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**( ( ) quantum bits conventional bit**

off <=> V <=> 0 quantum mechanical bit (qubit) | 0 <=> <=> | 1 <=> <=> 1 ( superposition: a1 a2 ( ) a1| 0 + a2| 1 =

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**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

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**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

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**quantum gates X ≡ X = X X-1 = 1-bit logic gate: NOT**

(a1| 0 + a2| 1 ) = a1|1 + a2| 0 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 X X-1 = 1

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**quantum parallelism a1 |00> + a2 |01> a3 |10> a4 |11>**

input b1 |00> + b2 |01> b3 |10> b4 |11> = output a1 F |00> + a2 F |01> a3 F |10> a4 F |11> F

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**how to create superposition**

≡ | 0 ≡ 1 start in ground state manipulation with a unitary transformation Hadamard Gate H = 1 -1 √2 H |0 = 1 √2 (|0 + |1)

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**quantum computing |Y0 U H H-1 Y|A|Y classical bit**

1 ON – 5.5 V 0 OFF – 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

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**Bloch Sphere |y = cos( ) |0 + eij sin( )|1 |0 NOT: |1 |0 H:**

the 2 dimensional Hilbertspace of a single qubit can be represented by the Bloch-Sphere operations on a single qubit are represented by rotations on this sphere |y = cos( ) |0 + eij sin( )|1 q 2 |0 |1 source: NOT: |1 |0 H: |0 |0 + |1 |1 |0 - |1

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**Bloch Sphere y |y = 1 |y = a1|0 + a2|1 |a1|2 + |a2|2 = 1**

|y = r1eij|0 + r2eiJ|1 polar coordinates: multiply with global phase e-ij: |y = r1|0 + r2eif|1 f = (J - j) |y = r1|0 + r2eif|1 = r1|0 + (x + iy)|1

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**Bloch Sphere 3 dim unit sphere |y = z |0 + (x + iy)|1**

|y = r1|0 + r2eif|1 = r1|0 + (x + iy)|1 with normalization constraint: |r1|2 + | x + iy |2 = r12 + (x – iy) (x + iy) = r12 + x2 + y2 = 1 3 dim unit sphere |y = z |0 + (x + iy)|1 = cos q |0 + sin q (cos f + i sin f)|1 = cos q |0 + eif sin q |1 x = r sin q cos f y = r sin q sin f z = r cos q |y = cos( ) |0 + eif sin( )|1 q 2 0 ≤ q ≤ p, 0 ≤ f ≤ 2p

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**infinitesimal unitary transformation**

finite transformations can be decomposed in successive infinitesimal transformations U(e) = I + ieF ^ (I + ieF) ^ (I - ieF*) = I ^ ^ F hermitian, e infinitesimal small and real ^ with F can be determined by the change dL of an observable L ^ L’ = L + dL = U(e)LU(e)* = L + ie[F,L] ^ U = eieF ^

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**how to rotate a qubit ^ UY (x) = Y (R-1x) ≈ Y (x+ey, y-ex, z)**

rotation about z-axis ^ UY (x) = Y (R-1x) ≈ Y (x+ey, y-ex, z) ≈ Y (x,y,z) + e (y Y /x – x Y /y) = Y (1 – ie/ħ [xpy – ypx]) = Y (1 – ie/ħ J3) ^ ^ and finite angle q i ħ - qJ3 ^ ^ i q ħ n U(q) = (U(e))n = (1 – J3)n → e

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**spin as basis Sz = = Z Sx = = X Sy = = Y 1 ħ 2 -1 |0 ħ 2 1 -i ħ |1 2**

Pauli spin matrices form a complete base using spin as basis is very convenient for all implementations Sz = = Z 1 -1 ħ 2 |0 |1 Sx = = X 1 ħ 2 Sy = = Y i -i ħ 2 p 2 NOT : e = = e i ħ - pSx -i - i 1

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**Superposition of one and more qubits**

H = 1 -1 √2 e = i ħ (Sx+ Sz) p √2 1 -1 i√2 H2=H2H1 |00 = √2 (|0 + |1) = (|00 + |01 + |10 + |11) 1 2

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**entanglement Bell states |cQC = |Y1 |Y2**

√2 states that can be factorized: live in subspaces H1 and H2 Bell states 1 √2 | c → (|01 - |10) states that cannot be factorized: live in product space HQC only

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Bell states The Bell State has the property that upon measuring the 1st qubit one obtains two possible results. - 0 with probability ½ leaving the post measurement state - 1 with probability ½ leaving the post measurement state - The measurement of the 2nd qubit always gives the result depending on the measurement of the 1st qubit. - ie: The measurements are CORRELATED |Y‘ = |01 |Y‘ = |10

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**Bell basis |01 |10 |11 |00 |Y = (|01 |10) |Y = (|00 |11)**

superpositions connected by the outline Bell states connected by diagonals |Y = (|01 |10) 1 √2 |Y = (|00 |11) 1 √2

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**Bell’s inequalities -1 1 1 1 1 1 -1 -1 -1 1 a’**

restrictions due to assumption of hidden classical variables inequalities are violated by quantum mechanics z’ x’ z x no influence between measurements, they are done at different spacetime points g,g’ = 1 a,a’ = 1 f := (a+a’)g – (a-a’)g’ -1 1 1 1 1 (a and a’ are either equal or opposite) f := S p(a,a’,g,g’) f ≤ 2 1 -1 -1 -1 1 a’ ag + a’g – ag’ + a’g’ ≤ 2 a’ a’ a g’ g g’ g g a

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**Bell’s inequalities |Y = | A | A | G | G | A | A | G | G 1 √2**

z’ x’ z x g,g’ = 1 a,a’ = 1 ag = Y | a g |Y = 1 √2 a’g = a’g’ = 1 √2 ag’ = 1 √2 ag+a’g – ag’+ a’g’ = 2 √2 1 -1 a = a’ = 1 -1 g = – √2 g’ =

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**experiment source of entangled photons**

(use spontaneous parametric down conversion of a non-linear, birefringent crystal)

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**experiment spacetime separation measurement apparatus**

measurement time: 100 ns physical random number generator G. Weihs et al, Phys. Rev. Lett. 81, 5039 (1998)

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**uncorrelated measurements**

random number generator measurement apparatus measurement apparatus

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**boolean algebra and logic gates**

2-bit logic gates: x y x OR y x y x AND y y x OR y 1 x y x AND y 1 x all other operations can be constructed from NOT, OR, and AND x XOR y = (x OR y) AND NOT (x AND y)

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**classical binary addition**

two one-bit digit in, one two-bit digit out: a0 + b0 = c0 + c1 X & a0 b0 c0 (add mod2) c1 (carry bit) fanout 0 1 truth table + X 1 2 3 4 a3 b1 a2 b2 a1 b3 a0 b0 c1 c2 c0 c3 c4 more than one bit...

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**2 qubit gates |00 = |01 = |10 = |11 = 1 CNOT: 1 00 01 10**

base vectors of a two–qubit register: |00 = 1 |01 = |10 = |11 = a, a b a, b 00 01 10 11 1 CNOT:

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**2 qubit gates switch on the interaction Hamiltonian**

use free evolution of the system |1 |0 00 00 01 01 10 11 11 10 CNOT: source:

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the CNOT gate control target i ħ p 2 e Sy Sy i ħ p 2 e Sz

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**create entanglement 1 |00 = |01 = |10 = |11 = Ca= |1 |a |b**

Cb= |1 H 1 √2 (|0 - |1) |a |b 1 UCNOT = = - √2 |00 = 1 |01 = |10 = |11 = |a |b → (|01 - |10) no factorization into product states possible 1 √2

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**no-cloning theorem 1 1 1 1 is a “no-copying theorem”**

classical: copy with XOR y x XOR y 1 x (x,0) → (x,x) 1 1 1 1 quantum mechanical: copy with CNOT ?

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**no-cloning theorem |Y |0 = a0 |0 |0 + a1 |1 |0 1 =**

“control” qubit is used as source “target” qubit is initialized to |0 try to copy |Y = a0 |0 + a1|1 |Y |0 = a0 |0 |0 + a1 |1 |0 Bell state 1 = a1 a0 a1 a0 = a0 |0 |0 + a1 |1 |1 ≠ |Y |Y

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**toward n qubits |Y = S ai |i |Y = a0|00 + a1|01 + a2|10 + a3|11**

two–qubit state: |Y = a0|00 + a1|01 + a2|10 + a3|11 n–qubit state: 2 -1 n |Y = S ai |i Hilbertspace: 2n 2n i=0 e.g., n = 5: |00101 = |5 |00110 = |6

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**universal computing |a |b |c |c(a b) Toffoli gate**

all possible operations can be done by using 1-qubit-rotations, phase-shifts and the CNOT gate → this set of gates is therefore called “universal” (in a classical computer NOT and NAND are a universal set) single universal gate: Toffoli gate (3 qubits) |a |b |c |c(a b) Toffoli gate

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**Toffoli gate 1 UTF = Table of Truth 000 Matrix 001 010 011**

|c |c(a b) Toffoli gate a, b, c (ab) a, b, c 000 001 010 011 100 110 111 101 Table of Truth Matrix 1 UTF =

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