Measuring progress in Shor’s factoring algorithm
Thomas Lawson
Télécom ParisTech,
Paris, France
1 / 58
Shor’s factoring algorithm
What do small factoring experiments show?
Do they represent progress?
2 / 58
Overview
Picking the calculation
Shortcuts
A measure of success
3 / 58
| Shor’s factoring algorithm
Shor’s factoring algorithm
4 / 58
| Shor’s factoring algorithm
Factoring
To factor N
1
coprime x
2
order r, (xr mod N = 1)
3
factors are gcd(x 2 ± 1, N ).
r
N=21
3
Pick coprime x
Find order r
gcd
7
5 / 58
| Shor’s factoring algorithm
Order finding
Factoring N = 21 using coprime x = 4.
The order is given by 4r mod 21 = 1.
40 = 1
41 = 4
42 = 16
43 = 64
44 = 256
45 = 1024
46 = 4096 . . . .
6 / 58
| Shor’s factoring algorithm
Order finding
Factoring N = 21 using coprime x = 4.
The order is given by 4r mod 21 = 1.
40 mod 21 = 1
41 mod 21 = 4
42 mod 21 = 16
43 mod 21 = 1
44 mod 21 = 4
45 mod 21 = 16
46 mod 21 = 1
....
Here r = 3.
7 / 58
| Shor’s factoring algorithm
Factoring: an example
Factoring N = 21 using coprime x = 4.
The order is given by 4r mod 21 = 1.
40 mod 21= 1
41 mod 21= 4
42 mod 21= 16
43 mod 21 = 1
44 mod 21 = 4
45 mod 21 = 16
46 mod 21 = 1
....
Here r = 3.
8 / 58
| Shor’s factoring algorithm
Factoring: an example
The factors of N = 21 are
3
gcd(4 2 ± 1, 21)
= gcd(8 ± 1, 21)
= gcd(7, 21) and gcd(9, 21)
=7 and 3.
9 / 58
| Shor’s factoring algorithm
Quantum order finding
Quantum operators speed it up.
U |1i = |4i
U 2 |1i = |16i
U 3 |1i = |1i
10 / 58
| Shor’s factoring algorithm
Quantum order finding
U contains the whole order, r.
U |φk i = αk |φk i
U r |φk i = |φk i
αkr = 1
11 / 58
| Shor’s factoring algorithm
Quantum order finding
U contains the whole order, r.
U |φk i = αk |φk i
U r |φk i = |φk i
k
αk = e2πi r
12 / 58
| Shor’s factoring algorithm
Quantum order finding
The QOFA quickly finds αk .
Rn M n . . .
|100...0>
|+>
U
n
2
...
|+>
R2 M2
U²
|+>
R1 M1
U
k
= 0.M1 M2 M3 . . . Mn
r
13 / 58
| Shor’s factoring algorithm
Quantum order finding
The output of the quantum order finding algorithm.
Pr
...
1/r
2/r
3/r
4/r
5/r
Output
(for n → ∞).
14 / 58
| Picking the calculation
Picking the calculation
15 / 58
| Picking the calculation
Trivial calculations
Normally the distribution becomes fuzzy as n is reduced.
Pr
...
1/r
2/r
3/r
4/r
5/r
Output
16 / 58
| Picking the calculation
Trivial calculations
Normally the distribution becomes fuzzy as n is reduced.
Pr
...
1/r
2/r
3/r
4/r
5/r
Output
17 / 58
| Picking the calculation
Trivial calculations
But for trivial calculations this does not happen.
For order
r = 4,
Pr
0/4
1/4
2/4
3/4
Output
18 / 58
| Picking the calculation
Trivial calculations
But for trivial calculations this does not happen.
For order
r = 4,
Pr
0.000 0.010 0.100
0.110
Output
19 / 58
| Picking the calculation
Trivial calculations
A circuit that does this
|100...0>
|+>
R3 M3
U4
|+>
R2 M2
|+>
U²
R1 M1
U
Pr
0.000 0.010 0.100
0.110
Output
20 / 58
| Picking the calculation
Trivial calculations
A circuit that does this
|100...0>
|+>
H
0
I
|+>
R2 M2
|+>
U²
R1 M1
U
Pr
0.000 0.010 0.100
0.110
Output
21 / 58
| Picking the calculation
Trivial calculations
A circuit that does this
|100...0>
|+>
H
0
I
I
R2 0/1
I
U²
R1 0/1
U
Pr
0.000 0.010 0.100
0.110
Output
22 / 58
| Picking the calculation
Trivial calculations
This happens if r = 2p ,
because 1/r represented in binary.
N = 15 gives r = 2 or r = 4. This is always trivial.
23 / 58
| Picking the calculation
Nontrivial calculations
Nontrivial: N = 21 with x = 4 gives r = 3.
1
= 0.01010101 . . .
3
24 / 58
| Picking the calculation
Nontrivial calculations
a"
⇥0
⇥0
⇥0
⇥0
⇥0
⇥1
⇥0
H"
H"
2
c"
{I,R}" H"
H"
(n;1)"
Further"iteraBons"
to..."
(n;2)"
U"2" N = 21 with x U"
Factoring
=2" 4 (giving r = 3).
3
8
Probability"
b"
0
0
0
1
0
00
01
10
11
U"2
n=2"
U"
0.35
0.35
1
4
1
8
1
2
0"
U"2"
000
011
101
0
0000
0101
1011
Increasing"precision"
FIG. 1: The iterative order finding algorithm for factoring 21. a, Measurement of the control qubit after each controlled
unitary gives the next most significant bit in the output and the outcome is fed forward to the iterated (semi-classical) Fourier
transform, which applies either the identity operation I or the appropriate phase gate R, prior to the Hadamard H. b, As
the number of iterations increases the precision increases. c, For two bits of precision the controlled unitary operations can be
constructed with this arrangement of controlled-swap gates.
Fourier transform is constructive for states contributing
to the 00 term and boosts its probability of observation
to three times that of the probability for observing the
10 term, which experiences destructive quantum interference among its contributory states. Decoherence in the
two qubit control register, the single swap of U 2 is implemented with a controlled-NOT (CNOT) gate; U 1 is
realised with two swaps, the first of which is a CNOT
gate, while it is sufficient for the second swap to be un25 / 58
controlled. (See Appendix for details).
| Picking the calculation
Second cause of triviality
Lack of precision.
|100...0>
|+>
R3 M3
|+>
U4
R2 M2
U²
|+>
R1 M1
U
If r > 2n no interference happens.
26 / 58
| Picking the calculation
Second cause of triviality
Lack of precision.
R3 M3
|+>
R2 M2
|+>
R1 M1
|100...0>
|+>
If r > 2n no interference happens.
27 / 58
| Picking the calculation
Second cause of triviality
For n steps, only 2n states can be accessed.
R3 M3
|+>
R2 M2
|+>
R1 M1
|100...0>
|+>
If r > 2n no interference happens.
28 / 58
| Picking the calculation
Second cause of triviality
Lack of precision.
R3 M3
|+>
R2 M2
|+>
R1 M1
|100...0>
|+>
If r > 2n no interference happens.
29 / 58
| Picking the calculation
Idée reçue 1)
Short r are easy.
In fact
r = 212 is easier than r = 3,
r must be small.
30 / 58
| Experimental shortcuts
Experimental shortcuts
31 / 58
| Experimental shortcuts
Simplifying the circuit
Demonstrations use shortcuts:
Compiling
Circuit simplifications
Substitutions
32 / 58
| Experimental shortcuts
Compiling
Compiling removes the hardest part of the algorithm - making the unitary
n
operators, U 2 .
Rn M n . . .
|100...0>
|+>
U
n
2
...
|+>
R2 M2
U²
|+>
R1 M1
U
33 / 58
| Experimental shortcuts
Compiling
7
Rn M n . . .
|100...0>
|+>
U
n
2
...
|+>
R2 M2
U²
|+>
R1 M1
U
Figure: Niskanen et al. 2004.
34 / 58
| Experimental shortcuts
Compiling
All demonstrations have used compiling.
Rn M n . . .
|100...0>
|+>
U
n
2
...
|+>
R2 M2
U²
|+>
R1 M1
U
It is not scalable.
It needs knowledge of the calculation.
A part of the algorithm is missing.
35 / 58
| Experimental shortcuts
Compiling
All demonstrations have used compiling.
Rn M n . . .
|100...0>
|+>
U
n
2
...
|+>
R2 M2
U²
|+>
R1 M1
U
It is not scalable.
It needs knowledge of the calculation.
A part of the algorithm is missing.
36 / 58
| Experimental shortcuts
Simplifying the circuit
Demonstrations use shortcuts:
Compiling
Circuit simplifications
Substitutions
37 / 58
| Experimental shortcuts
Circuit simplifications
Unused qubits are removed.
R3 M3
|+>
R2 M2
|+>
R1 M1
|100...0>
|+>
This is fine (even if it needs knowledge of the calculation).
38 / 58
| Experimental shortcuts
Circuit simplifications
Unused qubits are removed.
R3 M3
|+>
R2 M2
|+>
R1 M1
|100...0>
|+>
This is fine (even if it needs knowledge of the calculation).
39 / 58
A<< 9D89 O:HF9 , FA F7 '.!C 87B O:HF9A - 87B W 8;< F7 8 ?FN9:;< >K '.! 87B
',!6 #D< ;<EFA9<; FA 9D:A F7 8 ?FN9:;< >K '...!C '.,.!C ',..! 87B ',,.!C >;
'.!C '-!C '+! 87B ']!6 #D< @<;F>BF=F9Q F7 9D< 8?@JF9:B< >K '%! FA 7>5 -C A>
" ! L!- ! + 87B E"="B""0+!- ! ,# ,\# ! W# \6 #D:AC <G<7 8K9<; 9D< J>7E
87B =>?@J<N @:JA< A<O:<7=< >K 9D< BFKM=:J9 =8A< RPFE6 +SC 9D<
B898 =>7=J:AFG<JQ F7BF=89< 9D< A:==<AAK:J <N<=:9F>7
Doing otherwise<N@<;F?<798J
is a waste
of resources.
>K VD>;UA 8JE>;F9D? 9> K8=9>; ,\6
!<G<;9D<J<AAC
Factor N = 15 giving
r =9D<;<
4. 8;< >HGF>:A BFA=;<@87=F<A H<95<<7 9D< ?<8AI
:;<B 87B FB<8J A@<=9;8C ?>A9 7>98HJQ K>; 9D< BFKM=:J9 =8A<6 $AF7E 8
7:?<;F=8J ?>B<JC 5< D8G< F7G<A9FE89<B 5D<9D<; 9D<A< B<GF89F>7A
| Experimental shortcuts
Circuit simplifications
|100...0>
|+>
H
a
(0)I
n
m
!0"
|+>
0
(1)
H
R2 M2
(2) U²
n
!1"
|+>
x
1
ax mod N
U (4)
Inverse
QFT
b
1:
2:
3:
4:
5:
6:
7:
T
e
m
p
o
r
a
l
a
v
e
r
a
g
i
n
g
H
H
90 H
H
5F9D $ ! , # -#.!( , C , !
K>JJ>5F7E >HA<;G89F>7A
AF7EJ< A@F7 B<A=;F@9F>7A
R,S b"1 R87B c1S <
=>??:9<T R-S 9D< *) K>;
8@@JF<B 9> 8;HF9;8;Q "T 87
R1 M1
(3)
x
87B @D8A< B8?@F7E Rc1
"!!!) ,
*. ! %
.
i
ω i /2π
T1,i
T2,
1
2
3
4
5
6
7
–22052.0
489.5
25088.3
–4918.7
15186.6
–4519.1
4244.3
5.0
13.7
3.0
10.0
2.8
45.4
31.6
1.3
1.8
2.5
1.7
1.8
2.0
2.0
45 90 H
H
A B
C
D
E
F
G
H
!"#$%& ' !"#$%"& '()'"(% *+) ,-+)./ #01+)(%-&2 (3 4"%0($5 +* %-5 6"#$%"& '()'"(%2 7()5/
)58)5/5$% 6"9(%/3
#$: 9+;5/ )58)5/5$%
+85)#%(+$/2 <(&5
05*% %+ )(1-%2 =>? @$(%(#0(A5
Figure:
Vandersypen
et1+5/
al.*)+&2001.
# B)/% )51(/%5) +* ! ! C"0+1C "# 6"9(%/ %+ %>! ! ! ! %>! =*+) /-+)% D>!? #$: # /5'+$:
!"#$%& * ,%)"'%")5 #$: 8)+85)%
85)X"+)+9"%#:(5$G0 ()+$ '+&805
&5#/")5: / ELU EON K#0"5/3 J5 '+
:(**5)5$% *)+& %-#% :5)(K5: ($ )5*2
=($ HA? #% EE2T <3 )50#%(K5 %+ # )5
EL
40 / 58
| Experimental shortcuts
Simplifying the circuit
Demonstrations use shortcuts:
Compiling
Circuit simplifications
Substitutions
41 / 58
| Experimental shortcuts
Substitutions
Technology which is not ready is replaced.
R2 M2
U²
|+>
R1 M 1
U
The global properties can be tested.
42 / 58
| Testing Shor’s algorithm
Testing Shor’s algorithm
43 / 58
| Testing Shor’s algorithm
Idée reçue 2)
The number factored is a good measure.
15 4
21 4
51
85
This does not represent progress.
44 / 58
| Testing Shor’s algorithm
Idée reçue 2)
The number factored is a good measure.
15 4
21 4
51
85
This does not represent progress.
45 / 58
| Testing Shor’s algorithm
Idée reçue 3)
Factoring (like love) should be blind.
Avoid trivial cases.
Factoring is physics.
46 / 58
| Testing Shor’s algorithm
Idée reçue 4)
Factoring a large number is a good test.
3 × 7 = 21
This shows the computer works. But it doesn’t say how well.
47 / 58
| Testing Shor’s algorithm
Idée reçue 4)
Factoring a large number is a good test.
3 × 7 = 21
Bad circuits occasionally get the factors
Perfect circuits often fail
48 / 58
| Testing Shor’s algorithm
A test of progress
We should test the order finding algorithm.
Pr
...
1/r
2/r
3/r
4/r
5/r
Output
49 / 58
| Testing Shor’s algorithm
A test of progress
The overlap improves with n.
|+>
R M |+>
U128
R M |+>
U64
R M |+>
U32
R M |+>
U16
R M |+>
U8
R M |+>
R M |+>
U2
U4
R M
U
Pr
...
1/r
2/r
3/r
4/r
5/r
Output
50 / 58
| Testing Shor’s algorithm
A test of progress
The overlap improves with n.
|+>
R M |+>
U128
R M |+>
U64
R M |+>
U32
R M |+>
U16
R M |+>
U8
R M |+>
R M |+>
U2
U4
R M
U
Pr
...
1/r
2/r
3/r
4/r
5/r
Output
51 / 58
| Testing Shor’s algorithm
A test of progress
The overlap improves with n.
|+>
R M |+>
U128
R M |+>
U64
R M |+>
U32
R M |+>
U16
R M |+>
U8
R M |+>
R M |+>
U2
U4
R M
U
Pr
...
1/r
2/r
3/r
4/r
5/r
Output
52 / 58
| Testing Shor’s algorithm
A test of progress
The overlap improves with n.
|+>
R M |+>
U128
R M |+>
U64
R M |+>
U32
R M |+>
U16
R M |+>
U8
R M |+>
R M |+>
U2
U4
R M
U
Pr
...
1/r
2/r
3/r
4/r
5/r
Output
53 / 58
| Testing Shor’s algorithm
Conclusion
To design a factoring experiment
1
choose a nontrivial case,
2
only leave out uninteresting parts of the circuit,
3
judge the quantum order finding algorithm.
54 / 58
| Thank you for your attention.
Thank you for your attention.
55 / 58
| Extras
Extras
56 / 58
| Extras
QFT vs iterative PEA
a)
#" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " #
#
H
...
|0!
H
#
H
#
•
|0!
..
.
#
#
#
|0!
...
#
H
•
...
#
•
#
•
...
Rn−1
R1
•
H
•
!!!
# ..!"
# .
#
!!!
# !"
#
!!!
# !"
#
Rn
...
R1
H
#
#
#
#
...
2
U
U2
#
# U
" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " "
|1!
n
b)
|0!
|1!
H
•
U2
H
n
!!!
!"
. . . |0!
...
H
φ
φ1
Rn−1
. . . R1 n−2
•
U2
H
!!!
!"
|0!
H
φ
Rnφ1 . . . R1 n−1
•
H
!!!
!"
U
57 / 58
| Extras
Odd r
The factors are
r
gcd(x 2 ± 1, N )
N = 21, x = 4 gives r = 3
3
gcd(4 2 ± 1, 21)
= gcd(23 ± 1, 21)
since x is a perfect square.
58 / 58