Quantum Computation and Quantum Information – Lecture 3

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

2. Motivation • Quantum computersare 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!

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. Z Y H 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:

10. X Y Z H Summary of Simple Gates

11. Reversibility Requirement • All quantum operations have to bereversible • 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. x x x x 0 y xÅy x A Qubit Cloning Circuit? • Using the XOR gate, it is possible to copy a classical bit: 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. H The Bell State Circuit x y

24. H The Bell State Circuit By Example ?

25. Quantum Teleportation Circuit M1 H M2 XM2 ZM1

26. Quantum Teleportation Circuit (2) M1 H M2 XM2 ZM1 (|10>+|01>)

27. Quantum Teleportation Circuit (3) M1 H M2 XM2 ZM1

28. Quantum Teleportation Circuit (4) M1 H M2 XM2 ZM1 00, 01, 10 or 11

29. Quantum Teleportation Circuit (5)

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

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

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

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

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

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