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Vorlesung Quantum Computing SS ‘08 1 quantum bits conventional bit on 3.2 - 5.5 V 1 off -0.5 - 0.8 V 0 quantum mechanical bit (qubit) | 0  | 1  1 0 (

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Presentation on theme: "Vorlesung Quantum Computing SS ‘08 1 quantum bits conventional bit on 3.2 - 5.5 V 1 off -0.5 - 0.8 V 0 quantum mechanical bit (qubit) | 0  | 1  1 0 ("— Presentation transcript:

1 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:

2 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 

3 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

4 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 =

5 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

6 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

7 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 

8 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( )     

9 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) 

10 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 

11 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 ^ ^

12 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 ^^

13 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

14 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

15 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

16 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  ‘   ‘ 

17 Vorlesung Quantum Computing SS ‘08 17 Bell basis    – superpositions connected by the outline – Bell states connected by diagonals  =  1 √2√2  =  1 √2√2

18 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

19 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  

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

21 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

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

23 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)

24 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...

25 Vorlesung Quantum Computing SS ‘ qubit gates base vectors of a two–qubit register:                     a, a  b a, b 00000000 0101 0101 10101111 11111010                 CNOT:

26 Vorlesung Quantum Computing SS ‘ qubit gates – switch on the interaction Hamiltonian – use free evolution of the system |1  |0  00  0000  00 01  0101  01 10  1110  11 11  1011  10 CNOT: source:

27 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

28 Vorlesung Quantum Computing SS ‘08 28 create entanglement Ca= 1Ca= 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                    

29 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 ?                

30 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

31 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

32 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) aa bb cc aa bb  c  (a  b)  Toffoli gate

33 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 =                                                 aa bb cc c  (a  b)  Toffoli gate bb aa


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