Quantum
Computing
Introduction ·
,2 Basic Concepts
2 1 .
Quantum bits
Classical bits : 0 ,
1
of 1
Quantum bit
/grbit") :
superposition 0 and
>
-
guantum state 147 is described as 14) =
x 103 +
111)
with < B = K ,
:
1x1 +
111 =
1
>
-
Mathematical description : 143=42 -
142 =
(p)
with 10 =
10) and 11) =
(i)
(x amplitudes)
the &
Qubits and cannot /in generall directly be observed/measured .
Instead : "Standard" measurement will results in 0 with
probability /x13
1 with probability1B1
The measurement also
hanges the qubit ( = warfunction collapse
If O 14) measurement
measuring the grbit will be 10) directly after
=
,
If -
1 -1- 147 113 - 1-
measuring
=
practise
-
:
one can estimate the probabilities IXK and IBK in
experiments by repeating
the many times
same
experiment
= repititions are called "trials' or "Shots . .
what is a
grbit physically ? 10) and 11 are . . .
Circuit notation
Miriation
:
·
two different polarisations of a
photon
of
·
alignment nuclear or electronic spir
·
ground or exited state of an atom
·
clockwise or counterclockwise loop current states in a
Josephson junction
Boch sphere representation
A useful
graphical depiction of
qubit the
a is :
If B to be real-valued
x and
happen ,
then one can find
angle JER , such that x =
cos() and
B sin()
=
1B
11mx
et cos()
ar
----- 1BR general
-
(x) in x
cospla
+ : =
ei8 sin()
+
v)
> X
sin()
cos()
B
=
1 = +
Re
1
j
=
and
with so-called phase angles & U
+
4
e)
ecos(E) e since) e(cos)(0) eisin(E) 13)
+
1
Then : 14) =
10) +
11) =
+
.
↑ be ...
ignored
can
, characterized U and Y
Thus 14)
by two
angles
=>
is
,
Block's
these
specify the
point defined as
sphere
N 4) ,
=
(insein ( on the surface of a
she are
why are 10) and 11) at the
poles ?
2 .
2
single qubit gate
me evolution
Principle of :
the grantim State IN) at current time + transitions to a new
guantum State 14)
Later time t t
at a
point
described IT) !
>
-
transition by a
complex unitary Matrix U :
14) =
U .
=>
a
preserves nor m
read from left to right,
circuit notation 143
Fangate 14 a
:
even
though (U(43) from -1. . .
(07)
I
EXAMPLES of NOT
quantum analogue classical
gate
:
the
·
Leads to Pauli-X
gate
: X = 0 =
(i) with Pauli matrices X Y, Z
,
Check X 10) =
(b)(i) =
(i) =
(1) . . .
e+c .
) and Pauli vector
8 =
(01 , 52 , 53) =
(x y
, , z)
·
Pauli-y-gate
:
Y = =
(ii)
·
Pauli-z-gate
:
z =
0
=
(i)
Cleaves 10) but
flips the of 11)
unchanged ,
sign
Recall the Block sphere representation : 14) =
cos() 103 +
esin()k)
14) =
costl 10
-
esinc) 117 =
cost) 103 + ele
+
)
sin() 11)
-
new Block sphere angles
: V' = & and Y =
Y +
It =
Rotation by 1800 around z-axis)
·
Hadermard gate
: H =E
-2) where <10 +
ple--a
·
phase gate s =
(i) i
T-gate T =
( e) where Th =
1 for j 1,
Pauli
satisfy op 3
The 2,
Matices = =
: ·
·
OjOn
= -
On Oj for all
j
+ K
·
[OjOn] : =
OjOn-OnEj
=
Zion for (j,k e) , cyclic permutations
the
introducing
Matrix
exponential
Ah with At Due
Computing
Introduction ·
,2 Basic Concepts
2 1 .
Quantum bits
Classical bits : 0 ,
1
of 1
Quantum bit
/grbit") :
superposition 0 and
>
-
guantum state 147 is described as 14) =
x 103 +
111)
with < B = K ,
:
1x1 +
111 =
1
>
-
Mathematical description : 143=42 -
142 =
(p)
with 10 =
10) and 11) =
(i)
(x amplitudes)
the &
Qubits and cannot /in generall directly be observed/measured .
Instead : "Standard" measurement will results in 0 with
probability /x13
1 with probability1B1
The measurement also
hanges the qubit ( = warfunction collapse
If O 14) measurement
measuring the grbit will be 10) directly after
=
,
If -
1 -1- 147 113 - 1-
measuring
=
practise
-
:
one can estimate the probabilities IXK and IBK in
experiments by repeating
the many times
same
experiment
= repititions are called "trials' or "Shots . .
what is a
grbit physically ? 10) and 11 are . . .
Circuit notation
Miriation
:
·
two different polarisations of a
photon
of
·
alignment nuclear or electronic spir
·
ground or exited state of an atom
·
clockwise or counterclockwise loop current states in a
Josephson junction
Boch sphere representation
A useful
graphical depiction of
qubit the
a is :
If B to be real-valued
x and
happen ,
then one can find
angle JER , such that x =
cos() and
B sin()
=
1B
11mx
et cos()
ar
----- 1BR general
-
(x) in x
cospla
+ : =
ei8 sin()
+
v)
> X
sin()
cos()
B
=
1 = +
Re
1
j
=
and
with so-called phase angles & U
+
4
e)
ecos(E) e since) e(cos)(0) eisin(E) 13)
+
1
Then : 14) =
10) +
11) =
+
.
↑ be ...
ignored
can
, characterized U and Y
Thus 14)
by two
angles
=>
is
,
Block's
these
specify the
point defined as
sphere
N 4) ,
=
(insein ( on the surface of a
she are
why are 10) and 11) at the
poles ?
2 .
2
single qubit gate
me evolution
Principle of :
the grantim State IN) at current time + transitions to a new
guantum State 14)
Later time t t
at a
point
described IT) !
>
-
transition by a
complex unitary Matrix U :
14) =
U .
=>
a
preserves nor m
read from left to right,
circuit notation 143
Fangate 14 a
:
even
though (U(43) from -1. . .
(07)
I
EXAMPLES of NOT
quantum analogue classical
gate
:
the
·
Leads to Pauli-X
gate
: X = 0 =
(i) with Pauli matrices X Y, Z
,
Check X 10) =
(b)(i) =
(i) =
(1) . . .
e+c .
) and Pauli vector
8 =
(01 , 52 , 53) =
(x y
, , z)
·
Pauli-y-gate
:
Y = =
(ii)
·
Pauli-z-gate
:
z =
0
=
(i)
Cleaves 10) but
flips the of 11)
unchanged ,
sign
Recall the Block sphere representation : 14) =
cos() 103 +
esin()k)
14) =
costl 10
-
esinc) 117 =
cost) 103 + ele
+
)
sin() 11)
-
new Block sphere angles
: V' = & and Y =
Y +
It =
Rotation by 1800 around z-axis)
·
Hadermard gate
: H =E
-2) where <10 +
ple--a
·
phase gate s =
(i) i
T-gate T =
( e) where Th =
1 for j 1,
Pauli
satisfy op 3
The 2,
Matices = =
: ·
·
OjOn
= -
On Oj for all
j
+ K
·
[OjOn] : =
OjOn-OnEj
=
Zion for (j,k e) , cyclic permutations
the
introducing
Matrix
exponential
Ah with At Due
