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6COM1046
- Quantum Computing
Practice Assignment Part
I
This is the first of two assignments for this module. It involves questions based on algebra, complex
numbers, probability and linear algebra. These are fundamental concepts underlying our future work in
quantum computation
The assignment would be worth 20o/o of the overall mark for the course,
if it was an actual
assignment.
Detailed solutions for each part should be given. This means that I am interested in your calculations, not
just your answers. Work with little or no calculations will result in low marks.
I
Simplify the following expressions:
a)
l6x+4y +llxz -2lx =
b)
^\
36i
+l6i
4po
(-2pt)t
vt
d)
I
-l
=
(-p)'
((r+s) t t)+((25-f)t f)=
(7 Marks)
2
Let r = 4, s= 8 and t =2
3
Expand the following brackets:
4
Factorise the following expressions:
5
.
Calculate the value
of (r
-s)' ,f
(2 marks)
a)
b)
5(4x -3y) :
obc(az +b3 +c)=
(2 Marks)
a)
b)
8z
+20y -12 =
mp+mq-mr -np-nQrrlr =
(2 Marks)
Solve the following equations:
a)
b)
12: -x +15
5l(z+2)=4/(z-2)
(3 Marks)
6
Reaffange the following formulae so that the given letter becomes the subject:
a)
b)
a=-b-c;
x=ny2)
c) ,=2r^E-i
\r+t
7
yb
t=
(4 Marks)
Solve the following simultaneous equations:
a)
b)
x+3y:6'3x-4y=J
m'-ll=2mn; m-n=6
(5 Marks)
8
a)
b)
Using the quadratic formula solve the equation xz = 6x -2
Express x' - 6x + 2in the form (x + a)2 + b and hence establish the minimum value for
the quadratic x2 -6x+2
(5 Marks)
9
a)
b)
Express lo9164 = 6 in index form
Express log24-logl2+ logSas a single logarithm
(3 Marks)
10
Express the following as a complex number in the form a + ib
the complex conjugate of i-t'
(3 + i) + (5 -L2i)
:
a)
b)
(5 Marks)
11 a)
b)
Draw on an Argand diagram the complex number
Hence, or otherwise, express
Ji
Ji -'fliund its conjugate.
-Jlii"exponential form.
(5 Marks)
12
Define the following terms from probability:
a) Probability space
b)
Var(X)
(4 Marks)
13
t4
A bag contains 3 blue balls, 2 red balls and an orange ball. Calculate the following probabilities:
a) Selecting a blue ball.
b) Selecting a blue ball, followed by an orange ball, given that the first ball selected is NOT
replaced prior to the second ball being drawn.
(3 Marks)
ret
a)
b)
lv,l=
[;]-, *l = [_i,]
f
Calculate the magnitude
of ly,) -lV)and
hence normalise the resulting vector.
Find the matrix representing a reflection in the line x
dimensional plane.
f y:
0, for points inthe 2
-
(5 Marks)
l5
a)
b)
State De Moivres Theorem.
Derive the cube roots of unity and show that their sum is zero
(6 marks)
Now check your work