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Daniel F. V. James Department of Physics University of Toronto

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 :.

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Daniel F. V. James Department of Physics University of Toronto

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  1. 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

  2. • Funding: DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 • My Shiny Brand New Group atToronto 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)

  3. Whither Quantum Computing? Roadmap Traffic-Light Diagram (Apr 2004) -updated Clock states, DFS 4. Quantum Gates 1. Scalable qubits 5. Measurement 2. Initialization 3. Coherence NMR Algorithmic cooling NIST gates NMR No known approach Trapped Ions Theoretical possibility Neutral Atoms Photons Experimental reality Solid State SET detectors (> 80%) Superconductors QLD, APL gates Cavity QED Entanglement at UCSB DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

  4. What can we do with them? DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 • Neat experiments like teleportation, Bell’s inequalities,... • Other applications quantum simulations, QKD repeaters,... • 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. • Outline: 1. RSA Encryption and Factoring 2. Simplifications for a Few Qubits 3. Linear Optics Quantum Computing (LOQC) 4. Factoring 15 with LOQC 5. Where next and conclusions

  5. Message M Alice • select two prime numbers: p,q • calculate: n = p.q; = (p-1).(q-1) • select e, with GCD(e,) = 1 public key: n,e • Message: M • calculate d, with e.d = 1 mod  encrypted message, E • calculate:          E = Me mod n • calculate:          Ed mod n = M mod n DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 1. RSA* Encryption and Factoring Bob • Easy to find the message if you know p and q • Security relies on difficulty of factoring n *Rivest, Shamir & Adelman, 1978; (also Clifford Cocks, 1973).

  6. • Either or is a factor of n. • Example: n = 77; c = 8; fn,c(x) From data, r = 10; x DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Factoring Numbers* • Chose a number, c, which is coprime with n i.e. GCD(c,n) =1 • The function fn,c(x) = cx mod n , is periodic, (period r). Period Finding  Factoring *P. Shor, Proc. 35th Ann. Symp. Found. Comp. Sci. 124-134 (1994); also: Preskill et al., Phys Rev A 54, 1034 (1996).

  7. Quantum Factoring i.e. the state of multiple qubits corresponding to x; e.g.if x=29, ⏐x〉=⏐11101〉 • Quantum parallelism: DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 • Classical factoring: evaluate fn,c(x) for a large number of values of x until you can find r. Large number of evaluations are replaced by one

  8. Quantum Factoring (cont.) Periodic Function Fourier Transform  ... ... 3/T 0 1/T T 2T 0 2/T • If2L/r=M, number of periods in the argument register: • Quantum Fourier Transform to argument register:

  9. Quantum Factoring (cont., again) • Discard the function register: the argument register is in a mixed state: DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 • Thus the state after the QFT is: • 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.

  10. Circuit Diagram initiation Fourier trans. modular exponentiation  QFT .... DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 readout argument function

  11. RSA 200: 27997833911221327870829467638722601621070446786955428537560009929326128400107609345671052955360856061822351910951365788637105954482006576775098580557613579098734950144178863178946295187237869221823983 3532461934402770121272604978198464368671197400197625023649303468776121253679423200058547956528088349 7925869954478333033347085841480059687737975857364219960734330341455767872818152135381409304740185467 x = Shors Algorithm~ L3 ~ 1012 operations: Hours ? DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Vulnerability of RSA to Quantum Computers? • RSA cryptosystem: • polynomial work to encrypt/decrypt • exponential work to break = factoring • BUT quantum factoring is only polynomial work # of instructions Classical ~ exp{AL} ~ 1020 instructions: 16 months (2003-05) # of bits, L, factored RSA200

  12. 2. Simplifications for a Few Qubits N=15, C=2 xfN,C(x) 01 14 21 34 xfN,C(x) 01 12 24 38 41 i.e. period 2 i.e. period 4 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 N=15, C=4

  13. Simplifications for a Few Qubits xfN,C(x) 00001 01100 10001 11100 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 N=15, C=4 N=15, C=2 xfN,C(x) 0000001 0010010 0100100 0111000 1000001 i.e. period 2 i.e. period 4 this is too profligate with qubits....

  14. Simplifications for a Few Qubits DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 N=15, C=4 N=15, C=2 xLogC[fN,C(x)] 0000 0101 1000 1101 xLogC[fN,C(x)] 00000 00101 01010 01111 10000 i.e. period 2 i.e. period 4

  15. Minimalist Period 2 Circuit cancel X Z xLogC[fN,C(x)] 0000 0101 1000 1101 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7   QFT  • Top rail cancellation occurs for all r =2n. • Two qubits, one quantum gate

  16. Slightly Less Minimalist Period 2 Circuit without Logarithm of fN,C(x) X DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7  X X • Three qubits, two gates.

  17. What about Period 4?   Z Z X X T xLogC[fN,C(x)] 00000 00101 01010 01111 10000   QFT 

  18. What about Period 4?  Z Z X X xLogC[fN,C(x)] 00000 00101 01010 01111 10000    • 4 qubits, 2 quantum gates

  19. 3. Linear Optics Quantum Computing DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Measurement-induced nonlinearity* * Knill, Laflamme & Milburn (“KLM”) , Nature409, 46 (2001)

  20. Linear Optics Quantum Computing (cont.) DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 • Non-deterministic gates • Don’t always work, but heralded when they do • Teleportation: moving information without measuring it • Teleport non-deterministic gates deterministic • Many non-deterministic gates proposed …

  21. Proposed Entangling Gates External ancillas • Simplified 2-photon Ralph, Langford, Bell, & White, PRA65, 062324 (2002) • Simplified 2-photonHofmann & Takeuchi, PRA66, 024308 (2002) • Linear-optical QND Kok, Lee & Dowling, PRA66, 063814 (2002) Gasparoni et al., PRL 93, 020504 (2004) Pittman et al., PRA 68, 032316(2004) Walther et al., Nature 434, 169 (2005) Internal ancillas • KLM 4-photon Knill, Laflamme, & Milburn, Nature409, 46 (2001) + teleportation + error correction = scalable QC • Simplified 4-photon Ralph, White, Munro, & Milburn, PRA65, 012314 (2001) • Efficient 4-photon Knill, PRA66, 052306 (2002) • Entangled ancilla 4-photon Pittman, Jacobs, and Franson, PRL88, 257902 (2002) • Entangled input 2-photon Pittman, Jacobs, and Franson, PRL88, 257902 (2002)

  22. Two Qubit Gate* DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND control qubit target qubit DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 CSIGN gate C C 0 0 C C 1 1 p phase shift T T 0 0 T T 1 1 *Ralph, Langford, Bell & White, PRA65, 062324 (2002)

  23. Two Qubit Gate* DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND control qubit target qubit DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 CSIGN gate -1/3 1/3 1/3 *Ralph, Langford, Bell & White, PRA65, 062324 (2002)

  24. Two Qubit Gate* DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND both transmitted both reflected DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 CSIGN gate -1/3 1/3 1/3 Non-deterministic CSIGN gate with probability 1/9 *Ralph, Langford, Bell & White, PRA65, 062324 (2002)

  25. DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND -1/3 Control out Control in 1/3 Target out Target in 1/3 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Interferometric Gate* CNOT gate *Ralph, Langford, Bell & White, PRA65, 062324 (2002)

  26. DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND Non-classical interference DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Interferometric Gate* C RH = 1/3 RV = 1 T *Ralph, Langford, Bell & White, PRA65, 062324 (2002)

  27. DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Beam-splitter Gate* C RH = 1/3 RV = 1 Non-classical interference T  No dual-path interferometers No adjustment if wrong splitting ratio Langford, Weinhold, Prevedel, Pryde, O’Brien, Gilchrist and White, PRL 95, 210504 (2005)

  28. DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Process Tomography of a Quantum Gate* Ideal Measured average gate fidelity: = 94 ± 2 % 1 gate works 90-95% of time; 2 gates should work 80-90% of time *Pryde, O’Brien, Gilchrist, James, Langford, Ralph, and White, PRL93 080502 (2004); Langford, et al., PRL95, 210504 (2005)

  29. 4. Factoring Experiment* DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND 4-photon source 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)

  30. DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND P000 = 27 ± 2% P100 = 24 ± 2% P010 = 23 ± 2% P110 = 27 ± 2% P00 = 52 ± 3% failure P10 = 48 ± 3% DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Measuring the output Order-finding algortihm uses mixed output state: non-deterministic order-2 order-4 add redundant bit then reverse argument bits F = 99.9 ± 0.3% SL= 99.9 ± 0.6% r=2 r=2 F = 98.5 ± 0.6% SL= 98.1 ± 0.8% r=6 GCD of Cr/2±1 and N GCD of 41±1 and 15 = 3,5 GCD of Cr/2±1 and N GCD of 41±1 and 15 = 3,5 GCD of 43±1 and 15 = 3,5 algorithm works near perfectly …?

  31. DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Measuring the output order-2 FGHZ = 59 ± 4% WGHZ = 9 ± 4% SL= 62 ± 4% joint state of argument and function registers is entangled & mixed independent photons F2Bell = 98.5 ± 0.6%= Tbd = 41 ± 5% Tce = 33 ± 5% SL= 98.1 ± 0.8% order-4 joint state of argument and function registers is highly entangled dependent photons

  32. 5. Where next: Period 3? xLogC[fN,C(x)] 00 11 22 30 4 1 5 2 xLogC[fN,C(x)] 00000 00101 01010 01100 100 01 101 10 i.e. period 3 i.e. period 3 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 N=21, C=4 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〉

  33. A bit less scary... flip qubit #2: ⏐〉= ( ⏐〉⊗ ⏐00〉+⏐〉⊗ ⏐0〉⏐〉⊗ ⏐0〉 where: ⏐〉⏐〉+⏐〉 ⏐〉⏐0〉+⏐0〉+⏐〉 ⏐〉⏐〉+⏐〉+⏐〉    √ √ √ 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?

  34. Period 3 Circuit X X QFT X X Ry() X H X X X R( H GHZ W W DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 • 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.

  35. Conclusions 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. • Complexity of quantum circuit depends on period r, rather than size of number. • Improving these results, step-by-step, is as good a route to practical quantum computers.

  36. BUT.... DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 • 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.76 3228 (1996).

  37. Quantum Factoring (cont.) • Assume that 2L=Mr (i.e. the size of the argument register is equal to a multiple of the unknown period, r): where: • Quantum Fourier Transform to argument register:

  38. DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND HH HV HD HR I I I X I Y I Z VH VV VD VR X I XX XY XZ DH DV DD DR Y I YX YY YZ RH RV RD RR Z I ZX ZY ZR |0+i1=|R |0= |H |0+1=|D DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Quantum Tomography* • State tomography: measure combinations of basis states 0 1 0 + 1 0 + i1 H VD R for 2 photon states, bi-photon Stokes parameters n qubit state requires 22n measurements • Process tomography: measure combinations of basis processes rotations on Poincare sphere I X Y Z n qubit gate requires 24n measurements • Reconstructed states and processes are unphysical: effect of uncertainties Maximum likelihood or Bayesian analysis required *James, Kwiat, Munro and White, PRA 64, 030302 (2001)

  39. DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Factoring Circuits* 4-photon source *Lanyon, Weinhold, Langford, Barbieri, James, Gilchrist, and White, PRL 99, 250505 (2007)

  40. DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Factoring Circuits* * order-2 order-2 order-4 *Lanyon, Weinhold, Langford, Barbieri, James, Gilchrist, and White, PRL 99, 250505 (2007)

  41. DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Factoring Circuits * order-2 order-4 order-2 order-4

  42. DEPARTMENT OF PHYSICS, UNIVERSITY OF QUEENSLAND Output state is: After logical measurement of argument register, function register is DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Circuit outputs 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 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 correlated

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