quantum computing |Y0 U H H-1 Y|A|Y quantum-bit (qubit) 0   0   1   a1 0 + a2 1 = a1 a2 preparation |Y0 calculation read-out time  U   H H-1 Y|A|Y  time

from classic to quantum we live in Hilbert Space H the state of our world is |y

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

double slit experiment classically: number of electrons measured has a broad distribution

double slit experiment quantum mechanically: coherent superposition |y = c1|y1 + c2|y2 wave function y = y (r,t) 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

double slit with electrons

double slit with electrons http://www.hqrd.hitachi.co.jp/global/movie.cfm

Double slit with larger objects O. Nairz, M. Arndt, and A. Zeilinger: Am. J. Phys. 71, 319 (2003)

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 *

cj |y  = y |cj * = c* j |y  the space of the world j |y1 + y2  = j |y1 + j |y2 the scalar product is distributive j |cy  = c j |y  and thus cj |y  = y |cj * = c* j |y  it is positive definite and real for y |y  ≥ 0 ,  

quantum computing |Y0 U H H-1 Y|A|Y quantum-bit (qubit) 0   0   1   a1 0 + a2 1 = a1 a2 preparation |Y0 calculation read-out time  U    H H-1 Y|A|Y time

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 = cnym|yn = cn dmn n cm = ym|j |j = |yn yn|j n

euclidic representation

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 = H1  H2  H3    HN    | HQC |cQC = cmno|ymno = cmno |ym1  |yn2  |yo3 m,n,o |cmnoQC = |ym1  |yn2  |yo3 = |ym1 |yn2 |yo3  we can find (or build) a quantum computer in our world

endohedral fullerenes 4 Å atom inside has an electron spin that can serve as qubit |+1/2 |-1/2 mS mI 10 Å source: K. Lips, HMI

quantum computing |Y0 U H H-1 Y|A|Y quantum-bit (qubit) 0   0   1   a1 0 + a2 1 = a1 a2 preparation |Y0 calculation read-out time  U    H H-1 Y|A|Y time

boolean algebra and logic gates classical (irreversible) computing gate in out 1-bit logic gates: identity NOT x Id 1 x NOT x 1 x NOT x

quantum logic gates X ≡ X = 1-bit logic gate: NOT (a1| 0  + a2| 1 ) = a1|1  + a2| 0  x NOT x 1 manipulation in quantum mechanics is done by linear operators operators have a matrix representation X ≡ 1 matrix representation for the NOT gate: X = 1 a1 a2

manipulation in our world because of the superposition principle |y = c1|y1 + c2|y2, mathematical instructions (operators) have to be linear: L (|y1 + |y2) = L |y1 + L |y2 ^ L (c1 |y1) = c1L |y1 ^ 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 X (f(x) + g(x))2 ≠ f2 + g2

[L,L] = [L,1] = [L,L-1] = 0, [L,aM] = a [L,M], linear operators (L + M) |y = L |y + M |y ^ (L M) |y = L (M |y) ^ however, generally L M |y ≠ M L |y , |y ^ commutator: [L,M] = L M – M L ^ [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 ^

vectors and operators  |j = |yn yn|j 1 = |yn yn| 1 L 1 = |ym ym| ( L |yn yn|) =   Lmn |ym yn| n m ^ with matrix elements Lmn = ym| L |yn ^

quantum dynamics movement of ion-qubits in a trap free particle wave packet traveling in a potential http://jchemed.chem.wisc.edu/JCEWWW/Articles/WavePacket/WavePacket.html movement of ion-qubits in a trap

quantum dynamics ħ  ^ pr = i r 2 + V(r) iħ  t the state vector |y (r,t) follows the Schrödinger equation: iħ  t |y (r,t) = + V(r) |y (r,t) ^ pr2 2m pr = ħ  i r ^ analogue to mechanical wave equations instead of the Hamilton Function H = T + V, the Hamilton Operator is used ħ2 H = + V(r) = - ^ pr2 2m 2 + V(r)

time evolution ?  U(t) = e how does a state evolve in time  iħ t i |y(t) = cn(t) |yn n evolve in time |y(t) = U(t) |y(0) ^ U(t): time evolution operator ^ insert into Schrödinger equation: U(t) |y(0) = H U(t) |y(0) ^ iħ  t = - H i ħ U’(t) U(t) ^  U(t) = e ħ - i H t ^ ^

unitary operators  quantum computing is reversible! U-1 = 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 U+(t) U(t) = e e = e0 = 1 ^ ħ - i H(t – t0) i H(t – t0) ^ ^ manipulation in quantum computing is done by unitary operations  quantum computing is reversible! (as long as one does not measure)

logic operations X ≡ X = X X-1 = 1-bit logic gate: NOT (a1| 0  + a2| 1 ) = a1|1  + a2| 0  X ≡ 1 matrix representation for the NOT gate: X = 1 a1 a2 X X-1 = 1

quantum computing |Y0 U H H-1 Y|A|Y classical bit 1  ON  3.2 – 5.5 V 0  OFF  -0.5 – 0.8 V quantum-bit (qubit) 0   1   a1 0 + a2 1 = a1 a2 preparation |Y0 calculation read-out time  U    H H-1 Y|A|Y time

measurement j | A+|y  = y | A |j * a physical observable is described by a hermitian operator A ^ an adjoint (hermitian conjugated) operator is defined by: ^ |y  = A |j  ^ y | = j | A+ ^ ^ j | A+|y  = y | A |j * ^ ^ for a hermitian operator: A+ = A

measurement = 0 = 1 |y  = a1 0 + a2 1 ^ probability that the measurement outcome is 0 or 1: p(0) = y | A0 | y  = |a1|2 p(1) = y | A1 | y  = |a2|2 ^ state after the measurement: A0 | y  = 0 |a1| a1 A1 | y  = 1 |a2| a2

hermitian operators ħ  ^ px = i x an example: the momentum operator ħ  i x ^ px = pxj |y  =  dx (pxj)* y =  dx ( j)* y =  dx (─ j*) y = ─ j*y | +  dx j*( y) = j |pxy  ħ  i x ħ i ∞ - ^ wavefunctions vanish at infinity

measurement  y | A | y  = a y |y  Ay |y  = a* y |y  ^ a physical observable is described by a hermitian operator A ^ a state |y  is an eigenstate of an operator A, if A |y  = a |y  ^ vector is invariant under sheer transformation → eigenvector of the transformation the eigenvalues a are real y | A | y  = a y |y  Ay |y  = a* y |y  ^ 0 = (a – a*) y |y  

For [A,B] ≠ 0, [A,B] is a measure for the uncertainty of a and b: measurement the mean value of A is given by y | A | y  ^ y | A | y  =  an |cn|2 = A ^ n ^ |y  = cn|yn n the probability measuring eigenvalue an is given by |cn|2 y | A | y  is the mean value of 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: Da·Db ≥ ½ | [A,B]y | ^

measurement  (an – am) ym|yn = 0  ym|yn = 0 a physical observable is described by a hermitian operator A ^ eigenvectors of different eigenvalues are orthogonal A |ym  = am |ym  ^ A |yn  = an |yn  ^ an ym|yn = ym|Ayn = Aym|yn = am ym|yn ^  (an – am) ym|yn = 0 an ≠ am  ym|yn = 0 hermitian operators share a set of eigenvectors if they commute [A,B] = 0 ^  A and B are diagonal in the same base ^