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Quantum Algorithms Towards quantum codebreaking. Artur Ekert. More general oracles. Quantum oracles do not have to be of this form. e.g. generalized controlled-U operation. n qubits. m qubits. Phase estimation problem. n qubits. m qubits. Phase estimation algorithm. - PowerPoint PPT Presentation
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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