Quantum bits and Gates

Sunita Kumari( 20122035)

November 2022

1 Introduction and vector q is given by:

 
All the speedups of quantum comput- α
ers are possible because of the charac- β
teristics of quantum bits.Quantum bit
coefficients in α|0> + β|0> qbit
is known as Qbit.Computers we are
are complex numbers and this is called
using these days are classic comput-
a superposition state. so a qubit can be
ers.Basic unit of these classic comput-
in a superposition of two basic states
ers is called bit.A bit can store either
zero and one. in quantum computing
a value of 0 or a value of 1.qubit is the
a qbit is not a definite state but rather
smallest unit of quantum information
it has the probability to be either in
just as the smallest bits of classical in-
the zero state or in the one state at
formation are 0 and 1. but there is
any given time and this probability can
a very big difference between bits and
actually be measured accurately if we
qbits.in classical computing we only have
know the values of coefficients a and b.
two states whereas in quantum com-
probability of being in the zero state is
puting we have qbits that are lies on
|α|2 and being in the one state is |β|2 .
scale {0,1}.

2 Qbit Single qbits are the basic units of

computation in any kind of quantum
Quantum bits can be in states that are computing system.
loosely equivalent to |0> and |1>. where single qubit is nothing but a vector
|0> is called a Zero state and |1> is in complex two-dimensional Hilbert space
called a One state. |0> and |1> are and it can be represented by very gen-
thought of unit vectors rather than states.erally as an arbitrary ket which is the
A qubit can be more than |0> and |1> linear combination of our orthonornal
though it can be a linear combination basis state zero and one whose inner
of a product is 0 and coefficients happen to
|0⟩ be complex numbers.
when we actually look at qubits they
and probabilistic collapse to either of two
|1⟩ states, the state |0> or the state |1>
. |q> = α|0> + β|1>. which is the collapse of the wave func-
tion.

1
|q> = α|0> + β|1>
probability of being a state |0> is :
p
α/ (α2 + β 2 ).

probability of being a state |1> is :

p
β/ (α2 + β 2 ).

Each vector on the bloch sphere can

<q|q> = 1 is the condition of nor- be represented in two basis: θ and ϕ.
malization and here |q> is called nor- The first is which is the angle between
malized state. Probabilities |α|2 and the vector and the z-axis. The second
|β|2 , summing to 1 follows quantum is which is the angle between the vec-
physical requirements and here |α|2 and tor and the positive x-axis measuring
|β|2 are probability amplitudes. counter-clockwise. From the image, it
is clear that we can achieve all possible
vectors in the Bloch sphere using these
3 Qbit representation two angles. Here, θ is between [0, π]
and ϕ is between [0, 2π].
Bloch Sphere
the Bloch sphere is a geometrical rep-
|q> =
resentation of the pure state space of
cos(
a two level quantum mechanical sys-
tem(qubit) θ/2)|0> + exp(iϕ) sin(θ/2)|1>
Two qubit states |0〉 and |1〉 are
represented by z and -z axes respec-
tively. |0〉 state denotes upward spin
of electron and |1〉 spin denotes down- .
ward spin of electron. Any point |q>
on this sphere is represented by equa- |θ,ϕ> =
2
tion |q> = α|0> + β|1>, where |α| 
cos θ/2
is probability of of electron having up-
(exp(iϕ/2))sin θ/2
ward spin and |β|2 is probability of elec-
tron having downward spin. As these
are probabilities we can also say that North pole corresponds to |0> state.
2 2 south pole(θ=π and ϕ=0) corresponds
|α| + |β| =1.
to |1> state.
The Bloch Sphere is is a generaliza-
tion of the representation of a complex Assume that θ = 0, This means
2 that: |q> = 1(|0〉)+exp(jϕ)(0)|1〉 =
number z with |z| = 1 as a point on
the unit circle in the complex plane. |0〉.
Now assume that θ = π, we get:

2
|q> = (0)|0〉+exp(jϕ)(1)|1〉 = exp(jϕ)|1>of the qubits, say the first qubit, and
= |1>. you can probably guess how this works:
measuring the first qubit alone gives 0
The point where the positive x-axis with probability |α00|2 +|α01|2 ,
meets the equator,
√ θ = π/2 and ϕ =√0. The post measurement state of this
Then, |q> = (1/
√ 2)|0>+(exp(j01/ 2))|1> = would then be this state nor-
system
(|0> + |1>)/ 2, malized appropriately mainly this di-
√
vided by —α00|2 +|α01|2 . Now there
T hepointwherethenegativex−axismeetstheequator,ϕ
= π, we get:
√ √ is a very important way in which the
|q> = (1/ 2)|0>+(exp(jπ/
√ 2)|1> = two qubit state and of course in prin-
(|0> − |1>)/ 2. ciple the multiqubit states they differ
from a single qubit state.

Now the point is this that depend-

4 Multiple Qbits ing upon the values of these constants
α which in general are complex,  
quantum bits it can take a linear super- 1
position of these. And in fact it is this As we know the state —0> was
0
linear superposition which provides the  
0
quantum computing with enormous par- and state —1> is
1
allel computing capability, Suppose we , now I can use this to define a ba-
have two qubits. If these were two clas- sis for two qubits or for that matter
sical bits, then there would be four pos- for any number of qubit. So in this
sible states, 00, 01, 10, and 11. Cor- notation the state —00> which is the
respondingly, a two qubit system has short hand notation for |0> state di-
four computational basis states denoted rect product with |0> and so therefore
|00〉, |01〉, |10〉, |11〉. A pair of qubits I have to simply multiply the matrix
can also exist in superpositions of these 1

four states, so the quantum state of 0
two qubits involves associating a com-  
1
plex coefficient – sometimes called an and
0
amplitude – with each computational itself. Now notice is this is not a usual
basis state, such that the state vector matrix multiplication but it is what we
describing the two qubits is called as a Kronecker multiplication or
—q〉 = α00|00〉 + α01|01〉 + α10|10〉 chronicle product. Now
+ α11|11〉.
Similar to the case for a single qubit,
the measurement result x (= 00, 01, 10
or 11) occurs with probability |(αx)2 |,
with the state of the qubits after the
measurement being —x>. The condi-
tion that probabilities sum to one is
therefore expressed by the normaliza-
tionP condition that
—(αx)2 | = 1. For a two qubit
system, we could measure just a subset