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Quantum Computers

Quantum Computers. The basics. Introduction. 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

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Quantum Computers

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  1. Quantum Computers The basics

  2. Introduction Dušan Gajević

  3. Introduction • Quantum computers use quantum-mechanical phenomenato represent and process data • Quantum mechanicscan be described with three basic postulates • The superposition principle- tells uswhat states are possible in a quantum system • The measurement principle - tells ushow much information about the state we can access • Unitary evolution - tells ushow quantum system is allowed to evolve from one state to another Dušan Gajević

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

  5. The superposition principle Dušan Gajević

  6. The superposition principle • The superposition principle statesthat 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 Dušan Gajević

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

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

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

  10. The measurement principle Dušan Gajević

  11. The measurement principle • The measurement principle saysthat measurement on the k state systemyields only one of at most k possible outcomesand alters the stateto be exactly the outcome of the measurement Dušan Gajević

  12. The measurement principle • It is saidthat quantum state collapses to a classical stateas a result of the measurement Dušan Gajević

  13. The measurement principle If we try to measurethis state... …the system will end up inthis state… …and we will also get itas a result of the measurement The probability of a system collapsing to this state is given with Dušan Gajević

  14. The measurement principle • This means: • We can tell the state we will readonly with a certain probability • Repeating the measurementwill 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 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 evolvefrom one state to another Does the equation appear more natural now? Dušan Gajević

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

  17. Qubit Dušan Gajević

  18. Qubit • Isolating two individual levels in our hydrogen atomand the qubit(quantum bit) is born Dušan Gajević

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

  20. Vector reprezentation Dušan Gajević

  21. Vector representation • Pure states of a qubitcan be interpreted as orthonormal unit vectorsin a 2 dimensional Hilbert space • Hilbert space– N dimensional complex vector space Reminder: Another way to write a vector – as a column matrix Dušan Gajević

  22. Vector representation • Column vectors (matrices) qubit state pure states a little bit of math Reminder: Scalar multiplication Reminder:Adding matrices Dušan Gajević

  23. Vector representation • System with k energy levelsrepresented as a vector in k dimensional Hilbert space system state pure states Dušan Gajević

  24. Entanglement Dušan Gajević

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

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

  28. Entanglement • Cross product in Dirac notationis often written in a bit different manner • The joint state of A and B in Dirac notation Dušan Gajević

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

  30. Entanglement • States like the one from the previous exampleare called entangled statesand 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 happenif we try to measure only a single qubitof an entangled quantum system? Dušan Gajević

  31. Entanglement Let’s take a look at the same example once again amplitudes probabilities Dušan Gajević

  32. Entanglement measuring the first qubit value of the first qubit measuring the second qubit value of the second qubit This remains trueno matter how large the distance between qubits is! Dušan Gajević

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

  34. Unitary evolution Dušan Gajević

  35. Unitary evolution • Unitary evolution meansthat transformation of the quantum system statedoes not change the state vector length • Geometrically,unitary transformation is a rigid body rotation of the Hilbert space Dušan Gajević

  36. Unitary evolution • It comes downto mappingthe old orthonormal basis states to new ones • These new statescan be described as superpositions of the old ones 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 Dušan Gajević

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

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

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

  41. Reversibility Dušan Gajević

  42. Reversibility • Reversibility is an important propertyof unitary transformation as a function –knowing the output it is always possible to determine input • What makes an operation reversible? • AND circuit • NOT circuit output input 1 A=1 B=1 ? 0 irreversible output input 1 A=0 0 A=1 reversible Dušan Gajević

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

  44. Reversibility • Similar to AND circuitapplies to OR, NAND and NOR,the usual building blocks of classical computers • Hence, in general,classical computers are not reversible Dušan Gajević

  45. Offtopic: Landauer’s principle • Again, an irreversible operation • NAND circuit We say information is “erased” every time output of NAND is 1 Whenever output of NAND is 1 – input cannot be determined Dušan Gajević

  46. Offtopic: Landauer’s principle • Landauer’s principle saysthat energy must be dissipated when information is erased,in the amount • Even if all other energy loss mechanisms are eliminatedirreversible operations still dissipate energy • Reversible operationsdo not erase any information when they are applied Boltzman's constant Absolute temperature Dušan Gajević

  47. References • University of California, Berkeley,Qubits and Quantum Measurement and Entanglement, lecture notes,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 documentUniversity 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. Dušan Gajević

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