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Quantum Cryptography

Quantum Cryptography. 1e29 = 29996224275833 x 3475385758524527 (LOCK) (MESSAGE/CODE) (KEY) Classical computer doing 20 GFLOPS needs 10 15 /20x10 9 ~ 0.5 day to solve this Largest prime has 22,338,618 digits  4000 years !.

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Quantum Cryptography

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  1. Quantum Cryptography 1e29 = 29996224275833 x 3475385758524527 (LOCK) (MESSAGE/CODE) (KEY) Classical computer doing 20 GFLOPS needs 1015/20x109 ~ 0.5 day to solve this Largest prime has 22,338,618 digits  4000 years!

  2. Classical Computation

  3. Inverter/NOT gate in out out in NOT 0 1 1 0

  4. 2-input Logic Gates 0 0 0 0 0 0 0 0 1 1 1 0 AND OR 1 1 0 0 0 1 1 1 1 1 1 1

  5. 2-input Logic Gates 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 0 NAND NOR XOR 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1 0 0 0 SUM

  6. NAND is universal A A A B NAND NOT AND = NOT(NAND) OR made of NAND and NOT A + B = NOT[NOT(A+B)] = NOT[NOT(A) and NOT(B)]

  7. NAND is universal A.B A A.B B NAND NOT A A A.B = A + B NOT B B NOT

  8. 2 Bit Half-Adder A 0 0 0 CARRY = AND B 0 1 0 SUM = XOR 1 0 0 A B CARRY SUM 1 1 1 0 0 1 1 1 1 0 2

  9. Full-Adder A Sum S B Carry Cout Cin S = A + B + Cin Cout = A.B + B.Cin + Cin.A

  10. Encoder: n inputs, 2n outputs (Identifies each A-B possibility) Enable D0 A D1 B D2 D0 = A NOR B = NOT(A) AND NOT(B) D1 = NOT(A) AND B D2 = A AND NOT(B) D3 = A AND B D3

  11. Decoder: 2n inputs, n outputs Collapses possibilities Enable D0 A D1 B D2 A = D2 + D3 B = D1 + D3 D3

  12. 2x1 and 4x 1 Multiplexer (Selector) S1 S0 Control S A A B F F C B D S=0, F=A S=1, F=B F = S.A + S.B (S0,S1) = (0,0), F=A (0,1), F=B (1,0), F=C (1,1), F=D F = S0.S1.A + S0.S1.B + S0.S1.C + S0.S1.D

  13. 8 x 1 Multiplexer DEMUX S1 S0 S2 S1 S0 S2 D0 D1 D0 D2 D1 D3 D2 F D4 D3 F D5 D4 D6 D5 D7 D6 D7

  14. 8 x 1 DEMUX

  15. STORAGE/MEMORY (FLIP-FLOPS) Need feedback S Q R Q

  16. How do we realize these gates?

  17. Build a “NOT” with CMOS inverters PMOS NMOS • voltage turns • PMOS on, shorts • to power supply voltage + voltage turns NMOS on, shorts to ground ON Gain OFF

  18. Build a NAND with CMOS inverters 0 0 1 0 1 1 NAND 1 0 1 1 1 0

  19. Build a NOR with CMOS inverters 0 0 1 0 1 0 NOR 1 0 0 1 1 0

  20. Build an oscillator with CMOS inverters

  21. Build memory with CMOS (DRAM, SRAM, Flash)

  22. Energetics of Computation

  23. Dissipation in CMOS PMOS Pdiss = IOFFVDD + aCVDD2f NMOS • Leakage high with scaling IOFF • Reduce VDD Steep devices • Reduce C/ION ratio  FinFETs • F2D = 0.5√L • F3D = 0.5√(L/Nlayers)

  24. Dissipation in binary switching “0” Barrier large enough to prevent spontaneous backflip “1”

  25. Dissipation when electron falls downhill eVAB A VAB B N els: (N+2)kBTln2 We’re already pretty efficient for on-chip dissipation! (Zhirnov) 3kBTln2 Perr = e-qDV/kT < ½  qDV > kBTln2 kBTln2

  26. Reversible Switching (“Moonwalking potential”) When we know where electron sits SLIDE When we don’t know where electron sits OOPS !! KNOWLEDGE CONNECTED TO DISSIPATION !

  27. Information entropy DF = -TDS • S = kBlnW • = <kBln(1/p) > • = -SikB(pilnpi)

  28. Dissipation in CMOS DF = -TDS S = -SikB(pilnpi) When we know where electron sits (read and copy) Si = 0 Sf = 0 When we don’t know where electron sits Sf = 0 Si = -kBln1/2 = kBln2

  29. Dissipation in AND AND 0 0 0 DF = -TDS 0 1 0 S = -SikB(pilnpi) 1 0 0 1 1 1 Si = -4kB(1/4ln1/4) Sf = -kB(3/4ln3/4 + 1/4ln1/4) Squeezing of phase space (2 inputs, 1 output)

  30. No Dissipation (CNOT) 0 0 0 A A A B Aout Bout B Flip B iff A = 1 0 1 0 0 1 0 1 1 Preserve phase space Bout is XOR ! 1 1 1 1 0 SUM

  31. CNOT is reversible ! Aout A A Bout B B

  32. CSWAP (Fredkin gate) A A B Flip B and C if A = 1 C A B Aout Bout C Cout 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 0 1 1 1 1 1

  33. CCNOT (Toffoli gate) A Flip A iff B = 1 and C = 1 B B A B Aout Bout C Cout C C 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 Set C = 1 Get A XOR B ! (Same as SUM bit) 0 1 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 1 1 1

  34. Quantum Computation

  35. PRIME FACTORIZATION Use Superposition + Entanglement + reversible gates for one-shot computation • Shor’s Algorithm • Grover’s Algorithm • Kane’s Nuclear spin QC

  36. What is Entanglement ?

  37. EPR Paradox: Pion decay  two entangled states in a singlet (e and e+ with opposite spins) |00> + |11>  Move far far away Measure bit 1  0 Bit 2 must collapse to 0 instantly !! (Nonlocality since light needs finite time)

  38. Is entanglement real? Could uncertainty accommodate a hidden variable? Is uncertainty just in knowledge or fundamental ??

  39. Bell’s Inequality |00> + |11>  Separate particles Run each through a rotation Uq1 x Uq2 Uq1= Uq2 = U+(p/2-q2) cosq1 -sinq1 sinq1 cosq1 1 0 |0> = |1> = 0 1

  40. Result -sin(q1-q2) [|00> + |11>]  Coincidence + cos(q1-q2) [ |10> - |01>]  Non-coinc P(AC = 1) = sin2(q1-q2) P(AC = -1) = cos2(q1-q2) P(AC = -1) – P(AC = 1) = P(01) + P(10) – P(00) – P(11) = correlation = cos[2(q1-q2)]

  41. Result (Alain Aspect etc)

  42. Bell

  43. Proof: Bell’s inequality Own vs Rent: O and O Car vs No Car: C and C TV vs No TV: T and T OCT + OCT + OCT + OCT + OCT + OCT + OCT + OCT = 1 If only two questions on a survey – coalesce info thus O O T T C C C C O O T T ? ? ? ? ? ? ? ? ? ? ? ? Jay Sulzberger

  44. Proof: Bell’s inequality Test for accuracy: OCT + OCT + OCT + OCT + OCT + OCT + OCT + OCT = 1 TO + TO + OC + OC + CT + CT ≤ 2 Proof: O O T T C C C C O O T T ? ? ? ? ? ? TOC + TOC + TOC + TOC + OCT + OCT + OCT + OCT + CTO + CTO + CTO + CTO = 2 [ 1 – OCT – OCT] ≤ 2 LHS: ? ? ? ? ? ?

  45. Bell

  46. P ( A & [not B] ) + P ( B & [not C] ) ≥ P ( A & [not C] ) TO + TO + OC + OC + CT + CT ≤ 2 http://arxiv.org/pdf/0704.2529v2.pdf

  47. Entanglement is real Electrons are in superposition A classical measurement decouples them by entangling with an apparatus. Decoherence kills phase information

  48. Entanglement is real How can we use this for QC ?

  49. Prime Factorization

  50. Useful for Quantum Cryptography 1e29 = 29996224275833 x 3475385758524527 (LOCK) (MESSAGE/CODE) (KEY) Classical computer doing 20 GFLOPS needs 1015/20x109 ~ 0.5 day to solve this Largest prime has 22,338,618 digits  4000 years! Create superposition of all states related to key as input Entangle with gateable output Measure output  input collapses (don’t peek yet !) Use gates to do Fourier Transform on input  input further collapses Now Read input From input + output reads, we can get factor !!!

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