quantum computing quantumbit qubit 0 a a 1

quantum computing quantum-bit (qubit) 0 a a 1 0 + a 2 1 = a 1 2 1 preparation Y 0 Vorlesung Quantum Computing SS ‘ 08 calculation H U read-out H-1 time Y|A|Y time 1

from classic to quantum we live in Hilbert Space H the state of our world is |y Vorlesung Quantum Computing SS ‘ 08 2

can you see? Don Eigler (IBM, Almaden) 48 Fe atoms on Cu(111) http: //www. almaden. ibm. com/vis/stm/gallery. html Vorlesung Quantum Computing SS ‘ 08 3

double slit experiment classically: number of electrons measured has a broad distribution Vorlesung Quantum Computing SS ‘ 08 4

double slit experiment quantum mechanically: wave function y = y (r, t) coherent superposition |y = c 1|y 1 + c 2|y 2 probability density: probability of finding a particle at sight r r(r, t) = |y(r, t)|2 interference pattern is observed → particles are described as waves Vorlesung Quantum Computing SS ‘ 08 5

double slit with electrons Vorlesung Quantum Computing SS ‘ 08 6

double slit with electrons http: //www. hqrd. hitachi. co. jp/global/movie. cfm Vorlesung Quantum Computing SS ‘ 08 7

Double slit with larger objects O. Nairz, M. Arndt, and A. Zeilinger: Am. J. Phys. 71, 319 (2003) Vorlesung Quantum Computing SS ‘ 08 8

state and space of the world a particle is described by a vector |y in Hilbert–Space complex functions of a variable, y(r), form the Hilbert–Space: y *(r) y(r) dr = y |y < ∞ H is a linear vector space with scalar (inner) product j |y = j *(r) y(r) dr = a , a C y | j = j | y * = a * Vorlesung Quantum Computing SS ‘ 08 9

the space of the world the scalar product is distributive j |y 1 + y 2 = j |y 1 + j |y 2 j |cy = c j |y and thus cj |y = y |cj * = c* j |y it is positive definite and real for y |y ≥ 0 , Vorlesung Quantum Computing SS ‘ 08 10

quantum computing quantum-bit (qubit) 0 a a 1 0 + a 2 1 = a 1 2 1 preparation Y 0 Vorlesung Quantum Computing SS ‘ 08 calculation H U read-out H-1 time Y|A|Y time 11

vector bases every vector |j can be decomposed into linear independent basis vectors |yn : |j = cn|yn , cn C n orthogonality can be written as ym *(r) yn(r) dr = ym|yn = dmn ym|j = c y |yn = cn dmn m n n n c m = y m| j = | yn yn | j n Vorlesung Quantum Computing SS ‘ 08 12

euclidic representation Vorlesung Quantum Computing SS ‘ 08 13

our world H is normed with respect to finding a particle of state |j anywhere P = j *(r) j (r) dr = j (r) 2 dr = j |j = 1 can be divided into sub-spaces connected by the vector product | | H = H 1 H 2 H 3 HN HQC ym| y 1 mno y=n yo| y 3 m= 1 y m |1 y y | c mno cmno yoo 33 QC =cmno 2 2 |y QC = n n 2 m, n, o we can find (or build) a quantum computer in our world Vorlesung Quantum Computing SS ‘ 08 14

endohedral fullerenes atom inside has an electron spin that can serve as qubit 4Å m. S |+1/2 m. I |+1/2 |-1/2 |+1/2 10 Å source: K. Lips, HMI Vorlesung Quantum Computing SS ‘ 08 15

quantum computing quantum-bit (qubit) 0 a a 1 0 + a 2 1 = a 1 2 1 preparation Y 0 Vorlesung Quantum Computing SS ‘ 08 calculation H U read-out H-1 time Y|A|Y time 16

boolean algebra and logic gates classical (irreversible) computing in 1 -bit logic gates: out gate identity x 0 1 Id 0 1 NOT x 0 1 x Vorlesung Quantum Computing SS ‘ 08 NOT x 1 0 NOT x 17

quantum logic gates 1 -bit logic gate: x 0 1 NOT (a 1| 0 + a 2| 1 ) = a 1|1 + a 2| 0 NOT x 1 0 manipulation in quantum mechanics is done by linear 0 operators 1 matrix representation for the NOT representation gate: X≡ 1 0 operators have a matrix X Vorlesung Quantum Computing SS ‘ 08 a 1 a 2 = 0 1 a 1 1 0 a 2 = a 2 a 1 18

manipulation in our world because of the superposition principle |y = c 1|y 1 + c 2|y 2 , mathematical instructions (operators) have to be linear: ^ ^ |y + L^ |y L (|y 1 + |y 2 ) = L 1 2 ^ (c |y ) = c L ^ |y L 1 1 examples: (c + d/dx) (f(x) + g(x)) = cf + d/dx f + cg + d/dx g dx (f(x) + g(x)) = f dx + g dx () X (f(x) + g(x))2 ≠ f 2 + g 2 2 Vorlesung Quantum Computing SS ‘ 08 19

linear operators ^ + M) ^ |y = L ^ |y + M ^ |y (L ^ M) ^ |y = ^ ^ |y ) (L L (M however, generally ^M ^ |y ≠ M ^L ^ |y L ^ ^ =^ ^ –M ^ L ^ commutator: [L, M] LM ^^ =L ^M ^+M ^L ^ anticommutator: [L, M] ^^ = [L, M] , |y ^^ – [M, L] + ^ ^ = [L, 1] ^ ^ = [L, L ^ ^-1] = 0, [L, a. M] ^ ^ = a [L, M], ^ ^ [L, L] ^ +L ^ , M] ^ = [L ^ , M] ^ + [L ^ , M], ^ [L 1 2 ^ ^ ^ ^ ^ [L 1 L 2, M] = [L 1, M] L 2 + [L 2, M] L 1 Vorlesung Quantum Computing SS ‘ 08 20

vectors and operators | j = | yn yn | j n 1 = | yn yn | n ^ | y y | ) = ^ 1 = | y y | ( L 1 L m m n n m n with matrix elements Vorlesung Quantum Computing SS ‘ 08 m n Lmn |ym yn| ^ |y Lmn = ym| L n 21

quantum dynamics free particle wave packet traveling in a potential http: //jchemed. chem. wisc. edu/JCEWWW/Articles/Wave. Packet. html movement of ion-qubits in a trap Vorlesung Quantum Computing SS ‘ 08 22

quantum dynamics the state vector |y (r, t) follows the Schrödinger equation: 2 ^ p r iħ |y (r, t) = + V(r) |y (r, t) 2 m t ħ ^ pr = i r • analogue to mechanical wave equations • instead of the Hamilton Function H = T + V, the Hamilton Operator is used 2 2 ^ ^ p ħ 2 + V (r ) H = 2 mr + V(r) = - 2 m Vorlesung Quantum Computing SS ‘ 08 23

time evolution (t) |yn evolve in time? how does a state |y(t) = c n n ? ^(t) |y(0) |y(t) = U ^ (t): time evolution operator U insert into Schrödinger equation: ^ U(t) ^ |y(0) = H ^ |y(0) iħ U(t) t ^ U’(t) i. H ^ = ^ ħ U(t) ^ = U(t) Vorlesung Quantum Computing SS ‘ 08 ^t -i. H e ħ 24

unitary operators ^=1 ^+ U ^+ ^ -1 = U U U unitary operators transform one base into another without loosing the norm (e. g. , a rotation is a unitary transformation) ^ the time evolution operator is unitary because H is hermitian ^+(t) U(t) ^ = U ^ – t ) - i H(t ^ –t ) i H(t 0 0 ħ e = e 0 = 1 manipulation in quantum computing is done by unitary operations quantum computing is reversible! (as long as one does not measure) Vorlesung Quantum Computing SS ‘ 08 25

logic operations 1 -bit logic gate: NOT (a 1| 0 + a 2| 1 ) = a 1|1 + a 2| 0 matrix representation for the NOT gate: X a 1 a 2 = X Vorlesung Quantum Computing SS ‘ 08 0 1 a 1 1 0 a 2 X-1 = 1 0 0 1 = X≡ 0 1 1 0 a 2 a 1 26

quantum computing quantum-bit (qubit) 0 a a 1 0 + a 2 1 = a 1 2 1 classical bit 1 ON 3. 2 – 5. 5 V 0 OFF -0. 5 – 0. 8 V preparation Y 0 Vorlesung Quantum Computing SS ‘ 08 calculation H U read-out H-1 time Y|A|Y time 27

measurement ^ a physical observable is described by a hermitian operator A an adjoint (hermitian conjugated) operator is defined by: ^ |j |y = A ^+ y | = j | A ^ + | y = y | A ^ | j * j | A ^+ = A ^ for a hermitian operator: A Vorlesung Quantum Computing SS ‘ 08 28

measurement |y = a 1 0 + a 2 1 probability that the measurement outcome is 0 or 1: ^ | y = |a |2 p(0) = y | A 0 1 ^ | y = |a |2 p(1) = y | A 1 2 state after the measurement: A 0 | y a 1 = 0 |a 1| A 1 | y a 2 = |a | 1 |a 2| 2 Vorlesung Quantum Computing SS ‘ 08 29

hermitian operators an example: the momentum operator ħ ^ px = i x ħ * y = dx i xj j* y = dx ─ ħi x ħ ħ ∞ * * = ─ j y | + dx j i x i -∞ ( p^xj |y = dx (pxj)* ( ) y ) ( y) = j |p^xy wavefunctions vanish at infinity Vorlesung Quantum Computing SS ‘ 08 30

measurement ^ a physical observable is described by a hermitian operator A ^ if a state |y is an eigenstate of an operator A, ^ |y = a |y A the eigenvalues a are real vector is invariant under sheer ^ y | A |transformation y = a y→|y eigenvector of the transformation A^y |y = a* y |y 0 = (a – a*) y |y Vorlesung Quantum Computing SS ‘ 08 31

measurement ^|y the mean value of A is given by y | A ^ ^ | y = a | c | 2 = A y | A n n n |y = cn|yn n the probability measuring eigenvalue an is given by |cn|2 ^ | y is the mean value of A ^ y | A ^ ^ = 0, If the operators of two observables A and B commute, [A, B] they can be measured at the same time with unlimited precision. ^ ^ ≠ 0, [A, B] ^ ^ is a measure for the uncertainty of a and b: For [A, B] ^ ^ Da·Db ≥ ½ | [A, B] y Vorlesung Quantum Computing SS ‘ 08 32

measurement ^ a physical observable is described by a hermitian operator A eigenvectors of different eigenvalues are orthogonal ^ |y = a |y A m m m ^ |y = a |y A n n n ^y = A^y |y = a y |y an ym|yn = ym|A m n m n ( a n – a m ) y m| y n = 0 an ≠ a m hermitian operators share a set of eigenvectors if they commute ^ ^ =0 [A, B] ^ are diagonal in the same base A^ and B Vorlesung Quantum Computing SS ‘ 08 33