1 An Introduction to Quantum Computing Sabeen Faridi Ph 70 October 23, 2007

2 What is a Quantum Computer? Memory made up of quantum bits (“qubits”) Quantum mechanical phenomena used to perform operations on data: Superposition Entanglement

3 Qubits vs. Classical Bits

4 Qubits Two level quantum system Boolean states are represented by quantum states and The quantum states form a “computational basis” for Any pure state is represented by a superposition of the form

5 Quantum Logic Gates Analogous to classical logic gates Reversible Represented by unitary matrices

6 Hadamard Gate

7 Phase Shift Gate

8 Quantum Networks The Hadamard and phase shift gates can be combined to construct any unitary operation on a single qubit Example:

9 Quantum Registers In a system of two quantum bits the quantum states form a “computational basis for A state is represented by a superposition of the form

10 Quantum Registers vs. Classical Registers

11 Controlled NOT Gate CNOT is known as controlled NOT as the second (target) qubit flips if and only if the first (control) qubit is 1

12 CNOT Gate (continued)

13 Controlled Phase Shift Gate

14 Quantum Fourier Transform Discrete Fourier transform Unitary operation For an n-qubit system, the QFT is represented by

15 QFT (continued) For a one qubit system, the QFT is implemented using a Hadamard gate For a two qubit system, we need two Hadamard gates and a controlled phase shift gate Similarly for a three qubit and four qubit QFT

16 QFT (continued)

17 Applications: Shor’s Algorithm Used to factor integers Consists of two parts Reduce to an order-finding problem (classical) Use a quantum algorithm to find the order Probabilistic outcome Factors integer N in time

18 Shor’s Algorithm (contd) Can theoretically be used to “break” public-key cryptosystem RSA Seven qubit system used to successfully used to factor 15 into 5 and 3 in 2001

19 Applications: Grover’s Algorithm Used to search unsorted databases Searches N entries in time Probabilistic outcome

20 Problems & Practicality Issues Practical Requirements: Physically scalable Initializable qubits Quantum gates faster than decoherence time Universal gate set Easy to read