1. Shor's Algorithm with a Linear-Optics Quantum Computer
Daniel F. V. James
Department of Physics
University of Toronto
ICQI Rochester
IThB1/Thursday 14 June 2007
10:30 a.m. - 11:00 a.m. Sloane Auditorium
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

2. • My Shiny Brand New Group at Toronto:
Dr. René Stock (postdoc)
Mr. Charles Meunier (Ph.D. student)
Miss. Asma Al-Qasimi (M.Sc. student)
Miss. Lucy Zhang (M.Sc. student)
Miss. Hoda Hossein-Nejad (M.Sc. student)
Mr. Max Kaznadiy (B.Sc. student)
Miss. Arghavan Safavi (B.Sc. student)
Mr. Ardavan Darabi (B.Sc. student)
• Collaborators:
Prof. Rainer Blatt (Innsbruck)
Prof. Andrew White (Queensland)
Prof. Paul Kwiat (Illinois)
Prof. Emil Wolf (Rochester)
...
See also: IFB3
Friday, 11:15 a.m.
• Funds:
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

3. NMR Algorithmic cooling
Whither Quantum Computing?
Roadmap Traffic-Light Diagram
(Apr 2004)
-updated
1. Scalable qubits
4. Quantum Gates
5. Measurement
2. Initialization
NMR Algorithmic cooling
3. Coherence
Clock states,
DFS
NMR
No know approach
Trapped Ions
Theoretical possibility
Neutral Atoms
Photons
Experimental reality
Solid State
SET detectors
(> 80%)
Superconductors
QLD, APL gates
Cavity QED
Entanglement at UCSB

4. What can we do with them?
• Neat experiments like teleportation, demonstrations of error correction, Bell’s inequalities...
• Other applications quantum simulations, repeaters for QKD, metrology, ...
• Scalability: more qubits and logic gates, larger scale entanglement, connections between remote nodes, speed.
• Find a signal, then maximize it: do Shor’s algorithm for simplified, small scale cases, then progressively improve it.

5. Factoring Numbers*
• Chose a number, C, which is coprime with N i.e. GCD(C,N) =1
(‘easy’ to check with Euclid’s algorithm).
• The function fN,C(x) = Cx mod N , is periodic, (period r).
• Example:
N = 77; C = 8;
fN,C(x) x
• Either or is a factor of N.
From data, r = 10;
Period Finding  Factoring
*P. Shor, Proc. 35th Ann. Symp. Found. Comp. Sci (1994);
also: Preskill et al., Phys Rev A 54, 1034 (1996).
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

6. ~2L/2 operations replaced by 1 operation
Quantum Factoring
• Classical factoring: evaluate fN,C(x) for a large number of values of x until you can find r.
i.e. the state of multiple qubits corresponding to x; e.g.if x=29, ⏐x〉=⏐11101〉
• Quantum parallelism:
L = number of qubits in the argument register
~2L/2 operations replaced by 1 operation
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

7. Quantum Factoring (cont.)
• Quantum Fourier Transform to argument register:
• Assume that 2L=Mr (i.e. 2L is equal to a multiple of the unknown period, r); then, because fN,C(x) is periodic:
where:

8. Quantum Factoring (cont., again)
• Thus the state after the QFT is:
• Discard the function register: the argument register is in a mixed state:
• Measurement of the function register yields, with high probability a number which is a multiple of N/p; extracting p, you can find the factors.
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

9. Circuit Diagram  .... QFT argument function initiation
modular exponentiation
Fourier trans.
readout 
....
QFT
argument
function

10. BUT.... • Unless you get real lucky, N is not a multiple of r
- you can fix this using bigger registers, so the ‘periodic’ signal swamps the rest.
• How actually do you implement the unitary operations for modular exponentiation and quantum Fourier transform?
both can be done efficiently (i.e. in a polynomial number of operations)
break down complicated operations into simpler operations (e.g. multiplexed adders and repeated squaring), which can be performed by CNOTs and related multi-qubit quantum gates.
QFT can be simplified by dropping some operations, and by doing it ‘semi-classically’ by measurement and feed-forward*
*R. B.Griffiths and C.-S. Niu Phys. Rev. Lett (1996).
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

11. Simplifications for a Few Qubits
N=15, C=4
N=15, C=2
x fN,C(x)
0 1
1 2
2 4
3 8
4 1
i.e. period 4
x fN,C(x)
0 1
1 4
2 1
3 4
i.e. period 2
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

12. Simplifications for a Few Qubits
N=15, C=4
N=15, C=2
x fN,C(x)
00 001
01 100
10 001
11 100
x fN,C(x)
i.e. period 2
i.e. period 4
this is too profligate with qubits....
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

13. Simplifications for a Few Qubits
N=15, C=4
N=15, C=2
x LogC[fN,C(x)]
00 00
01 01
10 00
11 01
x LogC[fN,C(x)]
000 00
001 01
010 10
011 11
100 00
i.e. period 2
i.e. period 4
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

14. Minimalist Period 2 Circuit
QFT  X
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

15. Minimalist Period 2 Circuit
c a n c e l   Z  X
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

16. Minimalist Period 2 CircuitZ  X
• Top rail cancellation occurs for all r =2n.
• Two qubits, one quantum gate
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

17. What about Period 4?  QFT   X X
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

18. What about Period 4?   Z T   X Z  X
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

19. What about Period 4? Z T   X Z  X
• 4 qubits, 3 quantum gates (one of which can be replaced by measurement).
• Experimental realization using Linear Optics for period 2 and 4 (i.e. factoring of 15): See: IFB3/Friday, 11:15 a.m.
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

20. Where next: Period 3? N=21, C=4
x LogC[fN,C(x)]
000 00
001 01
010 10
011 00
100 01
101 10
i.e. period 3
After modular exponentiation, a three qubit argument register plus two qubit function register will be in the state:
⏐〉= (⏐000〉+⏐011〉+⏐〉)⊗ ⏐00〉
+(⏐00〉+⏐〉+⏐〉)⊗ ⏐0〉
(⏐00〉+⏐〉)⊗ ⏐0〉
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

21. Where next: Period 3? flip qubit #2:
⏐〉= ⏐〉⊗ ⏐00〉+⏐〉⊗ ⏐0〉⏐〉⊗ ⏐0〉
⏐〉⏐00〉+⏐01〉+⏐〉
⏐W〉⏐0〉+⏐〉+⏐〉
⏐〉⏐〉+⏐〉
where:
• Can we make this using established techniques to create W states and GHZ states?
• Period is not a power of 2, hence full QFT will have to be implemented.
• Size of the argument register will not be a factor of the period.
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

22. Conclusions
• Simplified versions of Shor’s Algorithm are accessible with today’s quantum technology technology.
• Improving these results, step-by-step, is as good a route to practical quantum computers as any.
• Experiment by Andrew White’s group at University of Queen: Sloane Auditorium, tomorrow, 11:15am.
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7