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Quantum Computing
Jeff Goymerac
Christine Wang
•Timeline
•
•
•
•
•
•
1981 - Feynman
suggests quantum
computer model
1985 - David Deutsch
describes first quantum
Turing machine
1994 - Shor's algorithm
created
1996 - Grover’s
algorithm discovered
1998 - First 3 qubit NMR
computer
2000 - 5-qubit NMR
quantum computer built
•
•
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•
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2000 - 7-qubit NMR
quantum computer built
2001 - Shor's algorithm
executed on 7 qubit
computer
2005 - First qubyte created
2009 - Yale creates solidstate quantum processor
2011 - D-Wave announces
commercial quantum
computer
2011 – Record
computation of 3x5=15
Quantum Computer:
A computation device that makes
direct use of quantum-mechanical
phenomenon such as superposition
and entanglement to perform
operations on data
•Qubit (Quantum bits)
•
Unit of quantum information
100 qubits can store 2100 numbers
simultaneously 1,267,650,600,228,229,401,496,703,205,376
• Classical bits have to be in one
state or another while qubits can
be in a superposition of both states at the
same time
• Superposition is best described by
Schrödinger’s thought Experiment
(Schrödinger’s cat)
•
•Quantum Gates
Basic quantum circuit operating on a small
number of qubits
• Reversible
• Represented by
unitary matrices
•
•Quantum Circuit
Model for quantum computation
• Sequence of quantum circuits
• Input: qubits
• Output: measurement of some or all of the
qubits
•
•Quantum Algorithms
Implementation of quantum circuits
• Often non-deterministic
• Provide the correct solution only with a
certain known probability
• Use quantum superposition, quantum
entanglement
• Quantum entanglement: phenomenon when
the quantum state of each particle cannot
be described independently
•
•Quantum Algorithm Techniques
• Quantum
•
Fourier Transform
Shor’s Algorithm
• Amplitude
•
Amplification
Grover’s Algorithm
•Quantum Fourier Transform
• Linear
transformation on qubits
• The
quantum Fourier transform on N
1 N 1 
points is defined by:
j 
 k
2 ijk
N
N
k 0
The best quantum Fourier transform
algorithms known today require only
O(nlg n) gates to achieve an efficient
approximation
•Shor’s Algorithm
•
•
Used for factoring integer numbers
Consists of two parts
1. A reduction of the factoring problem to the
problem of order-finding, which can be
done on a classical computer.
2. A quantum algorithm to solve the orderfinding problem
•Shor’s Algorithm
•
1.
2.
3.
4.
5.
6.
7.
Classical Part
Pick a pseudo-random number a < N
Compute gcd(a, N). This may be done using the
Euclidean algorithm.
If gcd(a, N) ¹ 1, then there is a nontrivial factor of N, so
we are done.
Otherwise, use the period-finding subroutine to find r,
x
the period of the following function: f (x) = a mod N
i.e. the smallest integer r for which f (x + r) = f (x) .
If r is odd, go back to step 1.
If a r/2 º -1(mod N), go back to step 1.
r/2
The factors of N are gcd(a ±1, N). We are done.
•Shor’s Algorithm
• Quantum
1.
2.
Part
Start with a pair of input and output
qubit registers with lg N1 qubits each,
and initialize them to N å x 0 where x
x
runs from 0 to N .
Construct f (x) as a quantum function
and apply it to the above state, to
obtain 1 å x f (x)
N
x
•Shor’s Algorithm
Quantum Part
4. Apply the quantum Fourier transform on the
input register. This leaves us in the following
1
state: å å e2pixy/N y f (x)
N x y
5. Perform a measurement. We obtain some
outcome y in the input register and f (x0 ) in
the output register. Since f is periodic, the
probability 2to measure some y is given by
•
1
N
å
x:f ( x )= f ( x0 )
e2p ixy/N
1
=
N
2
åe p
2 i( x0 +rb)y/N
b
Analysis now shows that this probability is
yr
higher, the closer
is to an integer.
N
•Shor’s Algorithm
•
Quantum Part
y
6. Turn
into an irreducible fraction, and
N
7.
8.
9.
extract the denominator r ¢, which is a
candidate for r.
Check if f (x) = f (x + r ¢ ) . If so, we are done.
Otherwise, obtain more candidates for r
by using values near y, or multiples of r ¢. If
any candidate works, we are done.
Otherwise, go back to step 1 of the
subroutine.
•Shor’s Algorithm
• Exponentially
faster than the best known
classical algorithm for factoring
1.9(lg N )1/3 (loglog N )2/3
3
• O((lg N) ) vs O(e
)
• Implies
that public key cryptography
might be easily broken, given a
sufficiently large quantum computer
•Amplitude Amplification
•
Allows the amplification of a chosen
subspace of a quantum state
Generalizes the idea behind the Grover’s
search algorithm
• Discovered in 1997 by Gilles Brassard and
Peter Hoyer
• Independently rediscovered by Lov Grover in
1998
• Can be used to obtain a quadratic speedup
over several classical algorithms
•
•Grover’s Algorithm
Used for searching an unstructured database
or unordered list
1 N
x
å
1. Initialize the system to the state s =
N
•
x=1
2.
Perform the following “Grover iteration” r(N )times.
The function r(N ), which is asymptotically O(N 1/2 ) Is:
1.
2.
Apply the operator Uw = I - 2 w w
Apply the operator Us = 2 s s - I
•Grover’s Algorithm
3.
Perform the measurement W. The
measurement result will be lw with
probability approaching 1 for N >>1.
From lw , w may be obtained.
•Grover’s Algorithm
• Quadratically
faster than the best
possible classical algorithm for the same
task
1/2
• O(N ) vs O(N )
• Uses
O(lg N) storage space
QUANTUM COMPUTING
The final state must be
measured. This
collapses the quantum
state down to a
classical distribution
• Comparison based
quantum sorting
algorithms, take
W(nlgn) steps
•
CLASSICAL COMPUTING
•
•
Can read the final state
Comparison based
classical sorting
algorithms, take
W(nlgn) steps
• Given sufficient computational resources, a
classical computer could be made to
simulate any quantum algorithm.
•Questions?