exposé quantum information and quantum algorithm.pptx

exposé quantum information and quantum algorithm.pptx

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  QUANTUM INFORMATION 1 Y. AADEL
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  2 Outline  Introduction  Qubit Quantum gate  No-cloning Theorem  Entanglement  Quantum Algorithm  Deutsch-Jozsa algorithm  Grover Algorithm
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  Size of thecomponents Number of the components Speed  Theoretical limitations reached in 2020  Appearance of quantum phenomena Take advantage of quantum effects Introduction 3
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  4 Introduction Richard Feynman was amongthe first to recognize the potential in quantum superposition for solving such problems much faster. Entanglement Superposition
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  5 1 0      1 2 2    Measure |0 > ||2 ||2 |1 > Qubit The elementary unit of quantum information is the qubit Normalization condition
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  6 Qubit Bloch sphere cos 0sin 1 2 2 i e       1 0      All points on the surface of the Bloch sphere correspond to a possible state of a qubit. However, the possible values of a classical bit values are restricted to two can be identified at both poles of the sphere. 2 { , }   
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  7 Quantum gates 1 00 0 0 1 0 0 0 0 0 1 0 0 1 0 CNOT              † † UU U U I   a b U c d        † a c U b d             Input = output  Reversible
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  8 No-cloning theorem it isimpossible to create an identical copy of an arbitrary unknown quantum state Quantum cryptography
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  9 Bell state Entanglement 1,2 12      Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently.
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  10 Entanglement & Teleportation    0 1 1 00 11 2 2 1 0 00 11 1 00 11 2                             2 1 0 1 1 0 00 01 0 1 1 2 11 0 10                      0 1      Teleportation of unknown state
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  Quantum circuits area collection of wires (qubits) and gates that depict the time evolution of a quantum algorithm. The qubits are prepared in a known state and introduced as inputs to the system. The qubits then undergo evolution depicted by the gate operations on them. The evolution ends when the system is subjected to a quantum measurement. 11 Quantum algorithm
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  12 Quantum algorithm
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  13 Hadamard Transform . . . u    . 0,1 1 1 2 n u x n x x    Special case   0,1 1 0 2 n n n H n x x        Quantum Parallelism
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  14 Quantum Parallelism Quantum FourierTransform QFT N QFT  2 i N e      2 0,1 1 n i xy QFT N y x e y N      2n N  Hadamard Transform is a special case of QFT
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  Most quantum algorithmsdeveloped to date are based on four general techniques:  QFT Quantum Fourier Transform: the Deutsch-Jozsa algorithm, Shor’s algorithm.  Amplitude amplification: Grover’s algorithm.  Quantum Simulation: approximating the Jones polynomial and solving linear equations.  Quantum Walk : • element distinctness (Ambainis) • NAND trees evaluation (Farhi, Goldstone, Gutmann) • Triangle finding (F.Magniez et al.) • Evaluating Boolean Formulas (Farhi et al.) • … 15 Quantum algorithm
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  The first algorithmto show that quantum computers are capable of solving certain computational problems much more efficiently than classical deterministic computers 16 The problem For N=2n we are given Function f which is one of two kinds, either f(x) is constant for all values of x or f(x) is balanced, that is, equal to 1 for exactly half of all the possible x, and 0 for the other half.       0,1 : 0,1 0,1 N N x f   The goal is to find out whether f(x) is balanced or constant Deutsch-Jozsa algorithm
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  17 Circuit Deutsch-Jozsa forone qubit     1 1 0 1 0 1 2            (0) (0) (1) 2 1 1 0 1 1 0 1 2 f f f                       (0) (1) (0) (1) 1 1 0 1 1 0 1 1 0 1 2 2 f f f f H           after measure the first qubit we conclure that f constant or balanced Deutsch-Jozsa algorithm
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  18 Deutsch-Jozsa Circuit forn qubit . ( ) 3 0 1 ( 1) 2 2 x z f x n z x z             With Deutsch-Jozsa algorithm we get the solution after one evaluation No application Exponential acceleration Classically we get the solution after 2n-1+1evaluations Deutsch-Jozsa algorithm
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  • Database searching: Findthe desired file indexed as “𝛽” among N files 19 Grover’s algorithm
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  • The problemaddressed by Grover’s algorithm can be viewed as trying to find a marked element in an unstructured database of size N. To solve this problem a classical algorithm need, on average O(N/2) evaluations and O(N) in the worst case. • But with using Grover’s algorithm, a quantum computer can realize the same task using only O( 𝑵) evaluations. 20 Grover’s algorithm
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  21 Grover’s algorithm Classically Quantumsearch   O N   O N
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  The protocol forthe Grover’s algorithm is described in Fig 1 for n qubits Grover’s search algorithm begins with the initialized state 0 n  The Hadamard transform is used to get an equal superposition state. 1 0 1 N x x N      The Grover operator 22 Grover’s algorithm
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  Oracle Hadamard Transform Phase shift operator 0 ( 1)x x x     23 Grover’s algorithm
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  Oracle :    ( ) : 0,1 0,1 n f x  , , ( ) x y x y f x   With f is a Boolean function   ( ) 0 1 0 1 1 2 2 f x O x x                  f U 24 Grover’s algorithm
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  Phase shift operator0 0 x x    0 x    2 0 0 2 0 0 2 n n n n n n H I H H H H H I              25 Grover’s algorithm
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  Geometric Visualization In fact,the Grover iteration can be viewed as a rotation in the two-dimensional space spanned by the starting state vector and the state consisting of a uniform superposition of solutions to the search problem.  1 0 1 N x x N      1 x N M     1 x M    N M M N N       26 Grover’s algorithm
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  Geometric Visualization After repeatedGrover iteration, the state vector gets close to  cos 2 N M N    Let cos sin 2 2          2 G I O     The Grover iteration The state after repeating the Grover iteration k times 2 1 2 1 cos sin 2 2 k k k G                      27 Grover’s algorithm
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  Algorithm & procedure 01 n  Initial state 2 1 0 0 1 1 2 2 n n x x            Apply the Hadamard Transform for initial state   2 1 0 0 0 1 1 2 2 2 0 1 2 n R n x I O x x                           Apply the Grover iteration R time 4 N R         0 x Measure the first n qubit 28 Grover’s algorithm
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  29 At roughly thesame time that Grover published his algorithm, Bennett, Bernstein, Brassard and Vazirani published a proof that no quantum solution to the problem can evaluate the function fewer than times, So Grover’s algorithm is asymptotically optimal. ( ) O N Grover’s algorithm
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  Special case 30 00 01 10 11 1 11 1 00 01 10 11 2 2 2 2        2 G I O     1 1 1 1 00 01 10 11 2 2 2 2 O        2 G I O         1 1 1 1 2 00 01 10 11 2 2 2 2 G I               11  N M M N N       Grover’s algorithm
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  Performance & drawback 21 2 1 cos sin 2 2 k k k G                      2 1 2 2 k     sin 2 M N         With Let 𝜆 = 𝑀/𝑁 and let CI(x) denote the integer closest to the real number x. Then repeating the Grover iteration arccos 2arcsin k CI            31 Grover’s algorithm
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  Performance & drawback Thesuccess probability curve of Grover’s algorithm The Grover’s algorithm is no longer useful when 𝜆 > 0.25 32 Grover’s algorithm
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  33 When the Grover’salgorithm is applied to search an unordered database, the probability of getting correct results usually decreases with the increase of marked items. Grover’s algorithm
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  34 Thank You