1Vorlesung Quantum Computing SS ‘08

quantum computing

H H-1

calculation

U

preparation

read-out

|A|

time

time

quantum-bit (qubit)

0 1

a10 + a21 =a1a2

2Vorlesung Quantum Computing SS ‘08

from classic to quantum

we live in Hilbert Space Hthe state of our world is |

3Vorlesung Quantum Computing SS ‘08

can you see?

http://www.almaden.ibm.com/vis/stm/gallery.html

Don Eigler (IBM, Almaden)

48 Fe atoms on Cu(111)

4Vorlesung Quantum Computing SS ‘08

double slit experiment

classically:

number of electrons measured has a broad distribution

5Vorlesung Quantum Computing SS ‘08

double slit experiment

quantum mechanically:

interference pattern is observed→ particles are described as waves

coherent superposition| = c1| + c2|

wave function = (r,t)

probability density: probability of finding a particle at sight r(r,t) = |(r,t)|2

6Vorlesung Quantum Computing SS ‘08

double slit with electrons

7Vorlesung Quantum Computing SS ‘08

double slit with electrons

http://www.hqrd.hitachi.co.jp/global/movie.cfm

8Vorlesung Quantum Computing SS ‘08

Double slit with larger objects

O. Nairz, M. Arndt, and A. Zeilinger: Am. J. Phys. 71, 319 (2003)

9Vorlesung Quantum Computing SS ‘08

state and space of the world

complex functions of a variable, (r), form the Hilbert–Space:

(r) (r) dr = | < ∞

a particle is described by a vector | in Hilbert–Space

H is a linear vector space with scalar (inner) product

|(r) (r) dr = a , a C||a

10Vorlesung Quantum Computing SS ‘08

the space of the world

|||the scalar product is distributive

|cc |and thus

c||cc|

it is positive definite and real for | ≥ 0 ,

11Vorlesung Quantum Computing SS ‘08

quantum computing

H H-1

calculation

U

preparation

read-out

|A|

time

time

quantum-bit (qubit)

0 1

a10 + a21 =a1a2

12Vorlesung Quantum Computing SS ‘08

vector bases

every vector | can be decomposed into linear independent basis vectors |n| = cn|ncn

Cn

m|cnm|ncnmnn n

cm = m|

||nn|n

orthogonality can be written as

m(r) n(r) dr = m|n = mn

13Vorlesung Quantum Computing SS ‘08

euclidic representation

14Vorlesung Quantum Computing SS ‘08

our world H

can be divided into sub-spaces connected by the vector product

H = H1 H2 H3 HN

is normed with respect to finding a particle of state | anywhere

P = (r) (r) dr = (r)2dr = |=

| |HQC

mnoQCm n o mn o

we can find (or build) a quantum computer in our world

|QC = cmno|mnocmno |m |n |om,n,o m,n,o

15Vorlesung Quantum Computing SS ‘08

endohedral fullerenes

4 Å

atom inside has an electronspin that can serve as qubit

10 Åsource: K. Lips, HMI

|+1/2

|-1/2

|-1/2

|+1/2

|-1/2

|+1/2

mS mI

16Vorlesung Quantum Computing SS ‘08

quantum computing

H H-1

calculation

U

preparation

read-out

|A|

time

time

quantum-bit (qubit)

0 1

a10 + a21 =a1a2

17Vorlesung Quantum Computing SS ‘08

boolean algebra and logic gates

classical (irreversible) computing

gateinout

1-bit logic gates: identity

x NOT x

0 11 0

x Id

0 01 1

NOT

x NOT x

18Vorlesung Quantum Computing SS ‘08

quantum logic gates

1-bit logic gate: NOT a1|1 + a2| 0 (a1| 0 + a2| 1 ) =

manipulation in quantum mechanics is done by linear operators operators have a matrix representationX ≡

0

01

1matrix representation for the NOT gate:

X =0

01

1a1

a2

a1

a2

=a2

a1

x NOT x

0 11 0

19Vorlesung Quantum Computing SS ‘08

manipulation in our world

because of the superposition principle | = c1| + c2|, mathematical instructions (operators) have to be linear:

L (| + |) = L | L |^ ^ ^

L (c1 |) = c1L | ^ ^

examples:

(c + d/dx)

dx

()2

(c + d/dx) (f(x) + g(x)) = cf + d/dx f + cg + d/dx g

dx (f(x) + g(x)) = f dx + g dx

(f(x) + g(x))2 ≠ f2 + g2X

20Vorlesung Quantum Computing SS ‘08

linear operators

(L + M) | = L | + M | ^ ^ ^ ^

(L M) | = L (M |) ^ ^ ^ ^

however, generally L M | ≠ M L | , | ^ ^ ^ ^

commutator: [L,M] = L M – M L^ ^ ^ ^ ^ ^

anticommutator: [L,M]+= L M + M L^^^^^^

[L,L] = [L,1] = [L,L-1] = 0, [L,aM] = a [L,M],

[L1 + L2,M] = [L1,M] + [L2,M],

[L1L2,M] = [L1,M] L2 + [L2,M] L1

^ ^ ^ ^ ^ ^ ^ ^ ^ ^

^^^^^^^

^ ^ ^ ^ ^ ^ ^ ^ ^

[L,M] = – [M,L]^ ^ ^ ^

21Vorlesung Quantum Computing SS ‘08

vectors and operators

||nn|n1|nn|n

1 L 1=|mm|( L |nn|) = Lmn |m n|nmm n

^ ^

with matrix elements Lmn = m|L |n ^

22Vorlesung Quantum Computing SS ‘08

quantum dynamics

http://jchemed.chem.wisc.edu/JCEWWW/Articles/WavePacket/WavePacket.html

free particle wave packet traveling in a potential

movement of ion-qubits in a trap

23Vorlesung Quantum Computing SS ‘08

quantum dynamics

the state vector |(r,t) follows the Schrödinger equation:

• analogue to mechanical wave equations

• instead of the Hamilton Function H = T + V, the Hamilton Operator is used

pr = ħ i r

^

ħ2HVrpr

2

2m 2m2 Vr^

iħ t

|(r,t)Vr|(r,t)pr2

2m

24Vorlesung Quantum Computing SS ‘08

time evolution

?

?how does a state |(t) = cn(t) |nnevolve in time

|(t) = U(t) |(0)^ U(t): time evolution operator^

insert into Schrödinger equation:

U(t) |(0) H U(t) |(0)^iħ t

^^

Hiħ

U’(t)

U(t)^

^^

^ U(t)e ħ- i H t^

25Vorlesung Quantum Computing SS ‘08

unitary operators

the time evolution operator is unitary because H is hermitian

U+(t) U(t)e e = e0 = 1 ^ ħ- i H(t – t0)

^ ħi H(t – t0)

quantum computing is reversible!

U-1 = U+^ ^ U+ U = 1^ ^

unitary operators transform one base into another without loosing the norm (e.g., a rotation is a unitary transformation)

manipulation in quantum computing is done by unitary operations

(as long as one does not measure)

^

^ ^

26Vorlesung Quantum Computing SS ‘08

logic operations

1-bit logic gate: NOT a1|1 + a2| 0 (a1| 0 + a2| 1 ) =

X ≡ 0

01

1matrix representation for the NOT gate:

X =0

01

1a1

a2

a1

a2

=a2

a1

X X-1 =1

10

0

27Vorlesung Quantum Computing SS ‘08

quantum computing

H H-1

calculation

U

preparation

read-out

|A|

time

time

classical bit

1 ON 3.2 – 5.5 V

0 OFF -0.5 – 0.8 V

quantum-bit (qubit)

0 1

a10 + a21 =a1a2

28Vorlesung Quantum Computing SS ‘08

measurement

an adjoint (hermitian conjugated) operator is defined by:

a physical observable is described by a hermitian operator A

|A| =|A | *^ ^

for a hermitian operator: A = A^ ^

^| = A | ^| = | A

29Vorlesung Quantum Computing SS ‘08

measurement

| = a10 + a21

probability that the measurement outcome is 0 or 1:

p(0) |A0 | |a1|2p(1) |A1 | |a2|2

^

^

A0 | 0|a1| |a1|a1

state after the measurement:

A1 | 1|a2| |a2|a2

30Vorlesung Quantum Computing SS ‘08

hermitian operators

an example: the momentum operator

px = ħ i x

^

px| dx (px)* dx ( )*

dx (─ *)

─ *| + dx *( ) |px

ħ i x

ħ i x

ħi

∞∞-

ħ i x

^

^

wavefunctions vanish at infinity

31Vorlesung Quantum Computing SS ‘08

measurement

a physical observable is described by a hermitian operator A

0 = (a – a*) |

the eigenvalues a are real

|A | = a | A| = a* |

^

^

a state | is an eigenstate of an operator A, if

A | =a|

^

^

vector is invariant under sheertransformation → eigenvector of the transformation

32Vorlesung Quantum Computing SS ‘08

measurement

^|A | an |cn|2A^n

| = cn|nn

the probability measuring eigenvalue an is given by |cn|2

|A | is the mean value of A^ ^

the mean value of A is given by |A | ^

If the operators of two observables A and B commute, [A,B] = 0, they can be measured at the same time with unlimited precision.

^ ^

For [A,B] ≠ 0, [A,B] is a measure for the uncertainty of a and b:a·b ≥ ½ | [A,B]

^ ^ ^ ^

^ ^

33Vorlesung Quantum Computing SS ‘08

measurement

a physical observable is described by a hermitian operator A

eigenvectors of different eigenvalues are orthogonal

A |m =am|m ^

A |n =an |n ^

(an – am) m|nan ≠ am

m|n

an m|nm|AnAm|nam m|n^ ^

hermitian operators share a set of eigenvectors if they commute

[A,B] = 0^ ^ A and B are diagonal in the same base^ ^