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Vorlesung Quantum Computing SS ‘08 1 quantum bits conventional bit on V 1 off V 0 quantum mechanical bit (qubit) | 0 | 1 1 0 ( ( 0 1 ( ( a 1 | 0 + a 2 | 1 = a1a1 a2a2 () superposition:

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Vorlesung Quantum Computing SS ‘08 2 quantum computing HH -1 calculation U preparation read-out |A| time quantum-bit (qubit) 0 1 a 1 0 + a 2 1 = a1a1 a2a2

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Vorlesung Quantum Computing SS ‘08 3 boolean algebra and logic gates classical (irreversible) computing gate in out 1-bit logic gates: identity x NOT x x Id NOT xNOT x

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Vorlesung Quantum Computing SS ‘08 4 quantum gates 1-bit logic gate: NOT a 1 |1 + a 2 | 0 (a 1 | 0 + a 2 | 1 ) = manipulation in quantum mechanics is done by linear operators operators have a matrix representation X ≡ matrix representation for the NOT gate: X = a1a1 a2a2 a1a1 a2a2 = a2a2 a1a1 X X -1 =

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Vorlesung Quantum Computing SS ‘08 5 quantum parallelism a 1 F |00> + a 2 F |01> + a 3 F |10> + a 4 F |11> a 1 |00> + a 2 |01> + a 3 |10> + a 4 |11> input b 1 |00> + b 2 |01> + b 3 |10> + b 4 |11> = output F

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Vorlesung Quantum Computing SS ‘08 6 how to create superposition start in ground state ≡ | 0 1 0 manipulation with a unitary transformation H = 1 √2√2 H =H = √2√2 Hadamard Gate

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Vorlesung Quantum Computing SS ‘08 7 quantum computing HH -1 calculation U preparation read-out |A| time classical bit 1 ON 3.2 – 5.5 V 0 OFF -0.5 – 0.8 V quantum-bit (qubit) 0 1 a 1 0 + a 2 1 = a1a1 a2a2

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Vorlesung Quantum Computing SS ‘08 8 NOT: |1 |0 Bloch Sphere the 2 dimensional Hilbertspace of a single qubit can be represented by the Bloch-Sphere H: |0 |0 |1 |1 |0 |1 |0 |1 source: operations on a single qubit are represented by rotations on this sphere | = cos( ) + e i sin( )

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Vorlesung Quantum Computing SS ‘08 9 Bloch Sphere | = a 1 + a 2 | |a 1 | 2 + |a 2 | 2 = 1 | = r 1 e i + r 2 e i polar coordinates: multiply with global phase e -i | = r 1 + r 2 e i ( | = r 1 + r 2 e i r 1 + (x + iy)

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Vorlesung Quantum Computing SS ‘08 10 Bloch Sphere | = r 1 + r 2 e i r 1 + (x + iy) with normalization constraint: |r 1 | 2 + | x + iy | 2 = r (x – iy) (x + iy) = r x 2 + y 2 = 1 3 dim unit sphere x = r sin cos y = r sin sin z = r cos | = cos( ) + e i sin( ) = z + (x + iy) = cos + sin (cos + i sin ) = cos + e i sin 0 ≤ ≤ 0 ≤ ≤ 2

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Vorlesung Quantum Computing SS ‘08 11 infinitesimal unitary transformation finite transformations can be decomposed in successive infinitesimal transformations U( ) = I + i F ^ ^ ^ (I + i F) ^^ (I i F * ) = I ^^ ^ F hermitian, infinitesimal small and real ^ with F can be determined by the change L of an observable L ^ ^ ^ L’ = L + L = U( )LU( ) * = L + i [F,L] ^ ^ ^ ^ ^ ^ ^ ^ ^ U = e i F ^ ^

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Vorlesung Quantum Computing SS ‘08 12 how to rotate a qubit rotation about z-axis U (x) = (R -1 x) ≈ (x+ y, y- x, z) ≈ (x,y,z) + (y / x – x / y) = (1 – i /ħ [xp y – yp x ]) = (1 – i /ħ J 3 ) and finite angle ^ ^ ^ U( ) = (U( )) n = (1 – J 3 ) n → e i ħ n i ħ J 3 ^^

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Vorlesung Quantum Computing SS ‘08 13 spin as basis Pauli spin matrices form a complete base using spin as basis is very convenient for all implementations S z = = Z ħ 2 ħ 2 S x = = X ħ 2 ħ 2 S y = = Y 0 0i -i ħ 2 ħ 2 |0 |1 2 NOT : e = = e i ħ S x 0 0-i i

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Vorlesung Quantum Computing SS ‘08 14 Superposition of one and more qubits H =H = √2√2 e = i ħ (S x + S z ) √2√ i√2i√2 H 2 =H 2 H 1 = √2√2 √2√2 = 1 2

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Vorlesung Quantum Computing SS ‘08 15 entanglement QC → 1 √2√2 1 √2√2 states that can be factorized: live in subspaces H 1 and H 2 1 √2√2 → states that cannot be factorized: live in product space H QC only Bellstates

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Vorlesung Quantum Computing SS ‘08 16 Bell states The Bell State has the property that upon measuring the 1 st 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 2 nd qubit always gives the result depending on the measurement of the 1 st qubit. - ie: The measurements are CORRELATED ‘ ‘

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Vorlesung Quantum Computing SS ‘08 17 Bell basis – superpositions connected by the outline – Bell states connected by diagonals = 1 √2√2 = 1 √2√2

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Vorlesung Quantum Computing SS ‘08 18 Bell’s inequalities restrictions due to assumption of hidden classical variables inequalities are violated by quantum mechanics z x a,a’ = 1 z’ x’ g,g’ = 1 no influence between measurements, they are done at different spacetime points f := (a+a’)g – (a-a’)g’ (a and a’ are either equal or opposite) f := p(a,a’,g,g’) f ≤ 2 a a’ g’ g ga’ a g’g ag + a’g – ag’ + a’g’ ≤ 2

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Vorlesung Quantum Computing SS ‘08 19 Bell’s inequalities z x a,a’ = 1 z’ x’ g,g’ = a = a’ = g = – 1 √2√2 1 g’ = 1 √2√2 A G A G 1 √2√2 A G ag = | a g | = 1 √2√2 a’g = a’g’ = 1 √2√2 ag’ = 1 √2√2 ag + a’g – ag’ + a’g’ = 2 √2√2

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Vorlesung Quantum Computing SS ‘08 20 experiment source of entangled photons (use spontaneous parametric down conversion of a non-linear, birefringent crystal)

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Vorlesung Quantum Computing SS ‘08 21 experiment G. Weihs et al, Phys. Rev. Lett. 81, 5039 (1998) spacetime separationmeasurement apparatus measurement time: 100 ns physical random number generator

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Vorlesung Quantum Computing SS ‘08 22 uncorrelated measurements measurement apparatus measurement apparatus random number generator

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Vorlesung Quantum Computing SS ‘08 23 boolean algebra and logic gates 2-bit logic gates: y x OR y x y x AND y x x y x OR y x y x AND y 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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Vorlesung Quantum Computing SS ‘08 24 classical binary addition two one-bit digit in, one two-bit digit out: a 0 + b 0 = c 0 + c 1 X & a0a0 b0b0 c 0 (add mod 2 ) c 1 (carry bit) fanout truth table X + + X + + X 1234 a3a3 b1b1 a2a2 b2b2 a1a1 b3b3 a0a0 b0b0 c1c1 c2c2 c0c0 c3c3 c4c4 more than one bit...

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Vorlesung Quantum Computing SS ‘ qubit gates base vectors of a two–qubit register: a, a b a, b 00000000 0101 0101 10101111 11111010 CNOT:

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Vorlesung Quantum Computing SS ‘ qubit gates – switch on the interaction Hamiltonian – use free evolution of the system |1 |0 00 0000 00 01 0101 01 10 1110 11 11 1011 10 CNOT: source:

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Vorlesung Quantum Computing SS ‘08 27 the CNOT gate control target i ħ 2 e SySy i ħ 2 e - S y i ħ 2 e SzSz

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Vorlesung Quantum Computing SS ‘08 28 create entanglement Ca= 1Ca= 1 C b = 1 H 1 √2√2 a b a b → no factorization into product states possible 1 √2√2 U CNOT = = 1 √2√2 1 √2√2 1 √2√2 1 √2√2

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Vorlesung Quantum Computing SS ‘08 29 no-cloning theorem is a “no-copying theorem” classical: copy with XOR y x XOR y x (x,0) → (x,x) quantum mechanical: copy with CNOT ?

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Vorlesung Quantum Computing SS ‘08 30 no-cloning theorem “control” qubit is used as source “target” qubit is initialized to try to copy a 0 a 1 a 0 a 1 = a1a1 a0a0 a1a1 a0a0 = a 0 a 1 ≠ Bell state

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Vorlesung Quantum Computing SS ‘08 31 toward n qubits a 0 a 1 a 2 a 3 two–qubit state: n–qubit state: Hilbertspace: 2 n 2 n a i i i=0 e.g., n = 5: 2 -1 n

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Vorlesung Quantum Computing SS ‘08 32 universal computing 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) aa bb cc aa bb c (a b) Toffoli gate

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Vorlesung Quantum Computing SS ‘08 33 Toffoli gate a, b, c (a b) a, b, c 000 001 010 011 100 110 111 101 100 101 111 110 Table of Truth Matrix U TF = aa bb cc c (a b) Toffoli gate bb aa

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