introduction to quantum computing lecture 2: quantum ... · 1 introduction to quantum computing...
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Introduction to Quantum ComputingLecture 2: Quantum Algorithms
Rod Van [email protected]
June 27-29WIDE University School of Internetwith numerous slides from E. Abe
Course Outline
● Lecture 1: Introduction● Lecture 2: Quantum Algorithms● Lecture 3: Devices and Technologies● Lecture 4: Quantum Computer Architecture● Lecture 5: Quantum Networking & Wrapup
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Outline
● Introduction● Deutsch-Jozsa● Grover's search algorithm● Shor's factoring algorithm● Brief look at Quantum Error Correction (QEC)
How Do Quantum Algos Work?
● Runs are begun by creating a superposition of all possible input values.
● Executing a function gives a superposition of answers of all possible inputs! The hard part is extracting the answer we want.
● Every part of the superposition works independently on the algorithm.
● They all work by using interference. The phase of parts of the superposition are arranged to cancel out and leave only the interesting answer.
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n⊗0
0 0 0 0 1
2
0 1
2
0 1
2
1
2 n 2 0 00 0 01 1 11
1
2 n 2 0 1 2 N 11
2 n 2x
N 1
x
H 1
2
1 11 1 2n N created superposition of all
possible inputs
Apply Hadamards to n-qubit register
NO! Final step is to measure the register, and the superposition collapses to ONE state! It's as if we did the computation exactly once.
Quantum Parallelism
12 n 2
For example, x=0,1,…,N-1 calculate f(x) for all values of x
Using the quantum superposition, we run the calculation all at once
So do we get all of the answers for f(x)?
H⊗ n
(|ƒ (0) > + ƒ (1) > + ƒ (2) > + - - - + | ƒ (N-1) > )
H⊗n
Entanglement and Superposition
● Two qubits can have dependent values
00 11
2
1
200 0 01 0 10 1
211
Measure either qubit. If youget 0, you know the otherqubit must be 0.Bit0 Bit1
0 0 50%0 1 0%1 0 0%1 1 50%
1
2
1001
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Graphic Representation
Each bar is the amplitude of the wave function, that is, the square root of the probability, of finding the system in a particular state.
1
400 1
401 1
410 1
411
1
400 1
401 1
410 1
411
Example: Search
1
nN 2=
Start with all equalprobabilities:
Flip the phase ofthe answer:
Flip all states aboutthe mean (average):
The analog nature of phase figuresin strongly!
β
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Quantum Algorithms● Deutsch-Jozsa(D-J)
– Proc. R. Soc. London A, 439, 553 (1992)● Grover's search algorithm
– Phys. Rev. Lett., 79, 325 (1997)● Shor's algorithm for factoring large numbers
– SIAM J. Comp., 26, 1484 (1997)
D. Deutsch R. Jozsa L. K. Grover P. W. Shor
Deutsch's ProblemDetermine whether a function is “constant” or “balanced”
quantumQuery once to becertain!
constant:f(x) is either 0 for all x, or 1 for all x
balanced: f(x) is 0 for half the values of x, 1 for half the values
e.g., n=2:
e.g., n=2:
classicalMust query 2n-1+1 times to be certain
f x
f(x) =(0,0,0,0) or f(x)=(1,1,1,1)
f(x) =(0,0,1,1)
f(0) =?
f(1) =?
f(2) =?
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D-J Circuit
0
0 n
f fS
register
work bit
H x 1
2 z
x 1
2n 2z
∵ H 00 1
2H 1
0 1
2
Hadamard Gate
w h e r e , x x1
x2
xn
z z1
z2
zn
x z x 1 z 1 x 2 z 2 x n z n
transforms:
f Gate
S Gate
H⊗ n
H⊗ n⊗
(-1)xz|z>
(-1)xz|z>
ƒ|x>|y>|x>|y ƒ(x)>⊗
ƒ|x>|0> = |x>|0 ƒ(x)>=|x>|f(x)>⊗
ƒ|x>|f(x)>=|x>|f(x) f(x)>=|x>|0>⊗
S | x > | y > = (-1)y | x > | y >
What Happens in D-J
0 n0 H n f1
2n 2x
x 0 1
2 n 2x
x f x
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D-J (More Efficient Version)
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D-J and GroverDeutsch-Jozsa takes advantage of interference in the phase to cancel out unwanted terms in the superposition.
But, D-J uses only +1 and -1 in the phase and essentially calculates parity.
Grover's algorithm takes advantage of the full continuous nature of phase to create interference...
(Note: Remember, phase applies to the whole term in the superposition, not just a single qubit! Shift the phase on any qubitand you shift it on the whole term in the superposition.)
Grover's Search
nN 2= items
N 2 n unordered items, we're looking for item “β”
Classically, would have to check about N/2 items
To use Grover's algorithm, create superposition of all N inputs
N repetitions of the Grover iterator G will produce “β”
Hard task!!
β
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Grover's Algorithm Circuit
oracle: sometimes called a “gate”, an unknown function that tells us if we have the answer
G Gate Analysis
O 2Ox x 2 x x x
H n H noracle 2 0 0 1
O 1 2
2 1k
k k2N k ,k' ,k''
k k' ' k' kk
k k
k2
kk ∵ α
k
αk
N
sign flip for the answer only!
“inversion about average”
0k 2
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G Gate Analysis(2)
2 1 2cos 2sin 2
cos 2 sin 21 00 1
cos sinsin cos
O 1 2
1 00 1
2 01
0 1 1 00 1
G 2 1 Ocos sinsin cos
1
N 1x
x10
01
2
O
flipped with respect to
θ
flipped with respect to
rotate by angle θ
G
let alpha be the superposition of the N-1
possibilities that are wrong (e.g., not β)
N 1N
1
N
cos 2sin 2
∵θ 2arctan 1 N 1
2
2-D plane
2 30
Execution Example, N=4
1
4sin
2
changes in planeEvery value starts with equal amplitude (probability)
12
00 01 10 11
OO
12
00 01 10 11
2 11
2 1
60
1 execution of G gives 100% chance of correct answer (rare case)
2 k
214
12
00 214
12
01 214
12
10 214
12
11
10
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Execution Example, N=8
1
8sin
2
G 5
281
28
Gθ41.4°
Every value starts with equal amplitude
2
G11
4 8 1
4 8
G
θ
Executing G twice takes us close to the answer; more takes us PAST the answer
changes in plane
Repeat CountHow many times do we have to repeat G to get the
answer?
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Grover and Shor
Grover uses interference in phase to cancel unwanted terms.
Shor goes a step further, broadening the range of conditions in which useful interference occurs, by doing a Fourier transform...
Shor's Factoring Algorithm66554087 ? 6703 9929
An “efficient” classical algorithm for this problem is not known.
Using classical number field sieve, factoring anL-bit number is
Using quantum Fouriertransform (QFT), factoring an L-bit number is
Superpolynomial speedup! (Careful, technicallynot exponential speedup, and not yet proven thatno better classical algorithm exists.)
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Quantum Fourier Transformj QFT
N
1
N k 0
N 1
exp 2 ijk N k
H R2
R3
H R2
HR
k1 00 exp 2 i 2k
Circuit for QFT8
QFT818
1 1 1 1 1 1 1 11 1 2 3 4 5 6 7
1 2 4 6 1 2 4 6
1 3 6 1 4 7 2 5
1 4 1 4 1 4 1 4
1 5 2 7 4 1 6 3
1 6 4 2 1 6 4 2
1 7 6 5 4 3 2 1
exp 2 i 8 iUnitary matrix for QFT8
i i 4 0
FFT in quantum form
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Order-finding
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Order-finding Process
0 43 H 4 14 x 0
15
x 3
Î 3
14 x 0
15
x x 3
14
0 4 8 12 3
14
1 5 9 13 1
14
2 6 10 14 7
14
3 7 11 15 0
measurementgives one of
0,4,8,12Run through classical computer to find r
14
0 4 8 12 3
14
0 i 4 8 i 12 1
14
0 4 8 12 7
14
0 i 4 8 i 12 0
QFT16
QFT16
1
16
1 1 1 11 1 2 15
1 2 4 14
1 15 14 1
exp 2 i 16i i 8 0
Shor's Algorithm Flow
ar 1 modL find smallest r
r e v e n
p gcd ar 2 1,Lq gcd ar 2 1,L
L pq
calculate
a 1 a LL is the number we wish to factor; pick a random
p , q L
1. even2. prime3. power of a prime
Algorithm fails if L is:
NO
NO
YES
YES
quantum order-findingcalculate
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Factoring L=15
1 7 4 131 1 2 1 3 1
4 1ar 2 1 48 gcd 48,15 3ar 2 1 50 gcd 50,15 5
a 7
1 141 1
2 1
ar 2 1 13 gcd 13,15 1ar 2 1 15 gcd 15,15 15
a 14
r2 r7 r8 r13 4
Failure!!
1 111 1
2 1
ar 2 1 10 gcd 10,15 5ar 2 1 12 gcd 12,15 3
a 11
r4 r11 r14 2
a 2,4,7,8,11,13,14
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Euclid's Method
494 133 3 95
133 95 1 38
95 38 2 19
38 19 2
95
133
3895
19
38
Algorithm for findinggreatest common
denominator
494
133
gcd(494,133)
Π Gate (Circuit)x 2n 1 x n 1 2 n 2 x n 2 2 x 1 x 0
a x mod L a2 n 1 x
n 1 a2n 2 x
n 2 a2 x
1 ax
0 mod L
a2 n 1 x
n 1 mod L a2 n 2 x
n 2 mod L ax
0 mod L
Î x y x axymodL
Î therefore,
U2jU20
U21U2n 1
x0
x1
xj
xn 1
U2jy a2j
y modL
can be constructedusing classical techniques
a2jmodL
Van Meter & ItohPRA 71, 052320 (2005)「xj is the “enable bit” for the
multiplication
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Back to Factoring L=15
a 7Step.1 Pick random a
© Colin P. Williams (1997)
Step.2 Order-findingQFT example (N =28=256)
0 50 100 150 200
1
0.1
0.01
0.001
0.0001
0.00001
0 64 128 192
Step.3 Measure to recover r
r 4
Step.4p gcd( )ar 2 1,Lq gcd( )ar 2 1,L
calculate
ar 2 1 48 gcd(48,15)=3ar 2 1 50 gcd(50,15)=5
15 3 5Done!
Other Order-Finding Algos.
● Abelian subgroup, discrete logarithm● QFT based, but very different in classical portion
of algorithm● Hidden subgroup problems in general
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Main Classes of Algorithms
● 1: Use the QFT to find periodicity● 2: Grover's algorithm and friends● 3: Simulating quantum physics● (D-J seems to fall outside these)
Architecture Affects Algorithm Efficiency
Long-distance gates
Neighbor onlyoperations requireswapping qubits
2-qubitgate
Single line layout, neighbor only
2-qubitgate
swapqubitvalues
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Quantum Error Correction
● Discovered by Shor and Calderbank● Important work done by Steane● Based on known classical techniques, but with
important extensions● Must correct both bit errors and phase errors
3-bit Code EncodingKeep a logical qubit in triplicate:
0 1 enc odeL
000 111
This protects against single bit-flip errors,same as in classical coding.
1
0
0} L
Encoding circuit:
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3-bit Code Correction
000 111 er r or 010 101
Bit flip error occurs:
How do you correct?Calculate parity pair-wise; results will tell you which bit is in error,then a simple NOT gate on that bit will correct.
1
2
0
Parity circuit:
This bit now holds parity of 1 & 2
Just measure that bottom bit and you get the parity of that pair;two parities can tell you which bit was in error.
Wait a Minute!You said measurement destroys the superposition, won'twe lose our important quantum state?
Not quite... you only lose as much superposition as youlearn about the state. This is called a “partial trace”. α| 000 > |0 + β | 111 > | 0 > parity
0 111 0 parity 010 parity 101 parity
Measuring this qubit...Tells us nothingabout these qubits.
But measuringthis one wouldtell us somethingand collapse the state!
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What About Phase?
So far, that gets us bit flip protection. Now we need phase flip...
This is a basis change, and protects us against the phase flip error:
This code can be combined with the bit flip code to create the9-bit Shor code.
Wrap-Up on Algorithms
● “Quantum” algorithms actually have both quantum and classical parts
● Use of quantum interference based on complex,analog phase is critical
● Period-finding algorithms work well(exponential speedup over best known algos, but not yet proven better)