1. Quantum Computing: What’s It Good For?
Scott Aaronson
Computer Science Department, UC Berkeley
January 10, 2002

3. Overview History and background The quantum computation model
Example: Simon’s algorithm
Other algorithms (Shor’s, Grover’s)
Limits of quantum computing, including recent work
The future

4. Richard Feynman (1981):
“...trying to find a computer simulation of physics, seems to me to be an excellent program to follow out...and I'm not happy with all the analyses that go with just the classical theory, because nature isn’t classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem because it doesn't look so easy.”

5. David Deutsch (1985):
“Computing machines resembling the universal quantum computer could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine … Complexity theory for [such machines] deserves further investigation.”

6. What Is Quantum Mechanics?

7. What Is Quantum Mechanics?
Traditional Physics View
Quantum Computing View
Framework for atomic-scale physical theories
Computational model with amplitudes instead of probabilities
Complicated (lots of integral signs)
Simple
Pessimistic (i.e. Heisenberg uncertainty relation)
Optimistic (i.e. Shor’s factoring algorithm)

8. The Model Computer has n bits of memory
Classical case: if n=2, possible states are 00, 01, 10, 11
Randomized case: States are vectors of 2n probabilities in [0,1]
i.e. Pr[00]=0.2 Pr[01]=0.2 Pr[10]=0.1 Pr[11]=0.5
Quantum case: States are vectors of 2n complex numbers called amplitudes

9. The Model (con’t) Dirac ket notation: We write state as, i.e.,
0.5 |00 |01 + 0.5i |10 - 0.5i |11
Superposition over basis states
Normalization: If state is ii|i, then i|i|2 = 1
(Why complex numbers? Why |i|2 and not i2?)

10. Measurement
When we measure state, see basis state |i with probability |i|2
Furthermore, state collapses to |i
Can also make partial measurements
Example: Measuring 1st bit of
yields |00 with ½ prob., (|10+|11)/2 with ½ prob.

11. Time Evolution Matrix U is unitary iff UU†=I, † conjugate transpose
Equivalently: U preserves norm
Can multiply amplitude vector by some unitary U (i.e. replace state | by U|)
Quantum analogue of Markov transitions

12. Example: Square Root of NOT
Hadamard matrix:
H|0 = (|0+|1)/2 H|1 = (|0-|1)/2
H(|0+|1)/2 = |0 H(|0-|1)/2 = |1

13. Quantum Circuits
Unitary operation is local if it applies to only a constant number of bits (qubits)
Given a yes/no problem of size n:
Apply order nk local unitaries for constant k
Measure first bit, return ‘yes’ iff it’s 1
BQP: class of problems solvable by such a circuit with error probability at most 1/3
(+ technical requirement: uniformity)

14. The Power of Quantum Computing
Bernstein-Vazirani 1993:
BPP  BQP  PSPACE
BPP: solvable classically with order nk time
PSPACE: solvable with order nk memory
Apparent power of quantum computing comes from interference
Probabilities always nonnegative
But amplitudes can be negative (or complex), so paths leading to wrong answers can cancel each other out

15. Simon’s Problem f(x) x Given a black box
Promise: There exists a secret string s such that f(x)=f(y)  y=xs for all x,y (: bitwise XOR)
Problem: Find s with as few queries as possible

16. Example Input x Output f(x) 000 4 001 2 010 3 011 1 100 101 110 111
Secret string s:
101
f(x)=f(xs)

17. Simon’s Algorithm Classically, order 2n/2 queries needed to find s
- Even with randomness
Simon (1993) gave quantum algorithm using only order n queries
Assumption: given |x, can compute |x|f(x) efficiently

18. Simon’s Algorithm (con’t)
1. Prepare uniform superposition
2. Compute f:
3. Measure |f(x), yielding
for some x

19. Simon’s Algorithm (con’t)
4. Apply to each bit of
Result:
where

20. Simon’s Algorithm (con’t)
5. Measure. Obtain a random y such that
6. Repeat steps 1-5 order n times. Obtain a linear system over GF2:
7. Solve for s. Can show solution is unique with high probability.

21. Schematic Diagram f(x) |0 |0 |0 |0 |0 |0 O b s e r v e O b s e r

22. Period Finding Given: Function f from {1…2n} to {1…2n}
Promise: There exists a secret integer r such that f(x)=f(y)  r | x-y for all x
Problem: Find r with as few queries as possible
Classically, order 2n/3 queries to f needed
Inspired by Simon, Shor (1994) gave quantum algorithm using order poly(n) queries

23. Example: r=5

24. Factoring and Discrete Log
Using period-finding, can factor integers in polynomial time (Miller 1976)
Also discrete log: given a,b,N, find r such that arb(mod N)
Breaks widely-used public-key cryptosystems: RSA, Diffie-Hellman, ElGamal, elliptic curve systems…

25. Grover’s Algorithm Unsorted database of n items
Goal: Find one “marked” item
Classically, order n queries to database needed
Grover 1996: Quantum algorithm using order n queries

26. Limits of Quantum Computing
Bennett et al. 1996: Grover’s algorithm is optimal
(Quantum search requires order n queries)
Beals et al. 1998: For all total Boolean functions f: {0,1}n{0,1},
if quantum algorithm to evaluate f uses T queries,
exists classical algorithm using order T6 queries.

27. Collision Problem Given: a function f: {1,…,n}{1,…,N}, n even
Promise: f is either 1-1 (i.e. 3,7,9,2) or 2-1 (5,2,2,5)
Problem: Decide which
Models graph isomorphism, breaking cryptographic hash functions
Classical algorithm needs order n queries to f
Brassard et al. 1997: Quantum algorithm using n1/3 queries

28. Collision Lower Bound
Can a quantum algorithm do better than n1/3? Previously couldn’t even rule out constant number of queries!
A 2001: Any quantum algorithm for collision needs order n1/5 queries
Shi 2001: Improved to order n1/3

29. The Future

30. The Future
When processor components reach atomic scale, Moore’s Law breaks down
Quantum effects become important whether we want them or not
But huge obstacles to building a practical quantum computer!

31. Implementation

32. Implementation
Key technical challenge: prevent decoherence, or unwanted interaction with environment
Approaches: NMR, ion trap, quantum dot, Josephson junction, optical…
Recent achievement: 15=35 (Chuang et al. 2001)
Larger computations will require quantum error- correcting codes

33. Quantum Computing: What’s It Good For?
Potential (benign) applications
Faster combinatorial search
Simulating quantum systems
‘Spinoff’ in quantum optics, chemistry, etc.
Makes QM accessible to non-physicists
Surprising connections between physics and CS
New insight into mysteries of the quantum