Vorlesung Quantum Computing SS ‘08
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quantum parallelism
a1 F |00>+
a2 F |01>+
a3 F |10>+
a4 F |11>
}{a1 |00>
+a2 |01>
+a3 |10>
+a4 |11>
}{input
b1 |00>+
b2 |01>+
b3 |10>+
b4 |11>
}{=
output
F
Superposition for input created by Hadamard gates
Functions are represented by unitary operators
Quantum state tomography – but how to get a “real” result?
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Deutsch algorithm: table of truthD. Deutsch, Proceedings of the Royal Society of London A 400 (1985), 97
simplest example: 1-bit-to-1-bit function f : x → {0,1}
case 1
0
0
case 2
1
1
case 3
1
0
case 4
1
0
constant balanced
f(1)
f(0)
real or
false?
false true
f(0)f(1) 0 0 1 1
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strength of quantum parallelism
calculate a function f(x) with x ∈ {0,…,K} for all x at once
classical quantum
function balanced(f:function)a=f(0)b=f(1)return a b
function balanced(Uf:function)1 = H 0 = H |2 = Uf 1
3 = Hx 2 = f(f(|Rreturn f(f(
use quantum computing only if f(x) is “expensive” to calculate(time, memory), e. g., spin dynamics for huge structures
problem: superposition cannot be read-out: it will always collapse to an eigenstate
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Deutsch algorithm: principle
input
eigen-state
read-out
eigen-state
super-position
transferto
eigen-state
or
test-function
??
??
one run only!
0 1 2 3
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quantum circuit
Uf
x x
y yf(x)
data register
target register
addition modulo 2= f(x) for y=0
0
√2
f(0)f(1)
√2
H2 =√2
= 12√2
√2n
1 xf(x)x
evaluation of function f(x) with n data qubits x and 1 target qubit
BUT: How to extract information?
√2
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Deutsch algorithm: quantum circuit
1
0Uf
x x
y yf(x)
H
H
H
0 1 2 3
0
1H2 =√2
= 12√2
2Uf 1
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true
2Uf 1(|0|1) (|f(0) - |1 f(0)) 12
Deutsch algorithm: find Uf
case 1
0
0
case 2
1
1
case 3
1
0
case 4
1
0f(1)
f(0)
Uf
Uf x,yx,yf(x)
Uf: |0,0 → |0,0 , |0,1 → |0,1 , |1,0 → |1,1 , |1,1 → |1,0
Uf =
CNOT
Uf: |0,0 → |0,1 , |0,1 → |0,0 , |1,0 → |1,0 , |1,1 → |1,1
Uf =
Z-CNOT
Uf: |0,0 → |0,1 , |0,1 → |0,0 , |1,0 → |1,1 , |1,1 → |1,0
Uf =
NOT
Uf: |0,0 → |0,0 , |0,1 → |0,1 , |1,0 → |1,0 , |1,1 → |1,1
Uf =
ID
2Uf 1(|0, f(0) - |0,1 f(0) + |1, f(1) - |1,1 f(1)) 12
2Uf 1(|0 + |1) (|f(0) - |1 f(0)) 12
false
information encoded in phase of x-qubit
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Deutsch algorithm: get the answer
3Hx 2
1√2
1√2
false : H2x =H =
= =
12
1
-11
11√2
1
1
12
f (0) f (1)
true : H2x = H =
= =
12
1
-11
11√2
1
-1
12
1√2
1√2
f (0) f (1)
f (0) f (1)| f (0) - |1 f (0)
√2
read-outread-out
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Deutsch algorithm: summary
evaluates a global property of a function f(x) with a single run
Example: f(x) → f(0) = 1, f(1) = 1
1
0Uf
x x
y yf(x)
H
H
H
0 1 2 3
01H2 =√2
= 12√2
2
2
12
1√2
H2x =H =
=
12
1
-11
11√2
1
1
12
2
2 Uf 1
12
-
-
--
12
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Deutsch algorithm: implementation
40Ca+
42S1/2
42P1/2
42P3/2
32D3/2
32D5/2
397 nm 729 nm
854 nm
866 nm
|0
|1
quad
rupo
letr
ansi
tion
used for Laser cooling
dete
ctio
n
D5
/2 o
ccup
atio
n P
D
D5
/2 o
ccup
atio
n P
D
red sideband blue sideband
Roos et al: Phys. Rev. Lett. 83, 4713 (1999)
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Deutsch algorithm: implementationGulde et al: Nature 421, 48 (2003)
case 1: Uf = ID
case 2: Uf = NOT
case 3: Uf = CNOT
case 4: Uf = Z-CNOT
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composite pulses
example: imprecise B1-field
rotate from z to -z:
1) – pulse at y (180y)
2) three pulse sequence: 90y180x90y
M. H. Levitt: Prog. Nucl. Magn. Reson. Spectrosc. 18 (1986)
for small deviations (180-2)y
for small deviations (90-)y(180-2)x(90-)y
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Deutsch algorithm: FidelityGulde et al: Nature 421, 48 (2003)
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more than 1 ionNägerl et al: Phys. Rev. A 60, 145(1999)
switching between ions within 14 s
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CNOT with 2 ionsSchmidt-Kaler et al: Nature 422, 408(2003)
3m
n=0 n=1
n=0 n=1
Control Target0 0
R+(,0)1 1CNOT: spin state changed for n=0
R+(,)e
e
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CNOT with 2 ionsSchmidt-Kaler et al: Nature 422, 408(2003)
3m
n=0 n=1
n=0 n=1
Control Target0 0
R+(,0)21
CNOT: spin state changed for n=0R+(,)e
e
R(,0) 1
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error sourcesSchmitd-Kaler et al: Nature 422, 408 (2003)
CNOT with 2 40Ca+ ions as qubits and vibration mode as coupling
source contribution
laser frequency noise~100 Hz (phase coherence) ~10%
laser intensity fluctuations ~1%
laser detuning error ~2%
residual thermal excitation ~4%
addressing error ~3%
off-resonant excitations(for tgate = 600 s ) ~4%
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Needle in a haystack
Grover algorithm: search in an unsorted database
“Quantum mechanics helps in searching for a needle in a haystack”
Phys. Rev. Lett. 79, 325 (1997)
Lov Grover,Bell labs
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Search in a database
Example: search for a specific number in a phonebook
• N = 2n entries with index x = 0…N-1.
• A “detector” function f(x):
f(x) = 0- Entry x is no solution:
f(x) = 1- Entry x is a solution:
We need:
0 0 0 0 0 1 0 0
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The oracle
Grover’s algorithm minimizes calls to “oracle”
0 0 0 0 0 1 0 0
Classical: on average N/2 calls to oracle.
Quantum: number of calls √N.
0 1 0 0
|x: adresses of data register
f(x)=1 if entrance is solution
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1
0 Hn
H
Hn
H
n01
1
Quantum circuit
Target register: oracle qubit |q prepared in 1 = |1
Data register: superposition of n=2 qubits |x 1 = |0
1 2 3
H2 =√2
= 12√2
f(x) = 1
R iterations
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Oracle operator
UO x q|x |q f(x) |q is flipped, if |x points to register with solution
|q0 2H =√2
UO |x |q0 = (-1)f(x) |x |q0
√2
|x √2
|x
√2
|x√2
= |x
UOf(x) = 0:
f(x) = 1:
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Grover’s algorithm
H2 =√2
= 12√2
Oracle qubit does not change: Look at data register only.
2 =
3.1 = UO 2 = (|00 - |01 + |10 + |11) 12
3.2 = H23.1 = 14 12
= (|00 + |01 - |10 + |11)
3.3 = C3.2 = (-1)x-0)-1 3.2 = (|00 - |01 + |10 - |11)12
3.4 = H23.3 = |01
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Grover’s algorithm
H
H
H
H
UO |x |q0 = (-1)f(x) |x |q0
1
-1 1 11
√2H = C
-
-
-
C
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Geometrical analysis
C = -1 + 2 |00| with 2 = Hnand2 = Hn
G= HnCHnUO= Hn(2|00|-1)HnUO=(2|22|-1)UO
|=√N-M
1 (1-f(x))xx
√M1 f(x)x
x|= Superpostion of solutions
Superpostion of “no-solutions”
|2= | + | N
N-M√ N
M√
= cos | + sin |2
2
|
|
|22
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Geometrical analysis
2
|
|
|2
UO 2 = cos | - sin |2
2
UO|2
|2
UO|2
(2|22|-1)=|22| - (1- |22|)
= P2 – P2┴
:relexion at |2
G|2
G 2 = cos | + sin |32
32
Gk 2 =cos | + sin |k+1
2k+12
G = cos
cos
- sin
sin
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Iterations
Necessary number R of iterations is CI to
2
2
= =
2
R CI = ≤
2
2
arcsinNM√
4 M
N√
NM << 1with error probability p ≤ sin2 =
2 NM
• For more iterations than R, error increases
=> One needs to know number M of solutions.
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ion trap quantum computing: a summary
• qubit representation– hyperfine states (9Be+, 43Ca+)– electronic states (40Ca+)– vibrational modes
• qubit manipulation: laser irradiation
• initial state preparation: – Doppler and sideband cooling
• read-out: fluoresence
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relaxation and operation
• electronic states: – energy relaxation time T1 ~ 1s– phase relaxation time T2 ~ 10 ms– gate operation time Tgate ~ 200 s– T2/Tgate ~ 50
• hyperfine states:– phase relaxation time T2 ~ 10s– gate operation time Tgate ~ 10 s– T2/Tgate ~ 106
source: Homepage group A. Steane http://www.physics.ox.ac.uk/users/iontrap/news.html
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scaling to more than 10 ionsKielpinski, Monroe, Wineland: Nature 417, 709 (2002)
C. Monroe group, Michigan ‘06