 (2015);
View online: https://doi.org/10.1063/1.4930633
View Table of Contents: http://aip.scitation.org/toc/apc/1677/1
Published by the American Institute of Physics
Braid Group Representation on Quantum Computation
Ryan Kasyfil Aziz 1, a) and Intan Muchtadi-Alamsyah2, b)
1
Department of Computational Sciences, Bandung Institute of Technology.
2
Algebra Research Group, Bandung Institute of Technology
a)
kasyfilryan@gmail.com
b)
ntan@math.itb.ac.id
Abstract. There are many studies about topological representation of quantum computation recently. One of diagram
representation of quantum computation is by using ZX-Calculus. In this paper we will make a diagrammatical scheme of
Dense Coding. We also proved that ZX-Calculus diagram of maximally entangle state satisfies Yang-Baxter Equation
and therefore, we can construct a Braid Group representation of set of maximally entangle state,
Keywords
: Entangle State, Qubit and Quantum Gate, Yang-Baxter equation, Braid group, ZX-Calculus
PACS
: 03.67.-a
INTRODUCTION
It is known in [1] that some aspects in quantum computation satisfy the structure of Temperley-Lieb Algebra. It
is also known in [2] and [3] that there is a relation between quantum group, braid group, and Temperley-Lieb
Algebra. However, there is no direct correlation between quantum group and quantum computation yet.
On the other hand, ZX-Calculus is introduced in [4] and [5] as topological form of quantum computation, and it
can be used to represent all aspects in quantum computation. In this paper, we make the diagram representation of
Dense Coding using ZX-Calculus. We also proved that set of maximally entangled state forms braid group structure
by using ZX-Calculus. We hope the diagrammatical result can be an opening to construct a quantum group structure
of quantum computation in the near future.
QUANTUM COMPUTATION
Different from classical computation, quantum computation depends on quantum bit or qubit, which exists
because of the energy caused by spinning electron. Mathematically, qubits are vectors in vector space ℂ! and its
basis are
1
0
0 =
and 1 =
.
0
1
Quantum gate is an operation on qubits. It can be represented by matrix form. There are two important gates,
which are CNOT gate and Hadamard Gate. These two gates can be represented by matrices form below.
1 0 0 0
1 1 1
.
!!"#$ = 0 1 0 0 ; !! =
0 0 0 1
2 1 −1
0 0 0 1
YANG BAXTER EQUATION AND BRAID GROUP
Let ! be a vector space over a field !. A linear automorphism ! of ! ⊗ ! is called an !-matrix if and only if ! is
solution of the following Yang-Baxter equation
The 5th International Conference on Mathematics and Natural Sciences
AIP Conf. Proc. 1677, 030011-1–030011-4; doi: 10.1063/1.4930633
© 2015 AIP Publishing LLC 978-0-7354-1324-5/$30.00
030011-1
! ⊗ !!! !!! ⊗ ! ! ⊗ !!! = !!! ⊗ ! ! ⊗ !!! !!! ⊗ ! .
Suppose ! ≥ 3. Let !! be a braid group and !! , ⋯ , !!!! be a generator of !! . The braid group !! has following
properties
!! !! = !! !! , for 1 ≤ !, ! ≤ ! − 1
!! !!!! !! = !!!! !! !!!! , for ! − ! > 1.
Elements of braid group are usually represented by braid diagrams and its operation is a concatenation of the
diagrams. The diagrams concatenation will be used for operation in ZX-Calculus in the next chapter. The next
lemma is due to [2].
Lemma 1. Let V be a vector space, c a linear automorphism of ! ⊗ !, and n>1. Let !! be a linear
automorphism of ! ⊗! defined by
! ⊗ !!! ⊗ !!! , !" ! = 1
!" 1 < ! < ! − 1 !! = !!! ⊗ !!! ⊗ ! ⊗ !!! ⊗ !!!!! ,
!!! ⊗ !!! ⊗ !, !" ! = ! − 1
then !! !!!! !! = !!!! !! !!!! is satisfied if and only if ! is a solution of Yang-Baxter equation.
ZX-CALCULUS
ZX-Calculus ([4]) is a graphical calculus used to represent systems and processes in quantum computation. This
method is not only graphical notation, but also has operation rules on it. ZX-Calculus is basically a graphic
representation of matrices in quantum computation. Some elements of ZX-Calculus are :
• The !-component (green dot) and !-component (red dot), with ! inputs, ! outputs and a phase
! ∈ 0,2! .
• The ℋ-component (yellow square) with one input and one output.
The !-component represents a map 0 ⋯ 0 ↦ 0 ⋯ 0 , 1 ⋯ 1 ↦ ! !" 1 ⋯ 1 , and another qubits will be sent to
0, while !-component represents a mapping + ⋯ + ↦ + ⋯ + , − ⋯ − ↦ ! !" − ⋯ − , and another qubits will
be sent to 0,. The ℋ-component simply represents Hadamard matrix !! . The graphical form of each component of
ZX-calculus can be seen in [4]. We also used it in the next section.
RESULT 1 : ZX-CALCULUS REPRESENTATION OF DENSE CODING
Dense coding [6] is a method in quantum computation that sends a classical bit using quantum channel. Angelina
will send a bit ! ∈ {00, 01, 10, 11}. She encodes the bit into corresponding qubits.
Bit
State after transformation
00
! ! = 1/ 2 (|00〉 + |11〉)
01
! ! = 1/ 2(|10〉 + |01〉)
10
− ! ! = 1/ 2(|10〉 − |01〉)
11
! ! = 1/ 2(|00〉 − |11〉)
Angelina sends the encoded qubit to Brad. After he gets the qubit, Brad will decode it using the CNOT gate
first. He will obtain the corresponce decoded qubit as shown in the next table. Note that the first qubit belongs to
Angelina and second qubit belongs to Brad. Since |0〉 corresponds to 0 and |1〉 corresponds to 1, Brad can determine
the bit Angelina had sent to him. In the language of ZX Calculus, we represent all the dense coding process in the
following diagram.
Received state
CNOT output
First qubit
Second qubit
!! |first qubit〉
(after decoding)
|! ! 〉
|0〉
|0〉
1/ 2(|00〉 + |10〉)
1/ 2(|0〉 + |1〉)
|! ! 〉
|0〉
|1〉
1/ 2(|11〉 + |01〉)
1/ 2(|1〉 + |0〉)
!
−|! 〉
−|1〉
|1〉
1/ 2(|11〉 − |01〉)
1/ 2(|1〉 − |0〉)
|! ! 〉
|1〉
|0〉
1/ 2(|00〉 − |11〉)
1/ 2(|0〉 − |1〉)
030011-2
RESULT 2 : BRAID GROUP REPRESENTATION ON QUANTUM COMPUTATION
Maximally entangle state has an important role in quantum information and computation. Let ! be a vector space
over complex field. We define purely entangled bipartite state by Ω = 00 〈00| + 11 〈11|.We construct
maximally entangle state from Ω as follow.
1 0 0 0
!
! = Ω |Ω〉 = 0 0 0 0 .
0 0 0 0
0 0 0 1
The ZX-Calculus diagram of maximally entangle state is represented as follow.
Theorem 1. Maximal entangle state satisfies Yang-Baxter equation.
Proof. We have to prove that ! ⊗ !!! !!! ⊗ ! ! ⊗ !!! = (!!! ⊗ !)(! ⊗ !!! )(!!! ⊗ !) and we can
show it by diagrammatical approaches below.
Hence we conclude that ! is a solution of Yang-Baxter equation.
∎
We will construct braid group representation on quantum computation using !. First, construct !! ∈ ! ⊗! by
⊗(!!!)
⊗(!!!!!)
⊗ ! ⊗ !!!
for ! = 1, ⋯ , ! − 1. The diagram of !! ∈ ! is represented as follow.
!! = !!!
Relation between quantum computation and braid group is stated in this theorem.
Theorem 2 (Main Theorem). Group ! = 〈 !! ! 〉 is a braid group.
Proof. It is sufficient to prove that ! satisfies braid group relations. Because ! is a solution of Yang-Baxter
equation, then by Lemma 1, !! !!!! !! = !!!! !! !!!! is satisfied. We proved that !! !! = !! !! for ! − ! ≥ 2 by
diagrammatical approaches below
030011-3
Hence we conclude that ! is a braid group.
∎
REFERENCES
1.
2.
3.
4.
5.
Y. Zhang, Contemporary Mathematics in Advance in Quantum Computation, 49-89, 482 (2009).
C. Kassel, Quantum Groups, (Springer-Verlag, 1991), pp. 166-178.
W.B.R. Rickorish , An Introduction To Knot Theory, Graduate Text in Mathematics, (Springer-Verlag,
1997).
B. Coecke and R. Duncan, Interacting Quantum Observable : Categorical Algebra and Diagrammatics,
New Journal of Physics 13 043016, (2011).
M. Nakahara and T. Ohmi, Quantum Computing From Linear Algebra to Physical Realization, (CRC
Press, Boca Raton, 2008).
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