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
Boulder 2/03
F. Chong -- QC

2. 7 qubit Quantum Computer
( Vandersypen, Steffen, Breyta, Yannoni, Sherwood, and Chuang, )
Bulk spin NMR: nuclear spin qubits
Decoherence in 1 sec; operations at 1 KHz
Failure probability = 10-3 per operation
Potentially KHz = 10-6 per op
pentafluorobutadienyl cyclopentadienyldicarbonyliron complex
Boulder 2/03
F. Chong -- QC

3. Silicon Technology ( Kane, Nature 393, p133, 1998 ) Boulder 2/03
F. Chong -- QC

4. Quantum Operations Manipulate probability amplitudes
Must conserve energy
Must be reversible
Boulder 2/03
F. Chong -- QC

5. Bit Flip Flips probabilities for |0> and |1>
X Gate 1 a = b + a 1 X
Bit-flip, Not 1 b
Flips probabilities for |0> and |1>
Conservation of energy
Reversibility => unitary matrix
(* means complex conjugate)
Boulder 2/03
F. Chong -- QC

6. Controlled Not Control bit determines whether X operates1 a
Controlled Not a 00 + b 01 + 1 b
Controlled X = 1 c d 10 + c 11
CNot X 1 d
Control bit determines whether X operates
Control bit is affected by operation
Boulder 2/03
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
Boulder 2/03
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
Boulder 2/03
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
Boulder 2/03
F. Chong -- QC

10. Shor’s Algorithm [Shor94]
m qubits H
r r
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> + …
Boulder 2/03
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) !!!
Boulder 2/03
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
Boulder 2/03
F. Chong -- QC

13. Quantum Error Correction
X12 X23 Error Type Action
no error no action
bit 3 flipped flip bit 3
bit 1 flipped flip bit 1
bit 2 flipped flip bit 2
(3-qubit code)
Boulder 2/03
F. Chong -- QC

14. Syndrome Measurment X X X Y Y Y Y 12 ' 2 2 ' 1 1 Boulder 2/03X 12 Y Y ' 2 2 Y Y ' 1 1
Boulder 2/03
F. Chong -- QC

15. 3-bit Error Correction X X A X X X A X Y X Y Y X Y Y X Y Boulder 2/031 01 X X A X 12 Y X Y ' 2 2 Y X Y ' 1 1 Y X Y '
Boulder 2/03
F. Chong -- QC

16. Concatenated Codes
Boulder 2/03
F. Chong -- QC

17. Error Correction Overhead
7-qubit code [Steane96], applied recursively
Recursion Storage Operations Min. time
(k) (7k) ( 153k ) ( 5k ) ,
,581,
, ,981,
, ,841,135,
Boulder 2/03
F. Chong -- QC

18. Recursion Requirements
Shor’s
Grover’s
Boulder 2/03
F. Chong -- QC

19. Clustering Recursive scheme is overkill
Don’t error correct every operation
[Oskin,Chong,Chuang IEEE Computer 02]
Boulder 2/03
F. Chong -- QC

20. Memory Hierarchy [Copsey,Oskin,Chong,Chuang,Abdel-Gaffar NSC02]
Boulder 2/03
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
Boulder 2/03
F. Chong -- QC

22. Narrow control wires don’t work
5nm access points contain only a handful of quantum states at temp < 1K
20 nm
Boulder 2/03
F. Chong -- QC

23. Perhaps a solution?
As two physical dimensions of the access point exceed 100nm thousands of electron states are held.
Classically, these states are restricted to the access point, however, quantum mechanically they tunnel downward, guided by the via, thus enabling control.
Boulder 2/03
F. Chong -- QC

24. Classical meets Quantum
Pitch-matching problem!
5nm
Classical access points
100nm
100nm
100nm
Natural architecture: linear arrays of qubits
Narrow-tipped
control
100nm
Phosphorus Atoms
20nm
20nm
Boulder 2/03
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
Boulder 2/03
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
Boulder 2/03
F. Chong -- QC

27. How to get longer wires?? Use “Quantum Teleportation” Sender |a>
Receiver
1 qubit
2 classical bits
Boulder 2/03
F. Chong -- QC

28. CAT State Two bits are in “lockstep” Named for Shrodinger’s cat
both 0 or both 1
Named for Shrodinger’s cat
Also “EPR pair” for Einstein, Podolsky, Rosen
Boulder 2/03
F. Chong -- QC

29. Teleportation Circuit
source H
|a>
|b>
EPR
Pair
(CAT)
CNOT
target X Z
|c>
|a>
Source generates |bc> EPR pair
Pre-communicate |c> to target with retry
Classical communication to set value
Can be used to convert between codes!
Boulder 2/03
F. Chong -- QC