Quantum ComputingJeff GoymeracChristine Wang

• 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

• 2000 - 7-qubit NMR quantum computer built

• 2001 - Shor's algorithm executed on 7 qubit computer

• 2005 - First qubyte created

• 2009 - Yale creates solid-state quantum processor

• 2011 - D-Wave announces commercial quantum computer

• 2011 – Record computation of 3x5=15

•Timeline

A computation device that makes direct use of quantum-mechanical phenomenon such as superposition and entanglement to perform operations on data

Quantum Computer:

•Qubit (Quantum bits)

• 100 qubits can store 2100 numberssimultaneously• Classical bits have to be in onestate or another while qubits canbe in a superposition of both states at the same time• Superposition is best described by Schrödinger’s thought Experiment

(Schrödinger’s cat)

• Unit of quantum information

1,267,650,600,228,229,401,496,703,205,376

• Basic quantum circuit operating on a small number of qubits• Reversible• Represented byunitary matrices

•Quantum Gates

• Model for quantum computation• Sequence of quantum circuits• Input: qubits• Output: measurement of some or all of the qubits

•Quantum Circuit

• 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 Algorithms

•Quantum Fourier Transform• Shor’s Algorithm

•Amplitude Amplification• Grover’s Algorithm

•Quantum Algorithm Techniques

•The quantum Fourier transform on N points is defined by:

The best quantum Fourier transform algorithms known today require only

gates to achieve an efficient approximation

• Linear transformation on qubits

•Quantum Fourier Transform

21

0

1 ijkN

N

k

j kN

• Consists of two parts1. 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 order-finding problem

• Used for factoring integer numbers•Shor’s Algorithm

1. Pick a pseudo-random number 2. Compute . This may be done using the

Euclidean algorithm.3. If , then there is a nontrivial factor of

N, so we are done.4. Otherwise, use the period-finding subroutine to

find , the period of the following function:i.e. the smallest integer for which .

5. If is odd, go back to step 1.6. If , go back to step 1.7. The factors of are . We are done.

• Classical Part•Shor’s Algorithm

1. Start with a pair of input and output qubit registers with qubits each, and initialize them to where runs from to .

2. Construct as a quantum function and apply it to the above state, to obtain

•Quantum Part•Shor’s Algorithm

4. Apply the quantum Fourier transform on the input register. This leaves us in the following state:

5. Perform a measurement. We obtain some outcome in the input register and in the output register. Since is periodic, the probability to measure some is given by

Analysis now shows that this probability is higher, the closer is to an integer.

• Quantum Part•Shor’s Algorithm

6. Turn into an irreducible fraction, and extract the denominator , which is a candidate for .

7. Check if . If so, we are done.8. Otherwise, obtain more candidates for r

by using values near , or multiples of . If any candidate works, we are done.

9. Otherwise, go back to step 1 of the subroutine.

• Quantum Part•Shor’s Algorithm

•Exponentially faster than the best known classical algorithm for factoring• vs

• Implies that public key cryptography might be easily broken, given a sufficiently large quantum computer

•Shor’s Algorithm

• 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

• Allows the amplification of a chosen subspace of a quantum state

•Amplitude Amplification

1. Initialize the system to the state2. Perform the following “Grover iteration”

times. The function , which is asymptotically Is:

1. Apply the operator2. Apply the operator

• Used for searching an unstructured databaseor unordered list

•Grover’s Algorithm

3. Perform the measurement . The measurement result will be with probability approaching 1 for . From , may be obtained.

•Grover’s Algorithm

•Quadratically faster than the best possible classical algorithm for the same task• vs

•Uses storage space

•Grover’s Algorithm

QUANTUM COMPUTING• The final state must be measured. This collapses the quantum state down to a classical distribution

• Comparison based quantum sorting algorithms, take

steps

• Can read the final state

• Comparison based classical sorting algorithms, take

steps

CLASSICAL COMPUTING

• Given sufficient computational resources, a classical computer could be made to simulate any quantum algorithm.

•Questions?