Quantum AlgorithmsTowards quantum

codebreaking

Quantum AlgorithmsTowards quantum

codebreaking

Artur Ekert

More general oracles

: 0,1 0,1n

f Quantum oracles do not have to be of this form

xU

n qubits

m qubits

xx u x U u

x

u

e.g. generalized controlled-U operation

x

xU u

Phase estimation problem

xU

n qubits

m qubits

x

u 2 'i p xe u

x

Phase estimation algorithm Suppose p is an n-bit number:

Recall Quantum Fourier Transform:

1 2 0

22 0. ...2

1 1:

2 2

n n n

ipx i p p p x

n n nx x

F p e x e x

Phase estimation algorithm

u

0n qubits

m qubits

H

xU

STEP 1:

1 2 0

0,1 0,1

2 (0. ... )2 '

0,1 0,1

1 10

2 2

1 1

2 2

n n

n n

n n

x

n nx x

i p p p xi p x

n nx x

u x u x U u

e x u e x u

1 2 0

22 0. ...2

1 1:

2 2

n n n

ipx i p p p x

n n nx x

F p e x e x

Recall Quantum Fourier Transform:

Phase estimation algorithm

u

0n qubits

m qubits

H

xU

STEP 2: Apply the reverse of the Quantum Fourier Transform

Fny

u

0 1 2 1... np p p p

But what if p’ has more than n bits in its binary representation ?

Phase estimation algorithm 00

00

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

Pro

babi

lity

Phase estimation - solution

u

0n qubits

m qubits

H

xU

Fny

u

0 1 2 1... np p p p

Order-finding problem

PRELIMINARY DEFINITIONS:

1,2,3... 1 : , 1N x N GCD x N

This is a group under multiplication mod N

For example

21 1,2,4,5,8,10,11,13,16,17,19,20

Order-finding problem

PRELIMINARY DEFINITIONS:

0 1 2 3 4 5 6

for consider all powers

, , , , , , .

of

..

N

a a a a a a a

a a

For example21

0 1 2

10

10 mod 21 1,10 mod 21 10,10 mod 21 16,.

1,10,16,13,4,19,1,10,16,13,4,19,1,10,16...

..

a

(period 6)

N

21

ORD minimum 1 such that 1mod

e.g. ORD 10 6

ra r a N

Order-finding problem

Order finding and factoring have the same complexity. Any efficient algorithm for one is convertible into an efficient algorithm for the other.

Solving order-finding via phase estimation

xU

n qubits

m qubits

x

y modxa y N

x

Suppose we are given an oracle that multiplies y by the powers of a

1 11 2 2

0

mod ,r i k i

kr r

k

u e a N U u e u

Solving order-finding via phase estimation

1 2u u

0 H

xU

Fny Estimate of p1 with prob. ||2

Estimate of p2 with prob. ||2

Solving order-finding via phase estimation

Solving order-finding via phase estimation

Shor’s Factoring Algorithm

1

02n qubits

n qubits

H

xU

F2ny

m

r

Quantum factorization of an n bit integer N

Wacky ideas for the future

• Particle statistics in interferometers, additional selection rules ?

• Beyond sequential models – quantum annealing?

• Holonomic, geometric, and topological quantum computation?

• Discover (rather than invent) quantum computation in Nature?

Beyond sequential models

…Interacting spins

configurations

ene

rgy

0 0 01 1 1 1 1

011101…01

annealing

Adiabatic Annealing

Initial simple Hamiltonian

Final complicated Hamiltonian

Coherent quantum phenomena in

nature ?

Further Reading

http://cam.qubit.org

Centre forQuantumComputation

University of Cambridge, DAMTP