Quantum Computation and Algorithms

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A Quick Safari Through
Quantum Computation and
Algorithms

M. Reza Rahimi,
Computer Science Department,
Sharif University of Technology,
Tehran, Iran,
August 2005.

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Outline
 Introduction
 Quantum Physics
 Quantum Physics Foundations
 Classical Computation Using Reversible Gates
 Quantum Gates and Universal Quantum Gates
 Quantum Complexity Class BQP
 Case Study: Grovers’ Search Algorithm
 Conclusion

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Introduction
 Computation is basically a physical fact. This is the
origin of Church-Turing-Markov thesis, which
implies that:
A Partial function is computable (in any accepted informal sense)
if and only if it is computable by some binary Turing machine.

 In this case, Church-Turing’s thesis is saying that
the universe can be simulated by a Turing machine .

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 So if we know the rules of the universe, we can
make good physical model for computation.
 One of the first questions that leaded us toward
quantum computation was ” What is the minimum
energy for computing of a special problem?”
 In this case we must analysis our program in
respect of power consumption.

Landauer’s Principle:
Erasure of information is necessarily a dissipative process.
If the process of erasure is isothermal then the work needed,
is at least:
W=KTLn2.

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 This rule tells us that any physical computation
process that erases information, is energy
consuming process. ( this fact is derived
according to thermodynamics laws.)
 Charles Bennett found another interesting
principle in 1973 that :
“Any computation that can be carried out in the reversible
process is dissipating no power.”

 So if we can make reversible gates, we can make
computers that dissipate no power.
 This phenomenon will be very important if we want
to make VLSI chip. The generated hit may damage
the chip.

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 As we know: “the universe is fundamentally
quantum mechanics, and the rules of quantum
mechanics are reversible in time, what kinds of
problems quantum machine can solve for us?”
 The break of this question was Peter Shor
Algorithm about integer factorization in
polynomial time.
 It was not known that integer factorization has a
classical polynomial time algorithm or not.
 The time complexity of Shor algorithm for L- digit
number is:
O ( L2 Log ( L) Log ( Log ( L)))
 The best known classical algorithm runs in:
1 2
O (exp(cL Log 3 L))
3

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 So studying quantum computation is useful,
For examples :
2. In chip design industry,
3. In cryptography,
4. ……
 In this talk I focuse on theoretical point of
quantum computation. At first general
principles of quantum physics and reversible
computation is reviewd, then quantum
complexity class is defined, and finally I focus
on Grover’s search algorithm.

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Quantum Physics
 Quantum physics phenomena are very odd. Let’s
take a look at an example.
 In figure 1 we have the wall with two slits on it
and one electron gun which shoots electrons
( Young’s Experiment).
 The first experience:
Cover one of the slits and compute the number
of electrons that collide the wall.
 Figure 1 shows the results according to our
expectation.

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Figure 1: The result of the first experience.

The second experience :

Now use both of the slits and count the number
of electrons that collide the wall.
What do you expect?

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Figure 2: The result of the second experience.

 Very interesting result! it seems that electrons
behave like wave.
 The third experience:
Now use one detector in one of the slits and see
the movement of electrons. what do you expect to
see on the wall?

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 The experience shows that in this case we have
the result expected in classical physics, that
means the similar result drown in figure 1.
 So it seems that classical physics rules can not
describe subatomic phenomena.
 We need physical framework for subatomic
physics.

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Quantum Physics Foundations
 States: A state is a complete description of a
physical system. In quantum mechanics a state is a
ray in Hilbert Space . We use the following Dirac
Ket Notation.
ϕ = ∑αi i , αi ∈C
i

 Observables: The observable is a property of a
physical system that in principle can be measured.
In quantum mechanics an observable is a self-
adjoint operator:
A = At

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 Measurement: In quantum mechanics the
numerical outcome of a measurement of
observable A is an eigenvalue of A, and the
state of it is eigenstate. Briefly we have:
2 2
ϕ =α 0 + β 1 α + β =1

 Which α2 and β are the probability of system
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to be in state 0 or 1 after measurement.
 Dynamics: Time evolution of the system is
unitary we have Schrödinger equation.
d
ϕ(t ) = −iH ϕ(t )  → ϕ(t ) = U (t ) ϕ(0)

dt
U (t ) t U (t ) = I

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Examples:
 State of the n-qubit quantum register is:
∑α ∑α
2
ϕ = s S s =1
n
s∈ 0 ,1} n
{ s∈ 0 ,1}
{

 Suppose that we observe one qubit of
quantum register and see it is 0. what is the
state of the register after observation?

1
∑α s 0 ⊗ S + βs 1 ⊗ S 0observed in qreg[1]→
is
    ∑α s 0 ⊗S
∑α
2
s∈{0 ,1}n −1 s∈{0 ,1}n −1
s
s∈{0 ,1} n −1

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 For 1- qubit system we have:
ϕ =α 0 +β 1 α + β =1
2 2

 The above expression means that:
Pr[ After measurement the 1-qubit is in state 0]= α
2

Pr[ After measurement the 1-qubit is in state 1]= β
2

 For 2-qubit system we have:
2 2 2 2
ϕ = α 00 00 + α 01 01 + α 10 10 + α 11 11 , α 00 + α 01 + α 10 + α 11 = 1
2
Pr[ After measurement the 2-qubit is in state 00]= α 00
2
Pr[ After measurement the 2-qubit is in state 01]= α01
2
2

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Classical Computation Using Reversible
Gates
 As stated before if we want to achieve
minimum energy, we must use reversible gates.
f : {0,1}n −1→ 0,1}n
 {
1

 For example classical AND and OR gates are
not reversible.
 One of the most popular reversible gates is
Fredkin gate.
 The definition of this gate is as follow:

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f(a, b, c) = (a, if(a) then b else c, if(a) then c else b).

 It is easy to check that F(F(a,b,c))=(a,b,c).
 AND, OR, and NOT gates can be easily made up of
Fredkin gate as follow:
a a
a^b
b
Fredkin Gate
0 ¬a^b

Figure 3: AND gate Implementation.

a a
0 ¬a
Fredkin Gate
1 a

Figure 4: NOT gate Implementation.

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 So we can implement any logical circuit with
Universal Fredkin gate. (in linear size with
some control input bits).

Input Bits Output Bits

Fredkin Circuit

Control Input Bits Some Junk output

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Quantum Gates and Universal Quantum
Gates
 As it was said quantum gates are unitary
matrices. For example:

 u00 u01 
 1-input quantum gate is: U = 
u  , UU t = I

 10 u11 

α0 0 + β0 1 u u01  α 0 + β 1
U =  00
u 
 10 u11 


U (α0 0 + β0 1 ) = (α 0 + β 1 ).

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 For 2-input quantum gate we have:

 u00 u01 u02 u03 
 
α 00 00 + α 01 01 + α10 10 + α11 11 u u11 u12 u13  β 00 00 + β 01 01 + β10 10 + β11 11
U =  10  , UU = I
t

u u21 u22 u23
 20 
u u31 u32 u33 
 30 

 Generally for n-input quantum gate the matrix
size is: 2 × 2
n n

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 There are some examples of famous quantum
gates:
1 0 0 0 0 0 0 0
 
0 0 1 0 0 0 0 0
0 1 0 0 0 0 0 0

0 0

1 1 1 
F =
0 0 1 0 0 0
0
H= 
1 − 1

0 0 0 0 1 0 0
2
0 0 0 0 0 1 0 0


 
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1 8×8
 

Fredkin Quantum Gate Hadamard Quantum Gate

 Note that the Fredkin gate is permutation
matrix and Hadamard gate has this property
that if input is in state 0 or 1 the output state
will be symmetric. (Both are Unitary Matrix).

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 Suppose we have n-qbit register and on the
first qbit U operates. what is the new state of
the system?
U

n-qbit n-qbit

(U ⊗ I )( 0 ⊗ ϕ 0 + 1 ⊗ ϕ 1 ) = U 0 ⊗ I ϕ 0 + U 1 ⊗ I ϕ 1

 a00 B a01B ...
 
A ⊗ B =  a10 B ... ... ϕ ⊗ φ = ∑αiβ j i ⊗ j
i, j
 ... ... ...
 

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 As it is clear the set of all quantum gates are
uncountable, So one may ask are there any
small sets of universal quantum gates?
 The answer to this question is Yes.
 Researchers have shown that there are some
universal quantum gates that we can make every
quantum circuit with good approximation. (for
example Tofolli and Hadamard Gates is
universal set)
 But for now we only use Hadamard and Fredkin
gates.

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Quantum Complexity Class BQP
 Definition: A language L ⊂ {0,1}∗ is in BQP iff there is a set of
quantum circuit {Cn }of size n k that:
2
x ∈ L ⇒ Pr{C ( x)1 = 1} ≥
3
1
x ∉ L ⇒ Pr{C ( x)1 = 1} ≤
3

 Also the circuit must be uniform which means that a
n
deterministic polynomial time Turing machine with input 1 writes
the description of the circuit {Cn } .
 Note!
the Turing machine writes the approximation of the circuit
because each gate can have complex numbers and for complex
numbers we need generally infinite precession.

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 Theorems:
1. P ⊆ BQP
2. BPP ⊆ BQP
3. BQP ⊆ PSPACE
 For proving the first one we know that every
language in P has Polynomial size circuit, we
can easily replace it with Fredkin gate.
 For the second one we know that BPP has
polynomial size circuit with random control
bits.

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Case Study: Grover's Search Algorithm

 Problem Statement:
 There is quantum space of size N we want the target
state a .
 This Problem is usually called Quantum Database Search.
 For solving this problem we use Grover Search
Algorithm.
 Before presentation of algorithm lets define
some basic unitary operators.

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1 0 ... 0
 
0 1 ... 0
 . . a = −1 0
 ii

 0 ... 0 1  N ×N
 

Phase shift operator which changes the sign of the i th element .
It is Obvious that this operator is unitary.

−1 0 ... 0
 
0 1 ... 0
D = HN  H N , H N = H 2× 2 ⊗ H 2× 2 ⊗ ... ⊗ H 2× 2
... ... ... ...   
  Log 2 N Times
0 0 0 1
 

D is called diffusion operator .

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 Lemma: Diffusion operator has two properties:
 It is unitary and can be efficiently realized.
 It can be interpreted as “inversion about the mean”.
Proof: 
−
 1 0 ... 0

0 1 ... 0
D =H N ... HN
... ... ... 
 
0 0 ... 1
 
 −
 2 0 ... 0 
  
 0
 0 ... 0 
=H N  ...
 +I  N
H
... ... ...
  

 0 0 0 0 
  
−
 2 N − N
2 ... − N
2 
 
−
 2 N − N
2 ... − N
2 
= +I
... ... ... ...
 
 2 N
− − N
2 − N
2 − N
2 
 
 2 N +
− 1 − N
2 ... − N 
2
 
 − N
2 − N +
2 1 ... − N 
2
= 
... ... ... ...
 
 − N
2 − N
2 ... − N + 
2 1


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 α 1   β1 
   
 α2   β2 
 .  =  .  → β = − 2 α +α = −2µ + α .
N
D
    i ∑ j i
N j =1
i

 .   . 
α   β 
 N  N

µ µ

αi αj αi αj

µ

βi βj

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1
1. Start state is ϕ =∑
x .
x N
2. Invert the phase of a using phase shift
operator.
3. Then invert about the mean using D.
4. Repeat step 2 and 3 N times.
5. Measure.

 According to the last relation it is obvious
that after N we can measure a with
probability at least 0.5.

Running Time of The Grover Search Algorithm = O ( N ).

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Conclusion
 In this talk we took a glance at quantum
computation.
 It is clear that quantum computing can solve
some problems that are hard for classical
computers.
 Some people may ask “what is the
philosophical source of the power for
quantum machines?”
 Really the sources of the power of quantum
machines are quantum superposition and
quantum entanglement.

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 Talking about these properties is a little long
and deep so for more information you can see
books in quantum mechanics.
 Nowadays researchers spend a lot of time
working on theoretical and practical aspects of
quantum machines.

The END

Quantum Computation and Algorithms

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