Quantum Bits (qubit)

1. Quantum Bits (qubit) • 1 qubit probabilistically represents 2 states |a> = C0|0> + C1|1> • Every additional qubit doubles # states |ab> = C00|00> + C01|01> + C10|10> +C11|11> • Quantum parallelism on an exponential number of states • But measurement collapses qubits to single classical values F. Chong -- QC

2. 7 qubit Quantum Computer ( Vandersypen, Steffen, Breyta, Yannoni, Sherwood, and Chuang, 2001 ) • Bulk spin NMR: nuclear spin qubits • Decoherence in 1 sec; operations at 1 KHz • Failure probability = 10-3 per operation • Potentially 100 sec @ 10 KHz = 10-6 per op • pentafluorobutadienyl cyclopentadienyldicarbonyliron complex F. Chong -- QC

3. SiliconTechnology ( Kane, Nature 393, p133, 1998 ) F. Chong -- QC

4. Quantum Operations • Manipulate probability amplitudes • Must conserve energy • Must be reversible F. Chong -- QC

5. Bit Flip X Gate 0 1 a = b 0 + a 1 X Bit-flip, Not 1 0 b • Flips probabilities for |0> and |1> • Conservation of energy • Reversibility => unitary matrix (* means complex conjugate) F. Chong -- QC

6. Controlled Not 1 0 0 0 a Controlled Not a 00 + b 01 + • Control bit determines whether X operates • Control bit is affected by operation 0 1 0 0 b Controlled X = 0 0 0 1 c d 10 + c 11 CNot X 0 0 1 0 d F. Chong -- QC

7. Universal Quantum Operations H Gate H Hadamard T Gate T Z Gate Z Phase-flip Controlled Not Controlled X CNot X F. Chong -- QC

8. Quantum Algorithms • Search (function evaluation) • Factorization (FFT, discrete log) • Adiabatic optimization (quantum simulated annealing) • Estimating Gauss sums, Pell’s equation • Testing matrix multiply • Key distribution • Digital signatures • Clock synchronization F. Chong -- QC

9. Quantum Factorization • For N = pq, where p,q are large primes, find p,q given N • Let r = Order(x,N), which is min value > 0 such that xr mod N = 1, x coprime N • Then (xr/2+/- 1) mod N = p,q • eg Order(2, 15) = 4 (x4/2+/- 1) mod 15 = 3,5 F. Chong -- QC

10. Shor’s Algorithm [Shor94] m qubits H r r 0 xr mod N FT s/r |0> n qubits j |0> • j=1: |r> = |0> + |4> + |8> + |12> + … • j=2: |r> = |1> + |5> + |9> + |13> + … • j=4: |r> = |2> + |6> + |10> + |14> + … • j=8: |r> = |3> + |7> + |11> + |15> + … F. Chong -- QC

11. Quantum Fourier Transform • r is in the period, but how to measure r? • QFT takes period r => period s/r • Measurement yields I*s/r for some I • Reduce fraction I*s/r => r is the denominator with high probability! • Repeat algorithm if pq not equal N • O(n3) instead of O(2n) !!! F. Chong -- QC

12. Error Correction is Crucial • Need continuous error correction • can operate on encoded data [Shor96, Steane96, Gottesman99] • Threshold Theorem [Ahanorov 97] • failure rate of 10-4 per op can be tolerated • Practical error rates are 10-6 to 10-9 F. Chong -- QC

13. Quantum Error Correction X12 X23 Error Type Action +1 +1 no error no action +1 -1 bit 3 flipped flip bit 3 -1 +1 bit 1 flipped flip bit 1 -1 -1 bit 2 flipped flip bit 2 (3-qubit code) F. Chong -- QC

14. X X Syndrome Measurment 0 X 12 ' Y Y 2 2 ' Y Y 1 1 F. Chong -- QC

15. X X X X 3-bit Error Correction A X 1 01 A X 0 12 ' Y Y X 2 2 ' Y Y X 1 1 ' Y Y X 0 0 F. Chong -- QC

16. Concatenated Codes F. Chong -- QC

17. Error Correction Overhead • 7-qubit code [Steane96], applied recursively Recursion Storage Operations Min. time (k) (7k) (153k ) (5k ) 0 1 1 1 1 7 153 5 2 49 23,409 25 3 343 3,581,577 125 4 2,401 547,981,281 625 5 16,807 83,841,135,993 3125 F. Chong -- QC

18. Recursion Requirements Shor’s Grover’s F. Chong -- QC

19. Clustering • Recursive scheme is overkill • Don’t error correct every operation [Oskin,Chong,Chuang IEEE Computer 02] F. Chong -- QC

20. Memory Hierarchy • [Copsey,Oskin,Chong,Chuang,Abdel-Gaffar NSC02] F. Chong -- QC

21. Fundamental Constraint: Quantum gates require classical control lines! ( Yablonovitch, 1999 ) ( Nakamura, Nature 398, p. 786, ‘99 ) ( Marcus 1997 ) • Quantum: 20 nm • Classical: 100’s of nm F. Chong -- QC

22. 20 nm Narrow control wires don’t work 5nm access points contain only a handful of quantum statesat temp < 1K F. Chong -- QC

23. Perhaps a solution? As two physical dimensions ofthe access point exceed 100nmthousands of electron states are held. Classically, thesestates are restrictedto the access point,however, quantummechanically theytunnel downward,guided by the via,thus enabling control. F. Chong -- QC

24. Classical meets Quantum 5nm • Pitch-matching problem! Classical access points 100nm 100nm 100nm • Natural architecture: linear arrays of qubits Narrow-tipped control 100nm Phosphorus Atoms 20nm 20nm F. Chong -- QC

25. What about wires?A short quantum wire • Short wire constructed from swap gates • Each step requires 3 CNOT ops (swap) • Key difference from classical: • qubits are stationary F. Chong -- QC

26. Why short wires are short • How long before error correction? • Limited by decoherence • Threshold theorem => distance • Theoretically  70 um (really 100X less) • Very coarse bounds so far F. Chong -- QC

27. 1 qubit Sender |a> Receiver |a> 2 classical bits How to get longer wires?? • Use “Quantum Teleportation” F. Chong -- QC

28. H CAT State • Two bits are in “lockstep” • both 0 or both 1 • Named for Shrodinger’s cat • Also “EPR pair” for Einstein, Podolsky, Rosen F. Chong -- QC

29. Teleportation Circuit source H |a> |b> • Source generates |bc> EPR pair • Pre-communicate |c> to target with retry • Classical communication to set value • Can be used to convert between codes! EPR Pair (CAT) CNOT target X Z |c> |a> F. Chong -- QC