Energy Performance Certificates Presented by: Jacquie Taylor.
Presented by: Erik Cox, Shannon Hintzman, Mike Miller, Jacquie Otto, Adam Serdar, Lacie Zimmerman.
-
Upload
bruce-powers -
Category
Documents
-
view
216 -
download
0
Transcript of Presented by: Erik Cox, Shannon Hintzman, Mike Miller, Jacquie Otto, Adam Serdar, Lacie Zimmerman.
Presented by:Erik Cox, Shannon Hintzman,
Mike Miller, Jacquie Otto,
Adam Serdar, Lacie Zimmerman
deadalivecat 2
1
What’s to come…
-Brief history and background of quantum mechanics and quantum computation
-Linear Algebra required to understand quantum mechanics
-Dirac Bra-ket Notation
-Modeling quantum mechanics and applying it to quantum computation
History of Quantum Mechanics
Sufficiently describes everyday things and events.
Breaks down for very small sizes (quantum mechanics) and very high speeds (theory of relativity).
Classical (Newtonian) Physics
Why do we need Quantum Mechanics?
In short, quantum mechanics describes behaviors that classical (Newtonian) physics cannot. Some behaviors include:
- The wave-particle duality of light and matter
- Discreteness of energy
- Quantum tunneling- The Heisenberg uncertainty principle
- Spin of a particle
Spin of a Particle- Discovered in 1922 by Otto Stern and
Walther Gerlach
- Experiment indicated that atomic particles possess intrinsic angular momentum, called
spin, that can only have certain discrete values.
The Quantum Computer
Idea developed by Richard Feynman in 1982.
Concept:Create a computer that uses the effects of quantum mechanics to its advantage.
Classical Computer
Information
Quantum Computer
Informationvs.
- Bit, exists in two states, 0 or 1
- Qubit, exists in two states, 0 or 1, and superposition of both
Why are quantum computers important?
Recently, Peter Shor developed an algorithm to factor large numbers on a quantum computer. Since factoring is key to current encryption, quantum computers would be able to quickly break current cryptography techniques.
In the beginning, there was Linear Algebra…
- Complex inner product spaces
- Linear Operators
- Unitary Operators
- Projections
- Tensor Products
Complex inner product spaces
njCzzzzzC jnn ..1,|,...,,, 321
An inner product space is a complex vector space , together with a map f : V x V → F where F is the ground field C. We write <x, y> instead of f(x, y) and require that the following axioms be satisfied:
V
,0,, xxVx and iff0, xx 0x
yzxzayaxzVzyxFa ,,,,,,,
*,,,, xyyxVyx
denotes complex conjugate*
(Positive Definiteness)
(Conjugate Bilinearity)
(Conjugate Symmetry)
Complex Conjugate:
iyxz iyxz *
1iwhere
njCzzzzzCV jnn ..1,,...,,, 321
Example of Complex Inner Product Space:
Cz Let
Vwv ,Let
nn wvwvwvwv *...**, 2211
Linear Operators
Example:
xgdxdxfdxdxgxfdxd ///
xfdxdcxcfdxd //
yAxAyxA ˆˆˆ (Additivity)
xAcxcA ˆˆ (Homogeneity)
Let and be vector spaces over , then is a linear operator if . The following properties exist:
V WWVA :ˆ
Vyx ,
C,Cc
Unitary Operators
tUU 1
IUUUU tt
t denotes adjoint
Properties:
Norm Preserving…
Inner Product Preserving…
AdjointsMatrix
Representation
nnnnn
n
n
aaaa
aaaa
aaaa
A
...
...
...
...
321
1322212
1312111
nnnnn
n
n
t
aaaa
aaaa
aaaa
A
*...***
...
*...***
*...***
321
1232221
1131211
Suppose VwvVVU ,,:
wvAwAv t Definition of Adjoint:
wvUUwUvU t
wvwvI
(Inner Product Preserving)
vvvvUvUvU (Norm
Preserving)
• In quantum mechanics we use orthogonal projections.
• Definition: Let V be an inner product space over F. Let M be a subspace of V. Given an element then the orthogonal projection of y onto M is the vector which satisfies
where v is orthogonal to every element .
Vy MPy
vPyy Mm
A projection operator P on V satisfies
We say P is the projection onto its range, i.e., onto the subspace
2PPP t
vPvVvW :
In quantum mechanics tensor products are used with :
• Vectors• Vector Spaces• Operators• N-Fold tensor
products.
If and , there is a natural mapping defined by
We use notation w v to symbolize T(w, v) and call w v the tensor product of w and v.
mCW nCV mnCVWT :
nmnnm yyxyyxyyxxT ,...,,...,,...,,...,,,..., 11111
nmmn yxyxyxyx ,...,,...,,..., 1111
4 Properties of Tensor Products
1. a(w v) = (aw) v = w (av) for all a in C;
2. (x + y) v = x v + y v;
3. w (x + y) = w x + w y;
4. w x | y z = w | y x | v .
Note: | is the notation used for inner products in quantum mechanics.
Property #1: a(w v) = (aw) v = w (av) for all a in C
Example in :
2C),,,()( 22122111 vwvwvwvwavwa
),,,( 22122111 vawvawvawvaw
Property #2:
(x + y) v = x v + y v
Example in :2C
vyyxxvyx )),(),(()( 2121
))),((),),((( 221121 vxxvxx
),,,( 22211211 vxvxvxvx
))),((),),((( 221121 vyyvyy
),,,( 22211211 vyvyvyvy
Property #3:
w (x + y) = w x + w y
Example in :2C
)),(),(()( 2121 yyxxwyxw
))),(()),,((( 212211 xxwxxw
),,,( 22122111 xwxwxwxw))),(()),,((( 212211 yywyyw
),,,( 22122111 ywywywyw
Property #4:
w x | y z = w | y x | z Example in :2C
zyxw |
)(*)()(*)( 21211111 zyxwzyxw
)(*)()(*)( 22221212 zyxwzyxw
),,,(|),,,( 2212211122122111 zyzyzyzyxwxwxwxw
Example in :2C zxyw ||
))(*)()(*)(( 2211 ywyw ))(*)()(*)(( 2211 zxzx
),(|),(),(|),( 21212121 zzxxyyww
)))((*)(*)(())(*))((*)(( 22211111 yzxwzxyw
)))((*)(*)(())(*))((*)(( 22221122 yzxwzxyw
)(*)()(*)( 21211111 zyxwzyxw )(*)()(*)( 22221212 zyxwzyxw
Dirac Bra-Ket NotationNotation
Inner Products
Outer Products
Completeness Equation
Outer Product Representation of Operators
Bra-Ket Notation Involves
Bra
<n| = |n>t
Ket
|n>
Vector Xn can be represented two
ways
z
y
x
w
v ***** zyxwv
*m is the complex conjugate of m
Inner ProductsAn Inner Product is a Bra multiplied by a Ket
<x| |y> can be simplified to <x|y>
<x|y> =
p
o
n
m
l
= ***** zyxwv***** pzoynxmwlv
Outer ProductsAn Outer Product is a Ket multiplied by a Bra
|y><x| =
p
o
n
m
l
=
*****
*****
*****
*****
*****
pzpypxpwpv
ozoyoxowov
nznynxnwnv
mzmymxmwmv
lzlylxlwlv
***** zyxwv
By Definition xvyvyx
Completeness Equation
vivivii ||||||
So Effectively
Iii ||
Let |i>, i = 1, 2, ..., n, be a basis for V
and v is a vector in V
is used to create a identity operator represented by vector products.
Proof for the Completeness Equation
Using Linear Algebra, the basis of a vectors space can be represented series of vectors with a one in each successive position and zeros in every other (aka {1, 0, 0, ... }, {0, 1, 0, ...}, {0, 0, 1, ...}, ...)
So |i><i| will create a matrix with a one in each successive position along the diagonal.
............
...000
...000
...001
............
...000
...010
...000
............
...100
...000
...000
etc.
Completeness Cont.Thus
|| ii =
............
...000
...000
...001
............
...000
...010
...000
............
...100
...000
...000
+ + + ... =
............
...100
...010
...001
= I
One application of the Dirac notation is to represent
Operators in terms of inner and outer products.
1
0,
n
ji
jijAi
jAiAij
and
• If A is an operator, we can represent A by applying the completeness equation twice this gives the following equation:
• This shows that any operator has an outer product representation and that the entries of the associated matrix for the basis |i are:
1
0,
n
ji
jijAi
jAiAij
Linear Algebra View
x
yv
u
yxv We can represent
graphically:
0yxUsing the rule of dot products we
know
ucx0cGiven that we can say
Linear Algebra View (Cont.)Using these facts we can solve
x
y
for and
uyucuv )(
uyuucuyuc )(
yucv
0yu�
Again using the rule of dot products2
ucuv We get
Linear Algebra View (Cont.)
2u
vuc
So
uu
vuvxvy
2
uu
vux
2
Plugging this back into the original equation
Gives us:
ucx
Projections in Quantum Mechanics
VW VvGiven that and
x
yvW
xPv Wxyxv This graph is a representation of
Given and
Projections in QM (cont.)
k,...2,1 being the full basis of W
}|,...1|,|,...2|,1{| nkk
We can regard the full basis of as being
V
nckc
kcccv
nk
k
|...1|
|...2|1|
1
21On Basis
Cc j For some
Projections in QM (cont.)
ncccv n 1...21111 21
Taking the inner products gives
n
kj
n
j
n
j
jvjjvjjvjv111
vjc j Therefore and more
generally
vc 11 So
Computational Basis (cont.)
2CA basis for will have basis vectors:
nV Called a computational basis
1...11,...,10...00,01...00,0...00
10...0001...00 Notation:
Quantum States
22,12,1 : Czzzz
Thinking in terms of directions
model quantum states by directions in a vector space
1
0 1z
2z
1,0
0,1
Associated with an isolated quantum system is an
inner product space called the “state
space” of the system. The system at any given
time is described by a “state”, which is a unit
vector in V.
nCV
• Simplest state space - or Qubit
If and form a basis for ,
then an arbitrary qubit state has the form , where a and b in
have .
• Qubit state differs from a bit because “superpositions” of an arbitrary qubit state are possible.
2CV 0| 1| V
1|0|| bax C1|||| 22 ba
The evolution of an isolated quantum system is
described by a unitary operator on its state space.
The state is related to the state by a
unitary operator i.e., .
)(| 2t)(| 1t
2,1 ttU )(|)(| 1,2 21tUt tt
Quantum measurements are described by a
finite set, {Pm}, of projections acting on the
state space of the system being measured.
• If the state of the system is immediately
before the measurement, then the probability that
the result m occurs is given by
.
|
||)( mPmp
• If the result m occurs, then the state of the
system immediately after the measurement is
)(
|
||
|2/1 mp
P
P
P m
m
m
The state space of a composite quantum system is
the tensor product of the state of its components.
If the systems numbered 1 through n are prepared
in states , i = 1,…, n, then the joint state of
the composite total system is .
)(| it
n || 1
Product vs. Entangled States
Product State – a state in Vn is called a product state if it has the form:
Entangled State – if is a linear combination of that can’t be written as a product state
si'
Example of anEntangled State
The 2-qubit in the state
Suppose:
|00 + |11 = |a |b for some |a and |b. Taking inner products with |00, |11, and |01 and applying the state space property of tensor products (states |i, i=1, …, n, then the joint state of the composite total system is |1 · · · |n) gives
0|a 0|b = 1, 1|a 1|b = 1, and 0|a 1|b = 0, respectively. Since neither 0|a nor 1|b is 0, this gives a contradiction
2/1100
Example of a 2-qubit
• A qubit is a 2-dimensional quantum system (say a photon) and a 2-qubit is a composite of two qubits
• 2-qubits “live” in the vector space 22 CC
Suppose that is an example of a 2 component system with being a linear combination of basic qubits with amplitude being the coefficients:
In which
11100100 3210 aaaa
12
3
2
2
2
1
2
0 aaaa
Measuring the 1st qubit• When we measure the first qubit in the composite
system, the measuring apparatus interacts with the 1st qubit and leaves the 2nd qubit undisturbed (postulate 4), similarly when we measure the 2nd qubit the measuring device leaves the 1st qubit undisturbed
• Thus, we apply the measurement ,in which
IP
IP
11
00
1
0
10 , PP
Leading to the probabilities and post measurement states…
2
1
2
01001 01000 aaaaPp
2
1
2
0
10
1
001
0100
0 aa
aa
p
P
Using postulate 3 the probability that 0 occurs is given by
If the result 0 occurs, then the state of the system immediately after the measurement is given by
Similarly we obtain the result 1 on the 1st qubit with probability…
2
3
2
211 1 aaPp
Resulting in the post-measurement state…
2
3
2
2
32
1
111
1110
1 aa
aa
p
P
In the same way for the second qubit…
,1
,0
2
3
2
12
2
2
2
02
aap
aap
2
3
2
1
3112
2
2
2
0
2002
1101
1000
aa
aa
aa
aa
After applying Quantum Measurement Techniques
2
11010 2211 pppp
11 ,00 12
11
02
01 vvvv
the post measurement states are
(A Perfectly Correlated Measurement)
The probabilities for each state for each qubit are all 1/2
Conclusion
• Brief History of Quantum Mechanics• Tools Of Linear Algebra
– Complex Inner Product Spaces– Linear and Unitary Operators– Projections– Tensor Products
Conclusion Cont.
• Dirac Bra-Ket Notation– Inner and Outer Products– Completeness Equation– Outer Product Representations– Projections– Computational Bases
Conclusion Cont. (again)
• Mathematical Model of Quantum Mech.– Quantum States– Postulates of Quantum Mechanics– Product vs. Entangled States
Bibliography
http://en.wikipedia.org/wiki/Inner_product_space
http://vergil.chemistry.gatech.edu/notes/quantrev/node14.html
http://en2.wikipedia.org/wiki/Linear_operator
http://vergil.chemistry.gatech.edu/notes/quantrev/node17.html
http://www.doc.ic.ad.uk/~nd/surprise_97/journal/vol4/spb3/
http://www-theory.chem.washington.edu/~trstedl/quantum/quantum.html
Gudder, S. (2003-March). Quantum Computation. American Mathmatical Monthly. 110, no. 3,181-188.
Special Thanks to:
Dr. Steve Deckelman
Dr. Alan Scott