Introduction to Quantum Computing

1. Introduction to Quantum Computing Lecture 1

2. OUTLINE • Why Quantum Computing? • What is Quantum Computing? • History • Quantum Weirdness • Quantum Properties • Quantum Computation

3. Why Quantum Computing?

4. Transistors per chip 109 ? 108 80786 Pentium Pro 107 80486 Pentium 106 80386 80286 105 8086 8080 104 4004 103 1970 1975 1980 1985 1990 1995 2000 2005 2010 Year Transistor Density

5. Electrons per device 104 (4M) (16M) (64M) 103 (256M) (Transistors per chip) (1G) 102 (4G) (16G) 101 ? 100 1 electron/transistor 10-1 1985 1990 1995 2000 2005 2010 2015 2020 Year Transistor Size

6. Why Quantum Computing? • By 2020 we will hit natural limits on the size of transistors • Max out on the number of transistors per chip • Reach the minimum size for transistors • Reach the limit of speed for devices • Eventually, all computing will be done using some sort of alternative structure • DNA • Cellular Automaton • Quantum

7. What is Quantum Computing?

8. Introduction • The common characteristic of any digital computer is that it stores bits • Bits represent the state of some physical system • Electronic computers use voltage levels to represent bits • Quantum systems possess properties that allow the encoding of bits as physical states • Direction of spin of an electron • The direction of polarization of a photon • The energy level of an excited atom

9. Spin up: Spin down: Spin States • An electron is always in one of two spin states • “spin up”– the spin is parallel to the particle axis • “spin down”– the spin is antiparallel to the particle axis • Notation:

10. 0 1 qubit • A qubit is a bit represented by a quantum system • By convention: • A qubit state 0 is the spin up state • A qubit state 1 is the spin down state

11. c0 + c1 0 1 1 0 |c0|2+|c1|2 = 1 Definitions • A qubit is governed by the laws of quantum physics • While a quantum system can be in one of a discrete set of states, it can also be in a blend of states called a superposition • That is a qubit can be in:

12. Measurement • If a qubit is realized by the spin of an electron, it is possible to measure the qubit value by passing the electron through a magnetic field • If the qubit encodes a |0> then it will be deflected upward • If the qubit encodes a |1> then it will be deflected downward

13. 2 2 Probability of Probability of 0 1 c0 c1 c0 + c1 1 0 Superposition Measurement • If the qubit is in a superposition state it cannot be determine if it will deflect up or down • However, the probability of each possible deflection can be found

14. Quantum Computing History

15. History • In the 1970’s Fredkin, Toffoli, Bennett and others began to look into the possibility of reversible computation to avoid power loss. • Since quantum mechanics is reversible, a possible link between computing and quantum devices was suggested • Some early work on quantum computation occurred in the 80’s • Benioff 1980,1982 explored a connection between quantum systems and a Turing machine • Feynman 1982, 1986 suggested that quantum systems could simulate reversible digital circuits • Deutsch 1985 defined a quantum level XOR mechanism

16. Existing Quantum Computers • liquid NMR quantum computers with 2 – 12 qubit registers. • Ion Trap method have achieved a single CONTROLLED NOT and 4 qubit entangled states • linear optics, • Superconductive Device…

17. Quantum Weirdness

18. Weird Measurement • One of the unusual features of Quantum Mechanics is the interaction between an event and its measurement • Measurement changes the state of a quantum system • Measurement of the superposition state of a qubit forces it into one of the qubit states in an unpredictable manner

19. Assumption Classical Quantum Comparison I • Compare qubits to classical bits A bit always has a definite value False, a qubit need not have a definite value until the moment after it is observed True A bit can only be 0 or 1 True False, a qubit can be in a superposition of 0 and 1 simultaneously A bit can be copied without affecting its value True False, a qubit in an unknown state cannot be copied without disrupting its state A bit can be read without affecting its value True False, reading a qubit that is initially in a superposition will change the value of the qubit

20. AssumptionClassical Quantum Comparison II Reading one bit has no effect on another unread bit False, if the qubit being read is entangled with another qubit reading one will affect the other True

21. Quantum Phenomena

22. Quantum Phenomena • There are five quantum phenomena that make quantum computing weird • Superposition • Interference • Entanglement • Non-determinism • Non-clonability

23. Superposition • The Principal of Superposition states if a quantum system can be measured to be in one of a number of states then it can also exist in a blend of all its states simultaneously • RESULT: An n-bit qubit register can be in all 2n states at once • Massively parallel operations

24. Interference • We see interference patterns when light shines through multiple slits • This is a quantumphenomena which isalso present in quantumcomputers • A quantum computercan operate on severalinputs at once, the results interfere with each otherproducing a collectiveresult

25. Entanglement • If two or more qubits are made to interact, they can emerge from the interaction in a joint quantum state which is different from any combination of the individual quantum states • RESULT: If two entangled qubits are separated by any distance and one of them is measured then the other, at the same instant, enters a predictable state

26. Non-Determinism • Quantum non-determinism refers to the condition of unpredictability • If a quantum system is in a superposition state and then measured, the measured state can not be predicted.

27. Non-Clonability • It is impossible to copy an unknown quantum state exactly • If you asked a friend to prepare a qubit in a superposition state without telling you which superposition state, then you could not make a perfect copy of the qubit • Useful in quantum cryptology

28. Quantum Computation

29. Quantum Computation Changes to a quantum state can be described using the language of quantum computation • Single Qubit Gates Classical Not Gate - Truth table Quantum Not Gate - Truth table

30. Quantum Computation Superposition of states? Not without further knowledge of the properties of quantum gates The quantum NOT gate acts LINEARLY… Linear behaviour is a general property of quantum mechanics Non-linear behaviour can lead to apparent paradoxes - Time Travel - Faster than light communication - Violates the 2nd Law of Thermodynamics

31. Quantum Computation NOT gate representation for any we get… to summarize…

32. Quantum Computation Are there any constraints on what matrices may be used as quantum gates? Of course! We require the normalization condition for and the result after the gate has acted The appropriate condition for this (of course) is that the matrix representing the gate is UNITARY That's it!!! Anything else is a valid quantum gate.

33. Quantum Computation Two more important gates… • Z gate • Hadamard Gate Note: Applying H twice to a state does nothing to it.

34. Quantum Computation Hadamard Gate: A most useful gate indeed!

35. X Z H Quantum Computation • Review: Important single-qubit gates

36. Quantum Computation • Arbitrary Single Qubit Quantum Gate - complete set from properties of a much smaller set Global Phase Factor Rotation about z Rotation Scaling Constant

37. NOT gate using NAND AND gate using NAND OR gate using NAND Quantum Computation • Classical Universal Gates (example) - The NAND gate is a classical Universal Gate. Why? • Universal Quantum Gates - An arbitrary quantum Computation on n qubits can be generated by a finite set of gates that are UNIVERSAL for quantum computation * Need to introduce some multiple quibit quantum gates

38. Multiple Qubit Gates • Controlled-NOT (CNOT) Gate - two input qubits: control and target - In General

39. CNOT quantum gate Any multiple qubit logic gate may be composed from CNOT and Single Qubit Gates

40. Other Computational Bases • Measurements - In terms of basis states - Generally any basis state can represent an arbitrary qubit state - If orthonormal then we can perform a measurement in keeping with probability interpretation

41. Quantum Circuits • Elements of a Quantum Circuit - each line in a circuit represents a "wire" * passage of time * photon moving from one location to another - assume the state input is a computational basis state - input is usually the state consisting of all s - no loops allowed ie: acyclic - No FANIN(not reversible therefore not Unitary) - FANOUT (can't copy a qubit)

42. x x Quantum Circuits • Quantum Qubit Swap Circuit

43. U Quantum Circuits • Controlled-U Gate - A Controlled-U Gate has one control qubit and n target qubits - where U is any unitary matrix acting on n qubits

44. Quantum Circuits • Measurement Operation - Converts a single qubit state into a probabilistic classical bit M

45. bit to be copied original bit scratch-pad initialized to zero copied bit Quantum Circuits • Can we make a Qubit Copying Circuit? - Copying a classical bit can be done with the Classical CNOT gate

46. Quantum Circuits • Can we make a Qubit Copying Circuit? - How about copying a qubit in an unknown state using a controlled-CNOT gate? bit to be copied Output State scratch-pad initialized to zero

47. Quantum Circuits • Can we make a Qubit Copying Circuit? - Does ? - Unless this does not copy the quantum state input - It is impossible to make a copy of the unknown quantum state - NO CLONING THEOREM -

48. H Quantum Circuits • Bell States, EPR States, EPR Pairs

49. Quantum Algorithms Final State Initial State Uf Target Register Data Register

50. Quantum Algorithms Eureka!!!! Both values of the function show up in the final state solution. This can be generalized to functions on arbitrary number of bits using the… HADAMARD TRANSFORM or WALSH-HADAMARD TRANSFORM