1. Quantum Computation and Quantum Information – Lecture 3
Part 1 of CS406 – Research Directions in Computing
Nick Papanikolaou

2. Motivation
Quantum computers are built from wires and logic gates, just as classical computers are
The potential of such devices stems from the ability to manipulate superpositions of states
Quantum algorithms solve problems which are not known to be solvable classically!

3. Lecture 3 Topics Quantum logic gates Simple quantum circuits
Quantum teleportation as a circuit
Deutsch’s quantum algorithm

4. Quantum vs. classical gates
The simplest boolean gate is NOT, with truth table:
Quantum gates have to be defined not only on the equivalents of 0 and 1, but on their superpositions too! in
out 1

5. Quantum NOT gate: Linearity
Suppose we define a quantum NOT gate as follows:
The action of the quantum NOT gate on a superposition must then be:
All quantum operations are linear

6. The NOT Gate as a Matrix
Because all quantum operations have to be linear, we can represent the action of a quantum gate by a matrix
The quantum NOT, or Pauli-X gate, is written:

7. Quantum State Vectors
Remember that a quantum state is represented by a vector
Notation:

8. Quantum NOT
We can express the NOT operation on a general qubit as matrix multiplication:

9. Other Single Qubit Gates
The Pauli-X gate works on only one qubit
Other common single qubit gates are:
Pauli-Z gate:
Pauli-Y gate:
Hadamard gate: Z Y H

10. Summary of Simple GatesX Y Z H

11. Reversibility Requirement
All quantum operations have to be reversible
Boolean operations are not necessarily so
A reversible operation is always given by a unitary matrix, i.e. one for which:

12. The Controlled NOT Gate
The CNOT gate is the standard two-qubit quantum gate
It is defined like this:

13. The Controlled NOT Gate (2)
CNOT is a generalisation of the classical XOR:
The CNOT gate is drawn like this:
“control qubit”
“target qubit”

14. The Controlled NOT Gate (3)
The matrix corresponding to the CNOT gate is:
The CNOT together with the single qubit gates are universal for quantum computing

15. Quantum Circuits
Using the conventions for control and target qubits, we can build interesting circuits
Example: A Qubit Swap Circuit

16. Qubit Swap Circuit

17. Features of Quantum Circuits
No loops are allowed; quantum circuits are acyclic
Fan-in is not allowed:
Fan-out is not allowed:

18. Generalised Control Gate
Any quantum gate U can be converted into a controlled gate:
One control qubit
n target qubits U
If the control qubit is “high,” U is applied to the targets. CNOT is the Controlled-X gate!

19. Quantum Measurement Measurement in a quantum circuit is drawn as: M
(classical bit representing outcome of measurement)
M = 0 with prob. or
M = 1 with prob. If
then:

20. A Qubit Cloning Circuit?
Using the XOR gate, it is possible to copy a classical bit: x y
xÅy
Can we build a quantum circuit that performs does this with qubits?

21. A Qubit Cloning Circuit? (2)
OK here
entangled!!

22. A Qubit Cloning Circuit? (3)
It is impossible to clone a qubit!
Also note that
unwanted terms!

23. The Bell State Circuit x y
Output x H y

24. The Bell State Circuit By Example?

25. Quantum Teleportation CircuitH M2
XM2
ZM1

26. Quantum Teleportation Circuit (2)H M2
XM2
ZM1

27. Quantum Teleportation Circuit (3)H M2
XM2
ZM1

28. Quantum Teleportation Circuit (4)H M2
XM2
ZM1
00, 01, 10 or 11

29. Quantum Teleportation Circuit (5)
If Alice obtains
Then Bob’s qubit is in state
So Bob applies gate
obtaining 00 I 01 X 10 Z 11
Y = ZX

30. What have we achieved?
The teleportation process makes it possible to “reproduce” a qubit in a different location
But the original qubit is destroyed!
Next topic: Quantum Parallelism and Deutsch’s quantum algorithm

31. Quantum Parallelism
Quantum parallelism is that feature of quantum computers which makes it possible to evaluate a function f(x) on many different values of x simultaneously
We will look at an example of quantum parallelism now – how to compute f(0) and f(1) for some function f all in one go!

32. Quantum Circuits for Boolean Functions
It is known that, for any boolean function
it is possible to construct a quantum circuit Uf that computes it
Specifically, to each binary function f corresponds a quantum circuit:
binary addition

33. Quantum Circuits for Boolean Functions (2)
What can this circuit Uf do? Example: x y
yÅf(x)

34. Quantum Circuits for Boolean Functions (3)
But what if the input is a superposition? x y
yÅf(x)
amazing! we’ve computed f(0) and f(1) at the same time!

35. Quantum Parallelism Summary
So, a superposition of inputs will give a superposition of outputs!
We can perform many computations simultaneously
This is what makes famous quantum algorithms, such as Shor’s algorithm for factoring, or Grover’s algorithm for searching
Simple q. algorithm: Deutsch’s algorithm

36. Deutsch’s Algorithm David Deutsch: famous British physicist
Deutsch’s algorithm allows us to compute, in only one step, the value of
To do this classically, you would have to:
compute f(0)
compute f(1)
add the two results
Remember:

37. Circuit for Deutsch’s Algorithmx y
yÅf(x) H H H

38. Circuit for Deutsch’s Algorithm (2)x y
yÅf(x) H H H

39. Circuit for Deutsch’s Algorithm (3)x y
yÅf(x) H H H
...and so we have computed

40. End of Lecture 3
Congratulations! If you are still awake, you have learned something about:
quantum gates (X, Y, Z, H, CNOT)
quantum circuits (swapping, no-cloning problem)
teleportation
quantum parallelism
and Deutsch’s algorithm