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Quantum computation and quantum information

Quantum computation and quantum information. Grover's algorithm. Complex numbers basics. Basics. Dirac (bra– ket ) notation ⟨φ|ψ ⟩ Introduced in 1939 by Paul Dirac Interpreted as the probability amplitude for the state ψ to collapse into the state φ. Basics.

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Quantum computation and quantum information

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  1. Quantum computation and quantum information Grover's algorithm Igor Ilijašević

  2. Igor Ilijašević

  3. Complex numbers basics Igor Ilijašević

  4. Basics • Dirac (bra–ket) notation ⟨φ|ψ⟩ • Introduced in 1939 by Paul Dirac • Interpreted as the probability amplitude for the state ψ to collapse into the state φ Igor Ilijašević

  5. Basics • - vectors must have same dimensions • - complex number not a vector - inner product Igor Ilijašević

  6. Basics • - Hermitian matrix (measurable) Igor Ilijašević

  7. Basics • - Identity (unit) matrix • - Unitary matrix • - Eigenvector (state of the system) ( - Eigenvalue (result (is a number))) Igor Ilijašević

  8. Basics • - orthonormal basis vector • Identity operator Igor Ilijašević

  9. Basics • Expressing a linear operator as a matrix • with respect to Igor Ilijašević

  10. Basics • Qubit • Has two states |0⟩ and |1⟩ - Computational basis states • Can also be in states other than |0⟩or |1⟩ • Can also form linear combinations of states – Superpositions • α and β are complex numbers • We can determine whether a qubit is in the state 0 or 1, but we cannot determine its quantum state (α and β) • We can get the result 0 with probability or 1 with probability , where + = 1 Igor Ilijašević

  11. More basics • Bloch sphere • Since + = 1 • A single qubit when measured gives us the following probabilities • The state is often denoted as • The state is often denoted as • What if we need more than 1 (qu)bit, say for example 2? • Classical bits 4 states: 00, 01, 10, 11 • Qubits 4 computational basis states: ,,, • But can also be in superpositions of these 4 states - amplitude • + + • Bell state – EPR pair Igor Ilijašević

  12. Quantum gates and circuit symbols Igor Ilijašević

  13. Quantum gates and circuit symbols Igor Ilijašević

  14. Quantum gates and circuit symbols Igor Ilijašević

  15. quantum gates • All quantum gates are reversible • Quantum gates can be easily represented using matrix form • Matrix must be unitary • U†U = I • That is the only constraint! • Quantum NOT gate acts linearly • Hadamard gate Igor Ilijašević

  16. Quantum gates Igor Ilijašević

  17. Quantum gates • There are as much single qubit gates as there are 2x2 unitary matrices • Arbitrary single qubit gate can be decomposed as a product of rotations • Rotation about the axis • Multi-quantum gates • CNOT Igor Ilijašević

  18. Quantum parallelism Igor Ilijašević

  19. Walsh–Hadamardtransform • Example: • Performing a function on bit input and 1 bit output • Prepare qubit states as • Apply the Hadamard transform to the first bits • Implement the quantum circuit for • As a result we get • Superposition over all states Igor Ilijašević

  20. Grover's algorithm • Quantum algorithm • Probabilistic • The probability of failure can be decreased by repeating the algorithm • Deutsch–Jozsa algorithm is a deterministic quantum algorithm • Searching an unsorted database with N entries in time using space • May be more accurate to describe it as "inverting a function" • “Only” a quadratic speedup compared to other quantum algorithms (exponential speedup) Igor Ilijašević

  21. Grover's algorithm • We have N entities • Database entries • We need an N-dimensional state space H which can be provided by qubits • Choose an observable, Ω, acting on H, with N distinct eigenvalues whose values are all known • Each of the eigenstates of Ω encode one of the entries in the database • We are provided with a unitary operator (quantum oracle) which acts as a subroutine that compares database entities • We need to identify the eigenstate or the eigenvalue that acts specially upon • Grover diffusion operator Igor Ilijašević

  22. Grover's algorithm • Perform the following "Grover iteration" r(N) times (asymptotically ) • Apply • Apply • Perform the measurement Ω which will give the result with probability approaching 1 for N≫1 • Get from Igor Ilijašević

  23. Grover's algorithm Igor Ilijašević

  24. Google Quantum Computing Playground Examples Igor Ilijašević

  25. Google Quantum Computing Playground Grover's algorithm implementation Igor Ilijašević

  26. References • “Quantum Computation and Quantum Information, 10th Anniversary Edition”, Michael A. Nielsen & Isaac L. Chuang • Google Quantum Computing Playground, http://qcplayground.withgoogle.com/ • http://twistedoakstudios.com/blog/Post2644_grovers-quantum-search-algorithm • http://en.wikipedia.org/wiki/Grover's_algorithm • http://www.quantiki.org/wiki/Main_Page • https://www.youtube.com/watch?v=T2DXrs0OpHU • https://www.youtube.com/watch?v=Xmq_FJd1oUQ Igor Ilijašević

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