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For computer scientists

Quantum computation. For computer scientists. Prof. Dorit Aharonov School of computer science and engineering Hebrew university, Jerusalem, Israel. What is a computation. La Segrada Familia (Barcelona) Architect: Gaudi. What is a computation?. Computation (Algorithm). Input 0111001.

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For computer scientists

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  1. Quantum computation For computer scientists Prof. DoritAharonov School of computer science and engineering Hebrew university, Jerusalem, Israel

  2. What is a computation La SegradaFamilia (Barcelona) Architect: Gaudi

  3. What is a computation? Computation (Algorithm) Input 0111001 Output 011000 B C A A Universal computation models: ≈ Q Turing machine, 1936 Uniform Circuits ≈ ≈

  4. Game of life Rules: A living site: stays alive if it has 2 or 3 live neighbors Otherwise dies A dead site: comes to life if it has exctly 3 living neighbors

  5. The Extended Church-Turing thesis (ECTT) A corner stone thesis in computer science: The Extended Church Turing thesis: “Any physically realizable computational model can be simulated efficiently by a randomized Turing machine” ≈ ≈ Quantum computation is the only computational model which credibly challenges the Extended Church Turing thesis

  6. Bird’s view on Quantum computation Inherently different from standard “classical” computers. We believe that it will be exponentially more powerful for certain tasks. Polynomial time Quantum algorithm for factoring Simon [‘94] Bernstein Vazirani[‘93] Deutsch Josza [‘92] Shor[’94] 6 6 Physics of many particles (non universal computations) Philosophy of Science Algorithms Cryptography technology

  7. About this school Goal: Intro to quantum computation & complexity Some important notions, results, open questions Note: We will not cover many important things… (A partial list will be provided & updated ) Two remarks: 1) The lectures are intertwined, not independent! 2) The TA sessions are mainly exercises. Do them! We rely on them in the next lecture.

  8. Intro Lecture: Qubits Part 1: The principles of quantum Physics Part2: The qubit Part 3: Measurements Part 4: Dynamics Part 5: Two qubits

  9. Part I: The principles of Quantum Physics

  10. The two slit experimentPart A: bullets

  11. Two slit experimentsPart B: Water waves

  12. Two slit experimentPart C: electrons Interference pattern for particles!

  13. Explanation: superpositions and measurements The particle passes through both paths simultaneously! If measured, it collapses to one of the options 1) The superposition principle 2) Measurement gives one option & changes the state

  14. Part II: The Qubit

  15. + + 1st quantum principle: Superposition + b a A quantum particle can be in a Superposition of all its possible “classical” states a b a b

  16. a + b The elementary quantum information unit: Qubit = Quantum bit The qubit can be in either one of the States:0,1 As well as in any linear combination!  a vector in a 2 dim Hilbert space |0 On the board: Dirac notation Vector notation Transpose Inner products density matrix We can also talk about qudits, Of higher dim. 0 1 |1

  17. Part III: Measurements

  18. a + b The quantum measurement 0 1 |a|2 |b|2 0 1 When a quantum particle is measured the answer is Probabilistic The Superposition collapses to one of its possible classical states 2ndprinciple measurement Those (weird!) aspects have been tested in thousands of experiments

  19. a + b Projective measurements A projective measurement is described by a Hermitian matrix M. M has eigenvectors (eigenspaces) with associated real eigenvalues The classical outcome is eigenvalue 𝛌 with probability = norm squared of projection on the corresponsing eigenspace & the state collapses to this projection and renormlized 0 On the board: Measure with respect to Z Prob=inner product squared Measure X: The +/- basis Probablity for 𝛱 a projection Expected value of measurement: Direct expression and as Tr(Mρ) Uncertainty principle 0 1 1

  20. Part IV: Dynamics

  21. Dynamics Schrodinger’s equation: The Hamiltonian (A Hermitian operator) Discrete time evolution: On the board: From the differential equation to unitary evolution (eigenvalues which are primitive roots of unity) Unitary as preserving inner product

  22. Quantum Gates On the board: Applying X,Z on computational basis states of a qubit Linearity Applying Hadamard on basis states and measuring (“coin flip”)

  23. Interference & path integrals |0 |0 |0 H H On the board: compute weights, repeat with measurement in the middle |1 |1

  24. Part V: Two qubits

  25. one two The superposition principle for more qubits three The state of n quantum bits is a superposition of all 2npossible configurations, each with its own weight!

  26. The space of two qubits The computational Basis for the two qubits space The EPR state:

  27. 1st ex. of entanglement: The CHSH game They win if: > 0.85!Pr(success) with EPR

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