Introduction to Quantum Computation

Leila Taghavi

August 14, 2021 Postulates of Quantum Mechanics Postulate 1

• Hilbert space: a complex vector space with inner product (state space of the system) • State vector of a qubit: a unit vector in

• Computational basis

3 Examples

• Superposition:

4 Hilbert space of a composite system

Examples:

entanglement

5 Postulate 2

• Evolution of a closed system Unitary transformation:

Example:

6 Postulate 3

• Measurement in computational basis

collapse

Examples: collapse

collapse

7 Question?

• Distinguishing two quantum state

collapse

8 Quantum Circuit Model Classical circuits

• Universal set of gates:

– AND AND OR NOT

0 0 0 0 1 0 1 0 1 – OR 1 0 0 1 0 1 1 1 1

– NOT

10 Classical circuits (cont.)

• Proof of universality of AND, OR, NOT by induction:

Therefore

OR is redundant:

11 Classical circuits - Half adder

Half adder

12 Quantum gates • Single-qubit gates Classical NOT

13 Quantum gates (cont.) • More single-qubit gates

https://qiskit.org/textbook/ch-states/single-qubit-gates.html

14 Quantum gates (cont.) • Two-qubit gates

15 Quantum gates (cont.)

• Another two-qubit gate

16 Quantum gates (cont.)

• Three-qubit gate

17 Embedding of classical circuits into quantum ones

• Classical gates are not reversible!

• Simulating AND using Toffoli:

{X, Toffoli} is complete

18 No cloning theorem Fan-out

Half adder

≠

19 Universality of quantum gates

• Uncountably many operators (Quantum gates)! • A finite set of gates: approximates any unitary transformation by any arbitrary precision. • {CNOT, Hadamard, phase, Toffoli}

20 Simple Quantum Circuits Bell state generator

H

Bell basis

22 Bell state generator

23 Quantum Fourier transform (a special case)

In general:

24 Teleportation

https://qiskit.org/textbook/ch-algorithms/teleportation.html Teleportation

• Alice wants to send a qubit to Bob • are two complex numbers! • Can she do it by sending two classical bits?! • Experimentally verified, record distance 143 km 26 Teleportation

H

send send

X Z

27 Superdense Coding

https://qiskit.org/textbook/ch-algorithms/superdense-coding.html Superdense coding

• Can Alice encode 2 classical bits into a single qubit and send it to Bob?

29 Superdense coding

30 Superdense coding • Sending 2 bits using a single qubit

Encoding send

H

Decoding

31 Deutsch-Jozsa algorithm (1992)

https://qiskit.org/textbook/ch-algorithms/deutsch-jozsa.html The problem • Given a function • Decide: is constant or balanced

0 or 1 on all inputs 1 for half, 0 for the other half

Classically best case: queries worst case: queries

First quantum algorithm that performs better than the best classical algorithm

33 The algorithm

0 or 1 on all inputs 1 for half, 0 for the other half • Is constant or balanced?

1 if constant 0 if balanced

Phase kickback

34 Why does it work?

Changes in the state of the first qubit:

Constant:

Balanced:

35 Simon’s Algorithm

https://qiskit.org/textbook/ch-algorithms/simon.html The problem

• Given a function • Promise:

Classically best case: queries worst case:

37 Period finding algorithm of Simon (1994)

• •

H H

H H

•

38 About the algorithm

• Motivation for QFT based algorithms: Shor’s factoring algorithm • Shor’s algorithm can break RSA

39