Quantum Computing: Whats It Good For? Scott Aaronson Computer Science Department, UC Berkeley January 10, 2002 aaronson.

Quantum Computing:What’s It Good For?

Scott Aaronson

Computer Science Department, UC Berkeley

January 10, 2002

www.cs.berkeley.edu/~aaronson

Overview

1. History and background

2. The quantum computation model

3. Example: Simon’s algorithm

4. Other algorithms (Shor’s, Grover’s)

5. Limits of quantum computing, including recent work

6. The future

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.”

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.”

What Is Quantum Mechanics?

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)

Traditional Physics View Quantum Computing View

What Is Quantum Mechanics?

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

The Model (con’t)

• Dirac ket notation: We write state as, i.e.,

0.5 |00 - 0.5 |01 + 0.5i |10 - 0.5i |11

Superposition over basis states

• Normalization: If state is ii|i, then i|i|2 = 1

(Why complex numbers? Why |i|2 and not i2?)

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.

1 1 100 10 11

2 22

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

Example: Square Root of NOT

• Hadamard matrix: 1 1

2 21 1

2 2

H

H|0 = (|0+|1)/2 H|1 = (|0-|1)/2

H(|0+|1)/2 = |0 H(|0-|1)/2 = |1

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:

1. Apply order nk local unitaries for constant k

2. 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)

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

Simon’s ProblemGiven a black box

x f(x)

Promise: There exists a secret string s such that f(x)=f(y) y=xs for all x,y (: bitwise XOR)

Problem: Find s with as few queries as possible

ExampleInput x Output f(x)

000 4

001 2

010 3

011 1

100 2

101 4

110 1

111 3

Secret string s:

101

f(x)=f(xs)

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

Simon’s Algorithm (con’t)

1. Prepare uniform superposition

/ 2

0,1

1

2 nn

x

x

2. Compute f:

/ 20,1

1

2 nn

x

x f x

3. Measure |f(x), yielding

for some x

1

2x x s f x

Simon’s Algorithm (con’t)

1

2x x s

1 12 2

1 12 2

H

4. Apply to each bit of

Result:

/ 2

0,1

11 1

22 n

x y x s y

ny

y

1

mod 2n

i ii

x y x y

where

Simon’s Algorithm (con’t)

5. Measure. Obtain a random y such that

0.x y x s y s y

7. Solve for s. Can show solution is unique with high probability.

6. Repeat steps 1-5 order n times. Obtain a linear system over GF2: 1

2

0

0

s y

s y

Schematic DiagramObserve

f(x)

Observe

nH nH |0|0

|0

|0|0

|0

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

Example: r=5

0123456789

10

0 1 2 3 4 5 6 7 8 9 10 11

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 arb(mod N)

• Breaks widely-used public-key cryptosystems: RSA, Diffie-Hellman, ElGamal, elliptic

curve systems…

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

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.

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

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

The Future

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!

Implementation

Implementation• Key technical challenge: prevent decoherence, or unwanted interaction with environment

• Approaches: NMR, ion trap, quantum dot, Josephson junction, optical…

• Recent achievement: 15=35 (Chuang et al. 2001)• Larger computations will require quantum error-correcting codes

Quantum Computing: What’s It Good For?

• Potential (benign) applications

- Faster combinatorial search

- Simulating quantum systems

• Makes QM accessible to non-physicists

• ‘Spinoff’ in quantum optics, chemistry, etc.

• Surprising connections between physics and CS

• New insight into mysteries of the quantum