1. Quantum Computers The basics

2. Introduction 2/47 Dušan Gajević

3. Introduction Quantum computers use quantum-mechanical phenomena to represent and process data Quantum mechanics can be described with three basic postulates – The superposition principle - tells us what states are possible in a quantum system – The measurement principle - tells us how much information about the state we can access – Unitary evolution - tells us how quantum system is allowed to evolve from one state to another 3/47 Dušan Gajević

4. Atomic orbitals - an example of quantum mechanics Introduction Electrons, within an atom, exist in quantized energy levels (orbits) A hydrogen atom – only one electron Limiting the total energy…...limits the electron to k different levels This atom might be used to store a number between 0 and k-1 4/47 Dušan Gajević

5. The superposition principle 5/47 Dušan Gajević

6. The superposition principle states that if a quantum system can be in one of k states, it can also be placed in a linear superposition of these states with complex coefficients Ways to think about superposition – Electron cannot decide in which state it is – Electron is in more than one state simultaneously The superposition principle 6/47 Dušan Gajević

7. The superposition principle State of a system with k energy levels “pure” states amplitudes “ket psi” Bra-ket (Dirac) notation Reminder: 7/47 Dušan Gajević

8. The superposition principle A system with 3 energy levels – examples of valid states 8/47 Dušan Gajević

9. “Very interesting theory – it makes no sense at all” – Groucho Marx 9/47 Dušan Gajević

10. The measurement principle 10/47 Dušan Gajević

11. The measurement principle The measurement principle says that measurement on the k state system yields only one of at most k possible outcomes and alters the state to be exactly the outcome of the measurement 11/47 Dušan Gajević

12. The measurement principle It is said that quantum state collapses to a classical state as a result of the measurement 12/47 Dušan Gajević

13. The measurement principle If we try to measure this state... …the system will end up in this state… …and we will also get it as a result of the measurement The probability of a system collapsing to this state is given with 13/47 Dušan Gajević

14. The measurement principle This means: – We can tell the state we will read only with a certain probability – Repeating the measurement will always yield the same result we got this first time – Amplitudes are lost as soon as the measurement is made, so amplitudes cannot be measured 14/47 Dušan Gajević

15. The measurement principle Probability of a system collapsing to a state j is given with – One might ask, if amplitudes come down to probabilities when the state is measured, why use complex amplitudes in the first place? Answer to this will be given later, when we see how system is allowed to evolve from one state to another Does the equation appear more natural now? 15/47 Dušan Gajević

16. “God does not play dice” – Albert Einstein “Don’t tell God what to do” – Niels Bohr 16/47 Dušan Gajević

17. Qubit 17/47 Dušan Gajević

18. Isolating two individual levels in our hydrogen atom and the qubit (quantum bit) is born Qubit 18/47 Dušan Gajević

19. Qubit Qubit state The measurement collapses the qubit state to a classical bit 19/47 Dušan Gajević

20. Vector reprezentation 20/47 Dušan Gajević

21. Pure states of a qubit can be interpreted as orthonormal unit vectors in a 2 dimensional Hilbert space – Hilbert space – N dimensional complex vector space Vector representation Reminder: Another way to write a vector – as a column matrix 21/47 Dušan Gajević

22. Vector representation Column vectors (matrices) qubit statepure states a little bit of math Reminder: Adding matrices Reminder: Scalar multiplication 22/47 Dušan Gajević

23. Vector representation System with k energy levels represented as a vector in k dimensional Hilbert space system statepure states 23/47 Dušan Gajević

24. Entanglement 24/47 Dušan Gajević

25. Entanglement Let’s consider a system of two qubits – two hydrogen atoms, each with one electron and two "pure" states 25/47 Dušan Gajević

26. Entanglement By the superposition principle, the quantum state of these two atoms can be any linear combination of the four classical states – Vector representation Does this look familiar? 26/47 Dušan Gajević

27. Entanglement Let’s consider the separate states of two qubits, A and B – Interpreting qubits as vectors, their joint state can be calculated as their cross (tensor) product Reminder: Tensor product 27/47 Dušan Gajević

28. Entanglement – Cross product in Dirac notation is often written in a bit different manner The joint state of A and B in Dirac notation 28/47 Dušan Gajević

29. Entanglement It’s impossible! all four have to be non-zero at least one has to be zero Let’s try to decompose to separate states of two qubits 29/47 Dušan Gajević

30. Entanglement States like the one from the previous example are called entangled states and the displayed phenomenon is called the entanglement – When qubits are entangled, state of each qubit cannot be determined separately, they act as a single quantum system – What will happen if we try to measure only a single qubit of an entangled quantum system? 30/47 Dušan Gajević

31. Entanglement Let’s take a look at the same example once again amplitudesprobabilities 31/47 Dušan Gajević

32. Entanglement measuring the first qubit measuring the second qubit value of the first qubit value of the second qubit This remains true no matter how large the distance between qubits is! 32/47 Dušan Gajević

33. “Spooky action at a distance” – Albert Einstein 33/47 Dušan Gajević

34. Unitary evolution 34/47 Dušan Gajević

35. Unitary evolution means that transformation of the quantum system state does not change the state vector length – Geometrically, unitary transformation is a rigid body rotation of the Hilbert space Unitary evolution 35/47 Dušan Gajević

36. It comes down to mapping the old orthonormal basis states to new ones – These new states can be described as superpositions of the old ones Unitary evolution 36/47 Dušan Gajević

37. Unitary evolution Unitary transformation of a single qubit – Dirac notation – Matrix representation Replace the old basis states… …with new ones Multiply unitary matrix… …with the old state vector 37/47 Dušan Gajević

38. Unitary evolution Example of calculus using Dirac notation Qubit is in the state… …applying following (Hadamard) transformation… …results in the state 38/47 Dušan Gajević

39. Unitary evolution Example of calculus using matrix representation Qubit is in the state… …applying Hadamard transformation… Reminder: matrix multiplication …results in the state 39/47 Dušan Gajević

40. Unitary matrices Unitary matrices satisfy the condition Conjugate-transpose of U “U-dagger” Reminder: Conjugate-transpose matrix Reminder: Complex conjugate Inverse of U Reminder: Inverse matrix Identity matrix Reminder: 40/47 Dušan Gajević

41. Reversibility 41/47 Dušan Gajević

42. Reversibility Reversibility is an important property of unitary transformation as a function – knowing the output it is always possible to determine input – What makes an operation reversible? AND circuit NOT circuit INPUTOUTPUT ABA and B 000 010 100 111 INPUTOUTPUT A not A 01 10 output 1 0 A=1 B=1 input ? irreversible output 1 0 input A=0 A=1 reversible 42/47 Dušan Gajević

43. Reversibility – Reversible operation has to be one-to-one – different inputs have to give different outputs and vice-versa Consequently, reversible operations have the same number of inputs and outputs Are classical computers reversible? 43/47 Dušan Gajević

44. Reversibility – Similar to AND circuit applies to OR, NAND and NOR, the usual building blocks of classical computers Hence, in general, classical computers are not reversible 44/47 Dušan Gajević

45. Offtopic: Landauer’s principle Again, an irreversible operation – NAND circuit INPUTOUTPUT ABA nand B 001 011 101 110 Whenever output of NAND is 1 – input cannot be determined We say information is “erased” every time output of NAND is 1 45/47 Dušan Gajević

46. Offtopic: Landauer’s principle Landauer’s principle says that energy must be dissipated when information is erased, in the amount – Even if all other energy loss mechanisms are eliminated irreversible operations still dissipate energy Reversible operations do not erase any information when they are applied Absolute temperature Boltzman's constant 46/47 Dušan Gajević

47. References University of California, Berkeley, Qubits and Quantum Measurement and Entanglement, lecture notes, http://www-inst.eecs.berkeley.edu/~cs191/sp12/ http://www-inst.eecs.berkeley.edu/~cs191/sp12/ Michael A. Nielsen, Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2010. Colin P. Williams, Explorations in Quantum Computing, Springer, London, 2011. Samuel L. Braunstein, Quantum Computation Tutorial, electronic document University of York, York, UK Bernhard Ömer, A Procedural Formalism for Quantum Computing, electronic document, Technical University of Vienna, Vienna, Austria, 1998. Artur Ekert, Patrick Hayden, Hitoshi Inamori, Basic Concepts in Quantum Computation, electronic document, Centre for Quantum Computation, University of Oxford, Oxford, UK, 2008. Wikipedia, the free encyclopedia, 2014. 47/47 Dušan Gajević