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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7. Factoring Numbers with a Linear-Optics Quantum Computer. Daniel F. V. James Department of Physics University of Toronto. • Funding :. - PowerPoint PPT Presentation
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Daniel F. V. JamesDaniel F. V. JamesDepartment of PhysicsDepartment of PhysicsUniversity of TorontoUniversity of Toronto
Factoring Numbers with a Linear-Optics Quantum Computer
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
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• Funding:
• My Shiny Brand New Group at TorontoToronto René Stock (postdoc) Asma Al-Qasimi (Ph.D.) Hoda Hossein-Nejad (Ph.D.) Arghavan Safavi (B.Eng.)MIT Felipe Corredor (B.Eng.)Stanford Max Kaznadiy (B.Sc.) Ardavan Darabi (B.Sc.) Rebecca Nie (B.Sc.)
• Collaborators:Prof. Rainer Blatt (Innsbruck)Prof. Andrew White (Queensland)Prof. Paul Kwiat (Illinois)Prof. Emil Wolf (Rochester)
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
NMR
Trapped Ions
Neutral Atoms
Photons
Solid State
Superconductors
Cavity QED
1. S
cala
ble
qubi
ts
2. In
itial
izat
ion
3. C
oher
ence
4. Q
uant
um G
ates
5. M
easu
rem
ent
Theoretical possibility
Experimental reality
No known approach
Whither Quantum Computing?Roadmap Traffic-Light Diagram(Apr 2004) -updated
Clock states, DFS
SET detectors(> 80%)
QLD, APL gates
NMR Algorithmic cooling
Entanglement at UCSB
NIST gates
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What can we do with them?• Neat experiments like teleportation, Bell’s inequalities,...
• 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.
• Other applications quantum simulations, QKD repeaters,...
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
• Outline:1. RSA Encryption and Factoring2. Simplifications for a Few Qubits3. Linear Optics Quantum Computing (LOQC)4. Factoring 15 with LOQC5. Where next and conclusions
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Alice
Bob
1. RSA* Encryption and Factoring
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
*Rivest, Shamir & Adelman, 1978; (also Clifford Cocks, 1973).
• select two prime numbers: p,q
• calculate: n = p.q; = (p-1).(q-1)
• select e, with GCD(e,) = 1
• calculate d, with e.d = 1 mod public key: n,e
• Message: M
• calculate: E = Me mod n
encrypted message, E
• calculate: Ed mod n = M mod n
• Easy to find the message if you know p and q
• Security relies on difficulty of factoring n
Message M
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• Example: n = 77; c = 8;
0
20
40
60
80
0 10 20 30 40a
f n,c(x
)
x
• The function fn,c(x) = cx mod n , is periodic, (period r). • Either or is a factor of n.
€
GCD cr / 2 −1,n( )
€
GCD cr / 2 +1,n( )
• Chose a number, c, which is coprime with n i.e. GCD(c,n) =1
GCD 85 +1,77( )=11
GCD 85 −1,77( ) =7
From data, r = 10;
Period Finding Factoring
Factoring Numbers*
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
*P. Shor, Proc. 35th Ann. Symp. Found. Comp. Sci. 124-134 (1994);also: Preskill et al., Phys Rev A 54, 1034 (1996).
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Large number of evaluations are replaced by one
Quantum Factoring
€
then you can do this : ˆ U f x 0x∑ ⎛
⎝ ⎜
⎞
⎠ ⎟= x fN ,C x( )
x∑
• Classical factoring: evaluate fn,c(x) for a large number of
values of x until you can find r.
€
if you can do this : ˆ U f x 0 = x fN ,C x( )
• Quantum parallelism:
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
i.e. the state of multiple qubits corresponding to x; e.g.if x=29, ⏐x =⟩ ⏐11101⟩
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€
φ x( ) =1
2L / 2fN ,C y( ) exp i
2π
2Lx.y
⎛
⎝ ⎜
⎞
⎠ ⎟
y =0
2 L -1
∑
Quantum Factoring (cont.)• Quantum Fourier Transform to argument register:
€
UQFT ⊗ I( ) x fN ,C x( )x∑ = x φ x( )
x∑
• If 2L/r=M, number of periods in the argument register:
€
φ x( ) = M ϕ s if x = sM s = 0,1,2....r −1( )
0 otherwise
⎧ ⎨ ⎩
0 T 2T
Periodic Function
... 0 1/T 2/T 3/T ...
Fourier Transform
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Quantum Factoring (cont., again)
€
x φ x( )x∑ = s2L /r ϕ s
s =0
r-1∑
• Thus the state after the QFT is:
• Discard the function register: the argument register is in a mixed state:
€
ρ final = s2L r s2L rs =0
r-1∑
• Measurement of the function register yields, with high probability a number which is a multiple of N/r; extracting r, you can find the factors.
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modular exponentiation Fourier trans.initiation
Circuit Diagram
argument
function
€
x 0x=0
2 L-1∑
€
x fN ,C x( )x=0
2 L-1∑€
C20
€
C21
€
C2L−1
....
QFT
€
sN /r ϕ ss =0
r-1∑
readout
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
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RSA cryptosystem:
• polynomial work to encrypt/decrypt
• exponential work to break = factoring
• BUT quantum factoring is only polynomial work
27997833911221327870829467638722601621070446786955428537560009929326128400107609345671052955360856061822351910951365788637105954482006576775098580557613579098734950144178863178946295187237869221823983
RSA 200:3532461934402770121272604978198464368671197400197625023649303468776121253679423200058547956528088349
7925869954478333033347085841480059687737975857364219960734330341455767872818152135381409304740185467
x=
Vulnerability of RSA to Quantum Computers?
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
RSA200
0 200 400 600 800 10001
10
100
10001 10
41 10
51 10
61 10
71 10
81 10
91 10
101 10
111 10
121 10
131 10
141 10
151 10
161 10
171 10
181 10
191 10
201 10
211 10
221 10
231 10
24
Classical ~ exp{AL}# of instructions
# of bits, L, factored
~ 1020 instructions: 16 months (2003-05)
Shors Algorithm~ L3
~ 1012 operations:Hours ?
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
2. Simplifications for a Few QubitsN=15, C=4
x fN,C(x)
0 1
1 4
2 1
3 4
i.e. period 2
N=15, C=2
x fN,C(x)
0 1
1 2
2 4
3 8
4 1
i.e. period 4
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Simplifications for a Few QubitsN=15, C=4
x fN,C(x)
00 001
01 100
10 001
11 100
i.e. period 2
N=15, C=2
x fN,C(x)
000 0001
001 0010
010 0100
011 1000
100 0001
i.e. period 4
this is too profligate with qubits....
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Simplifications for a Few QubitsN=15, C=4
x LogC[fN,C(x)]
00 00
01 01
10 00
11 01
i.e. period 2
N=15, C=2
x LogC[fN,C(x)]
000 00
001 01
010 10
011 11
100 00
i.e. period 4
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cancel
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Minimalist Period 2 Circuit
X
QFT
Z
• Top rail cancellation occurs for all r =2n.• Two qubits, one quantum gate
x LogC[fN,C(x)]
00 00
01 01
10 00
11 01
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XX
X
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Slightly Less Minimalist Period 2 Circuitwithout Logarithm of fN,C(x)
• Three qubits, two gates.
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X
X
What about Period 4?
QFT
Z
Z
T
€
T =exp iπ /8[ ] 0
0 exp −iπ /8[ ]
⎛
⎝ ⎜
⎞
⎠ ⎟
x LogC[fN,C(x)]
000 00
001 01
010 10
011 11
100 00
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X
X
What about Period 4?
Z
• 4 qubits, 2 quantum gates
x LogC[fN,C(x)]
000 00
001 01
010 10
011 11
100 00
Z
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Measurement-induced nonlinearity*
* Knill, Laflamme & Milburn (“KLM”) , Nature 409, 46 (2001)
•••
•••QUBITSQUBITSaa
LINEAROPTICAL
NETWORK
•••
•••
SINGLEPHOTONS
FAST FEEDFORWARDSINGLE PHOTON
DETECTION
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
3. Linear Optics Quantum Computing
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XZ|C⟩XZ|T⟩BXZ|φ⟩|φ⟩|CNOT⟩-NONDETERMIN
GATEB
• Non-deterministic gates• Don’t always work, but heralded when they do
• Many non-deterministic gates proposed …
• Teleport non-deterministic gates deterministic
NON-DETERMIN
GATE
•••
•••QUBITSQUBITS•••
•••
SINGLEPHOTONSSINGLE PHOTON
DETECTION&
FEEDFORWARD
• Teleportation: moving information without measuring it
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Linear Optics Quantum Computing (cont.)
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+ teleportation+ error correction= scalable QC
Internal ancillas
• Simplified 2-photonRalph, Langford, Bell, & White,PRA 65, 062324 (2002)
• Simplified 2-photon Hofmann & Takeuchi,PRA 66, 024308 (2002)
• Linear-optical QNDKok, Lee & Dowling, PRA 66, 063814 (2002)
External ancillasNON-
DETERMINGATE
QNDQND
Gasparoni et al., PRL 93, 020504 (2004)
Pittman et al., PRA 68, 032316 (2004)Walther et al., Nature 434, 169 (2005)
• KLM 4-photonKnill, Laflamme, & Milburn,Nature 409, 46 (2001)
• Entangled ancilla 4-photon Pittman, Jacobs, and Franson,PRL 88, 257902 (2002)
• Simplified 4-photon Ralph, White, Munro, & Milburn,PRA 65, 012314 (2001)
• Efficient 4-photon Knill, PRA 66, 052306 (2002)
• Entangled input 2-photon Pittman, Jacobs, and Franson,PRL 88, 257902 (2002)
NON-DETERMIN
GATE
Proposed Entangling Gates
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C0
C1
T0
T1
C0
C1
T0
T1
phaseshift
CSIGN gate
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Two Qubit Gate*
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
€
00
control qubit
€
⎧⎨⎩
€
⎧⎨⎩
target qubit
*Ralph, Langford, Bell & White, PRA 65, 062324 (2002)
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C0C1T0T1-1/3
1/3
1/3
CSIGN gate
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Two Qubit Gate*
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
€
00
€
+ 83 ϕ⊥
€
→ 13 00
*Ralph, Langford, Bell & White, PRA 65, 062324 (2002)
control qubit
€
⎧⎨⎩
€
⎧⎨⎩
target qubit
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both transmittedboth reflected
C0C1T0T1-1/3
1/3
1/3
CSIGN gate
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
*Ralph, Langford, Bell & White, PRA 65, 062324 (2002)
Two Qubit Gate*
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
€
10
€
→ 13 − 1− 1
3( ){ } 00
€
→ −13 00
€
+ 83 ′ ϕ ⊥
Non-deterministicCSIGN gate with probability 1/9
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C0C1T0T1HWPHWP
-1/3
1/3
1/3
Control in Control out
Target in Target out
C0C1T0T1HWPHWP
CNOT gate
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Interferometric Gate*
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
*Ralph, Langford, Bell & White, PRA 65, 062324 (2002)
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C
T
Non-classical interferenceRH = 1/3
RV = 1
C0C1T0T1HWPHWP
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
*Ralph, Langford, Bell & White, PRA 65, 062324 (2002)
Interferometric Gate*
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
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C
T
Non-classical interference
No dual-path interferometers
No adjustment if wrong splitting ratio
RH = 1/3
RV = 1
C0C1T0T1HWPHWP
Langford, Weinhold, Prevedel, Pryde, O’Brien, Gilchrist and White, PRL 95, 210504 (2005)
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Beam-splitter Gate*
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
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*Pryde, O’Brien, Gilchrist, James, Langford, Ralph, and White, PRL 93 080502 (2004); Langford, et al., PRL 95, 210504 (2005)
Ideal Measured
average gate fidelity:
1 gate works 90-95% of time; 2 gates should work 80-90% of time
Process Tomography of a Quantum Gate*
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
= 94 ± 2 %
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*Lanyon, Weinhold, Langford, Barbieri, James, Gilchrist, and White, PRL 99, 250505 (2007)DEPARTMENT OF PHYSICS,
UNIVERSITY OF QUEENSLAND
4-photonsource
4. Factoring Experiment*
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GCD of Cr/2±1 and NGCD of 41±1 and 15 = 3,5
P110 = 27 ± 2%
order-2 order-4
F = 99.9 ± 0.3%SL= 99.9 ± 0.6% F = 98.5 ± 0.6%
SL= 98.1 ± 0.8%
P000 = 27 ± 2%P100 = 24 ± 2%
GCD of Cr/2±1 and NGCD of 41±1 and 15 = 3,5
r=2
add redundantbit then reverseargument bits
P10 = 48 ± 3%
P00 = 52 ± 3% failure
r=6
GCD of 43±1 and 15 = 3,5
r=2
P010 = 23 ± 2%
algorithm works near perfectly …?
Order-finding algortihm uses mixed output state: non-deterministic
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Measuring the output
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
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order-2
order-4
FGHZ = 59 ± 4%WGHZ = 9 ± 4%
SL= 62 ± 4%
F2Bell = 98.5 ± 0.6%=
Tbd = 41 ± 5%Tce = 33 ± 5%
SL= 98.1 ± 0.8%
joint state of argumentand function registersis entangled & mixed
joint state of argumentand function registersis highly entangled
independent photons
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Measuring the output
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
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x LogC[fN,C(x)]
0 0
1 1
2 2
3 0
4 1
5 2
i.e. period 3
5. Where next: Period 3?
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
N=21, C=4x 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:⏐⟩= ⏐⟩⏐⟩⏐⟩⊗⏐⟩⏐⏐⟩⏐⟩⏐⟩⊗⏐⟩⏐⏐⟩⏐⟩⊗⏐⟩
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A bit less scary...
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
• Can we make this state using established techniques for making W states and GHZ states?
where: ⏐⟩⏐⟩⏐⟩√
⏐⟩⏐⟩⏐⟩⏐⟩√
⏐⟩⏐⟩⏐⟩⏐⟩√
flip qubit #2:⏐⟩= ( ⏐⟩⊗⏐⟩⏐⟩⊗⏐⟩⏐⟩⊗⏐⟩
€
18
€
3
€
3
€
2
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€
3 00 + 3 01 + 2 10 GHZ W W
Period 3 Circuit
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
H
X
X
X
Ry()
• 5 qubits, 8 gates + QFT (still pretty scary)• Period is not a power of 2; full QFT needed.• Size of the argument register will not be a factor of the period.
QFTXX
X
X
X
H
R(
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Conclusions
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
• 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.
• Complexity of quantum circuit depends on period r, rather than size of number.
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
BUT....• Unless you get real lucky, N is not a multiple of r
• How actually do you implement the unitary operations for modular exponentiation and quantum Fourier transform?
- you can fix this using bigger registers, so the ‘periodic’ signal swamps the rest.
- 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. 76 3228 (1996).
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€
φ x( ) =1
2L / 2fN ,C y( ) exp i
2π
2Lx.y
⎛
⎝ ⎜
⎞
⎠ ⎟
y =0
2 L -1
∑
Quantum Factoring (cont.)• Quantum Fourier Transform to argument register:
€
UQFT ⊗ I( ) x fN ,C x( )x∑ = x φ x( )
x∑
• Assume that 2L=Mr (i.e. the size of the argument register is equal to a multiple of the unknown period, r):
€
φ x( ) = M ϕ s if x = sM s = 0,1,2....r −1( )
0 otherwise
⎧ ⎨ ⎩
€
ϕ s = 1r
fN ,C y( ) exp 2πis.y /r( )y =0
r-1∑ ϕs ϕ t =δst
where:
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• Process tomography: measure combinations of basis processes
|0⟩|H⟩
|0+1⟩|D⟩
|0+i1⟩|R⟩
• State tomography: measure combinations of basis states
I X Y Z
n qubit gate requires 24n measurements
n qubit state requires 22n measurements
rotations on Poincare sphere
• Reconstructed states and processes are unphysical: effect of uncertainties Maximum likelihood or Bayesian analysis required
0 1 0 + 1 0 + i1H V D R
for 2 photon states, bi-photon Stokes parameters
I II XI YI Z
X IXXXYXZ
Y IYXYYYZ
Z IZXZYZR
HHHVHDHR
VHVVVDVR
DHDVDDDR
RHRVRDRR
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
*James, Kwiat, Munro and White, PRA 64, 030302 (2001)
Quantum Tomography*
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
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4-photonsource
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
*Lanyon, Weinhold, Langford, Barbieri, James, Gilchrist, and White, PRL 99, 250505 (2007)
Factoring Circuits*
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* order-2order-2 order-4
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Factoring Circuits*
*Lanyon, Weinhold, Langford, Barbieri, James, Gilchrist, and White, PRL 99, 250505 (2007)
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
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* order-2 order-4order-2 order-4
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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Factoring Circuits
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND
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Process and state tomography require 24n and 22n measurements:impractical for large circuits
Correlation measurements between registers require only 2n measurements:. logical measurement of argument register, Mij
Output state is: After logical measurement of argument register, function register is
order-2 order-4
{P01,P10} = {83 ± 4%, 59 ± 5%}{P00,P01,P10,P11} ={87 ± 3%, 84 ± 4%, 82 ± 5%, 67 ± 6%}
argument and function registers are highly correlatedQuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
Circuit outputs
DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND