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Quantum Search Algorithm and its Implementation using QisKit
- 1. https://dl.acm.org/doi/10.1145/237814.237866 Quantum Search Algorithmand its Implementation using QisKit To search in unstructured data with O( 𝐍) iterations Presented By: Zainab Abo-Hashima Grover's Algorithm : Grover, Lov K. "A fast quantum mechanical algorithm for database search." Proceedings of the twenty-eighth annual ACM symposium on Theory of computing. 1996.
- 2. 1 Geometric Interpretation Circuit of2-qubits Grover's algorithm Recap 2 3 5 Required Gates Grover's Search Algorithm 2 4 Agenda ZAS | 10/21/2022 | Implementation of 2-qubits Grover's algorithm
- 3. Grover's Search Algorithm 1-Recap: Unstructured Search , Quantum Oracle, and Amplitude Amplification 3 ZAS | 10/21/2022 |
- 4. Grover's Search Algorithm •Search problem: finding an element in an unstructured database • Suppose the search space consists of N elements • Find W minimum number of queries? • Classically, in the average case, we would have to evaluate all O(N/2) • In the worst case, we would have to evaluate all O(N) 4 10 55 …. w …. 3 99 0 1 2 3 4 5 6 N-1 Grover’s Algorithm: The Unstructured Search ZAS | 10/21/2022 | 4
- 5. Grover's Search Algorithm •In 1996, the computer scientist, Lov Grover, found a quantum algorithm for finding an item in an unstructured database of N items. Given : Assume we are given a function 𝑓(𝑥): {0,1}𝑛 → {0,1} Output : Find a target element (W) Grover’s algorithm claims to solve the problem with high probability in O( 𝑁) time. 4 10 55 ….. w ….. 3 99 0 0 0 0 1 0 0 0 Grover’s Algorithm: Overview ZAS | 10/21/2022 | 5
- 6. In simple terms,Grover’s algorithm consists of two parts: 1. Quantum Oracle 2. Amplitude Amplification ZAS | 10/21/2022 | 6
- 7. Grover's Search Algorithm 𝑓(𝑥)= 1, if 𝑥 = 𝑤 0, otherwise Grover’s Algorithm: Quantum Oracle Function • This oracle (black box) just flips the amplitude of the target element. • If the result is one, then it turns it negative direction • if the result is zero, then it turns it to positive direction (unchanged) • 𝑊ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑜𝑟𝑎𝑐𝑙𝑒 𝑖𝑠 |𝑥⟩ = (−1)𝑓(𝑥)|𝑥⟩ that satisfies: ZAS | 10/21/2022 | 7
- 8. Grover's Search Algorithm U X 𝑓(𝑥)= 1, if 𝑥 = 𝑤 0, otherwise q ⨁ f(x) X q Grover’s Algorithm: Quantum Oracle Function • This oracle (black box) just flips the amplitude of the target element. • If the result is one, then it turns it negative direction • if the result is zero, then it turns it to positive direction (unchanged) • 𝑊ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑜𝑟𝑎𝑐𝑙𝑒 𝑖𝑠 U|x〉|q〉 → |x〉|q ⨁ f(x)〉 that satisfies: ZAS | 10/21/2022 | 8
- 9. Grover's Search Algorithm Applythe oracle function (𝑈) 𝑈|x〉|q〉 → |x〉|q ⨁ f(x)〉 , |q〉 = 𝟏 𝟐 (∣ 𝟎⟩ −∣ 𝟏⟩ 𝑈 |x〉 |q〉 → |x〉| ( 1 2 (∣ 0⟩ −∣ 1⟩) ⨁ f(x)〉 = |x〉 ( 1 2 (∣ 0⟩⨁ f(x)⟩− ∣ 1⨁ f(x)⟩) =|x〉 ( 1 2 (∣ 0⨁ 1〉 −∣ 1⨁ 1⟩) =|x〉 ( 1 2 (∣ 1〉 −∣ 0⟩) = -|w〉 ( 1 2 (∣ 0〉 −∣ 1〉) 𝑖𝑓 𝑓 𝑥 = 1 𝑖𝑓 𝑓 𝑥 = 0 U|x〉 |q〉 → |x〉| ( 1 2 (∣ 0⟩ −∣ 1⟩) ⨁ f(x)〉 =|x〉 ( 1 2 (∣ 0⨁ f(x)〉 −∣ 1⨁ f(x)〉) =|x〉 ( 1 2 (∣ 0⨁ 0〉 −∣ 1⨁ 0〉) =|x〉 ( 1 2 (∣ 0〉 −∣ 1〉) Now, we can rewrite as U|𝑥⟩ = (−1)𝑓(𝑥)|𝑥⟩ Grover’s Algorithm: Quantum Oracle Function ZAS | 10/21/2022 | 9 Input Output X Y X ⊗ Y 0 0 0 0 1 1 1 0 1 1 1 0
- 10. Grover's Search Algorithm •Amplitude Amplification calculates the mean probability amplitude μ of all states and inverts the probability amplitudes around this mean. • Increases the amplitude of the target element, and decreases the amplitude of non target vector through iterations • Grover diffusion operator (D): Increasing the probability • Inversion about average • Rotating the vector Ψ towards 𝒘 (which is the solution). 𝐷 = 2|Ψ⟩⟨Ψ| − 𝐼 Grover’s Algorithm: Amplitude Amplification ZAS | 10/21/2022 | 10
- 11. Grover's Search Algorithm Grover’sAlgorithm: Idea 11 ZAS | 10/21/2022 | Example If number of items =4 Find the item |𝟏⟩ = 𝟎𝟏⟩ ? 00 01 10 11 00 01 10 11 00 01 10 11
- 12. Grover's Search Algorithm Theprocedure of Grover’s algorithm is as follows: Creating a uniform superposition Apply the oracle function (𝑈) Apply amplitude amplification ( Grover diffuser operator) (D) Repeat step U and D 𝜋 4 𝑵 iterations Apply the measurement 2 3 4 5 Grover’s Algorithm: Steps 12 ZAS | 10/21/2022 |
- 13. Grover's Search Algorithm 2-Geometric Interpretation: One Solution, Multiple Solutions, and Number of iterations 13 ZAS | 10/21/2022 |
- 14. Grover's Search Algorithm 𝛃 𝛂 Ψ= 1 𝑁 𝑥=0 𝑁−1 |x⟩ , 𝛂 = 1 𝑁 − 1 𝑥≠𝑤 |x⟩ 𝛃 = 𝒘 • Generate a rotation in a two-dimensional plane • A reflection around the axis 𝛂 • A reflection around the axis Ψ • θ is the angle between 𝛂 and Ψ Grover’s Algorithm: Geometric Interpretation Ψ 𝜃 U Ψ 𝜃 𝐷𝑈 Ψ 2𝜃 14 ZAS | 10/21/2022 |
- 15. Grover's Search Algorithm 𝛃 𝛂 Grover’sAlgorithm: Geometric Interpretation Ψ 𝜃 U Ψ 𝜃 𝐷𝑈 Ψ 2𝜃 15 ZAS | 10/21/2022 | 00 01 10 11 𝜳 00 01 10 11 𝑼 𝜳 00 01 10 11 𝐷𝑈 Ψ
- 16. Grover's Search Algorithm •One solution • |Ψ⟩ = 𝑁−1 𝑁 |𝛂⟩ + 1 𝑁 𝛃 • Ψ = cos 𝜃 |𝛂⟩ + sin 𝜃 𝛃 • Number of iterations 𝜋 4 𝑵 Grover’s Algorithm: Geometric Interpretation Ψ 𝛃 𝛂 𝑈 Ψ 𝜃 𝐷𝑈 Ψ 2𝜃 16 ZAS | 10/21/2022 | 𝜃
- 17. Grover's Search Algorithm •Multiple Solutions |Ψ⟩ = 𝑁 − 𝑀 𝑁 |𝛂⟩ + 𝑀 𝑁 𝛃 Ψ = cos 𝜃 |𝛂⟩ + sin 𝜃 𝛃 • Assume 𝐺 = 𝐷𝑈 • The first application of 𝐺 on the state 𝜓 𝐺 Ψ = cos 3𝜃 |𝛂⟩ + sin 3𝜃 𝛃 • The 𝑡th application of 𝐺 or the state |𝜓⟩, gives: • 𝐺t Ψ = cos(2𝑡 + 1)𝜃 |𝛂⟩ + sin(2𝑡 + 1)𝜃 𝛃 • Number of iterations 𝜋 4 𝑁 𝑀 Grover’s Algorithm: Geometric Interpretation Ψ 𝜃 𝛃 𝛂 𝑈 Ψ 𝜃 𝐷𝑈 Ψ 2𝜃 17 ZAS | 10/21/2022 |
- 18. Grover's Search Algorithm •Number of iterations • To maximize the probability of obtaining 𝛃 • Assume that θ= 1 𝑁 • sin((2t + 1)θ) ≈ 1 • (2t+ 1)θ ≈ 𝜋 𝟐 • 2t+1= 𝜋 𝟐θ • t= 𝜋 4θ - 1 𝟐 • t= 𝜋 4 𝑵 − 1 𝟐 = 𝜋 4 𝑵 Grover’s Algorithm: Geometric Interpretation Ψ 𝜃 𝛃 𝛂 𝑈 Ψ 𝜃 𝐷𝑈 Ψ 2𝜃 18 ZAS | 10/21/2022 |
- 19. Grover's Search Algorithm 3-Required Gates: Hadamard, X, Z , and CZ Gates 19 ZAS | 10/21/2022 |
- 20. Grover's Search Algorithm Grover’sAlgorithm: Hadamard Gate 20 ZAS | 10/21/2022 | Hadamard Gate Create an equal superposition of the two basis states Matrix Symbol IBM Quantum Symbol Input Output 𝟏 𝟐 𝟏 𝟏 𝟏 −𝟏 H H ∣ 0⟩ 1 2 (∣ 0⟩ +∣ 1⟩ H ∣ 1⟩ 1 2 (∣ 0⟩ −∣ 1⟩
- 21. Grover's Search Algorithm Grover’sAlgorithm: Z Gate 21 ZAS | 10/21/2022 | Z Gate Make π-rotation for quantum bit around the Z-axis Matrix Symbol IBM Quantum Symbol Input Output Z= 1 0 0 −1 Z Z ∣ 0⟩ ∣ 0⟩ Z ∣ 1⟩ −∣ 1⟩
- 22. Grover's Search Algorithm Grover’sAlgorithm: X Gate 22 ZAS | 10/21/2022 | X Gate Change quantum bit from state to another (Not gate) Matrix Symbol IBM Quantum Symbol Input Output 0 1 1 0 X X ∣ 0⟩ ∣ 1⟩ X ∣ 1⟩ ∣ 0⟩
- 23. Grover's Search Algorithm Grover’sAlgorithm: Identity Gate 23 ZAS | 10/21/2022 | Identity Gate Mapping each state to itself. Matrix Symbol IBM Quantum Symbol Input Output 1 0 0 1 𝑰 𝑰 ∣ 0⟩ ∣ 0⟩ 𝑰 ∣ 1⟩ ∣ 1⟩
- 24. Grover's Search Algorithm Grover’sAlgorithm: Controlled-Z Gate 24 ZAS | 10/21/2022 | Controlled-Z (CZ) Gate leaves the control qubit unchanged and performs a Z gate on the target qubit when the control qubit is in state |1⟩ Matrix Symbol IBM Quantum Symbol Input Output 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −1 CZ CZ ∣ 00⟩ ∣ 00⟩ CZ∣ 01⟩ ∣ 01⟩ CZ∣ 10⟩ ∣ 10⟩ CZ∣ 11⟩ -∣ 11⟩
- 25. Grover's Search Algorithm 4-Implementation of 2-qubits Grover’s Algorithm Using Qiskit 25 ZAS | 10/21/2022 |
- 26. Grover's Search Algorithm Grover’sAlgorithm: Implementation of 2-qubits 26 ZAS | 10/21/2022 | Number of items=N=4 Ψ = |00⟩ + |01⟩ + 10⟩ + |11⟩ Target item= w = 1 |01⟩ Number of qubits=n 2 Number of iterations=t 𝜋 4 𝑵= 𝜋 4 𝟒=1
- 27. Grover's Search Algorithm Theprocedure of Grover’s algorithm is as follows: Creating a uniform superposition Apply the oracle function (𝑈) Apply amplitude amplification ( Grover diffuser operator) (D) Repeat step U and D 𝜋 4 𝑵 iterations Apply the measurement 2 3 4 5 Grover’s Algorithm: Steps 27 ZAS | 10/21/2022 |
- 28. Grover's Search Algorithm Grover’sAlgorithm: Implementation of 2-qubits 28 ZAS | 10/21/2022 | Step1 : Creating a uniform superposition by applying hadamard gate |Ψ⟩ = 𝐻⊕𝒏 𝐻⊕𝟐 = 𝐻 ⊕ 𝐻 𝟏 𝟐 𝟏 𝟏 𝟏 −𝟏 ⊕ 𝟏 𝟐 𝟏 𝟏 𝟏 −𝟏 = 𝟏 𝟏 𝟏 𝟏 −𝟏 𝟏 𝟏 𝟏 𝟏 −𝟏 𝟏 𝟏 𝟏 𝟏 −𝟏 −𝟏 𝟏 𝟏 𝟏 −𝟏 = 𝟏 𝟐 𝟏 𝟏 𝟏 𝟏 𝟏 −𝟏 𝟏 −𝟏 𝟏 𝟏 −𝟏 −𝟏 𝟏 −𝟏 −𝟏 𝟏
- 29. Grover's Search Algorithm Grover’sAlgorithm: Implementation of 2-qubits 29 ZAS | 10/21/2022 | Step1 : Creating a uniform superposition by applying hadamard gate = 𝟏 𝟐 𝟏 𝟏 𝟏 𝟏 𝟏 −𝟏 𝟏 −𝟏 𝟏 𝟏 −𝟏 −𝟏 𝟏 −𝟏 −𝟏 𝟏 1 0 0 0 = 𝟏 𝟐 1 1 1 1
- 30. Grover's Search Algorithm Theprocedure of Grover’s algorithm is as follows: Creating a uniform superposition Apply the oracle function (𝑈) Apply amplitude amplification ( Grover diffuser operator) (D) Repeat step U and D 𝜋 4 𝑵 iterations Apply the measurement 2 3 4 5 Grover’s Algorithm: Steps 30 ZAS | 10/21/2022 |
- 31. Grover's Search Algorithm Grover’sAlgorithm: Implementation of 2-qubits 31 ZAS | 10/21/2022 | Step2 :Apply the oracle function (𝑈) for item |01⟩ by applying CZ and X gates 𝑰 ⊗ 𝐗 𝑪𝒁 𝑰 ⊗ 𝐗 1 0 0 1 ⊗ 0 1 1 0 𝟏 𝟎 𝟎 𝟎 𝟎 𝟏 𝟎 𝟎 𝟎 𝟎 𝟏 𝟎 𝟎 𝟎 𝟎 −𝟏 1 0 0 1 ⊗ 0 1 1 0 = 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
- 32. Grover's Search Algorithm Grover’sAlgorithm: Implementation of 2-qubits 32 ZAS | 10/21/2022 | Step2 :Apply the oracle function (𝑈) for item |01⟩ by applying CZ and X gates 𝑰 ⊗ 𝐗 𝑪𝒁 𝑰 ⊗ 𝐗 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 −1 0 0 𝑈= 1 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 1
- 33. Grover's Search Algorithm Theprocedure of Grover’s algorithm is as follows: Creating a uniform superposition Apply the oracle function (𝑈) Apply amplitude amplification ( Grover diffuser operator) (D) Repeat step U and D 𝜋 4 𝑵 iterations Apply the measurement 2 3 4 5 Grover’s Algorithm: Steps 33 ZAS | 10/21/2022 |
- 34. Grover's Search Algorithm Grover’sAlgorithm: Implementation of 2-qubits 34 ZAS | 10/21/2022 | Step3 :Apply Grover diffuser operator (D) by applying H , Z and CZ gates D= 2|Ψ⟩⟨Ψ| − 𝐼 = ((2|0⟩n⟨0|n−𝑰)) = ((2|0⟩2⟨0|2 − 𝑰) 2|0⟩2⟨0|2 − 𝑰=2 1 0 0 0 1 0 0 0 − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 = 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 = 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1
- 35. Grover's Search Algorithm Grover’sAlgorithm: Implementation of 2-qubits 35 ZAS | 10/21/2022 | Step3 :Apply Grover diffuser operator (D) by applying H , Z and CZ gates D= 2|Ψ⟩⟨Ψ| − 𝐼 (𝒁 ⊕ 𝒁) CZ=2|0⟩2⟨0|2 − 𝑰 𝟏 𝟏 𝟏 −𝟏 ⊕ 𝟏 𝟏 𝟏 −𝟏 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −1 = 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −1
- 36. Grover's Search Algorithm Grover’sAlgorithm: Implementation of 2-qubits 36 ZAS | 10/21/2022 | Step3 :Apply Grover diffuser operator (D) by applying H , Z and CZ gates D= 2|Ψ⟩⟨Ψ| − 𝐼 (𝒁 ⊕ 𝒁) CZ=2|0⟩2⟨0|2 − 𝑰 = 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −1 = 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1
- 37. Grover's Search Algorithm Grover’sAlgorithm: Implementation of 2-qubits 37 ZAS | 10/21/2022 | Step3 :Apply Grover diffuser operator (D) by applying H , Z and CZ gates D= 2|Ψ⟩⟨Ψ| − 𝐼 = 𝑯⊗𝒏((2|Ψ⟩n⟨Ψ|n − 𝑰) 𝑯⊗𝒏 Assume that |Ψ⟩=𝑯⊗𝒏|0⟩n ⟨Ψ|=⟨0|n 𝑯⊗𝑛 † 𝑯 †=𝑯 D= 2|Ψ⟩⟨Ψ| − 𝐼 =2𝑯⊗𝒏|0⟩n ⟨0|n ( 𝑯⊗𝑛 † −𝐼 =2𝑯⊗𝒏|0⟩n ⟨0|n𝑯⊗𝒏 − 𝑯⊗𝒏𝐼𝑯⊗𝒏 =𝑯⊗𝒏 (2|0⟩n ⟨0|n − 𝐼) 𝑯⊗𝒏
- 38. Grover's Search Algorithm 5-Circuit of 2-qubits Grover’s Algorithm: with IBM quantum Composer 38 ZAS | 10/21/2022 |
- 39. Grover's Search Algorithm Grover’sAlgorithm: Circuit diagram of 2-qubits 39 ZAS | 10/21/2022 | 1- Creating a uniform superposition 2- Apply the oracle function (𝑈) 3- Apply amplitude amplification (D) 4- Repeat U and D 𝝅 𝟒 𝑵 iterations 5- Apply the measurement • Circuit diagram of 2-qubits for oracle item 𝟎𝟏 = 𝟏
- 40. Grover's Search Algorithm Grover’sAlgorithm: Circuit diagram of 2-qubits 40 ZAS | 10/21/2022 | Grover’s algorithm of 2-qubits for oracle item 𝟎𝟏 = 𝟏 Remember that Ψ = 𝟏 𝟐 1 1 1 1 𝑈= 1 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 1 =𝑯⊗𝟐 ((2|0⟩2⟨0|2 − 𝑰) 𝑯⊗𝟐 𝑈|Ψ⟩ Ψ = 𝟏 𝟐 1 1 1 1 = 1 2 (|00⟩ + |01⟩ + 10⟩ + |11⟩) 𝑈|Ψ⟩= 1 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 1 𝟏 𝟐 1 1 1 1 = 𝟏 𝟐 1 −1 1 1
- 41. Grover's Search Algorithm Grover’sAlgorithm: Circuit diagram of 2-qubits 41 ZAS | 10/21/2022 | Grover’s algorithm of 2-qubits for oracle item 𝟎𝟏 = 𝟏 Remember that 𝑈|Ψ⟩= 𝟏 𝟐 1 −1 1 1 𝐻⊕𝟐 = 𝟏 𝟐 𝟏 𝟏 𝟏 𝟏 𝟏 −𝟏 𝟏 −𝟏 𝟏 𝟏 −𝟏 −𝟏 𝟏 −𝟏 −𝟏 𝟏 =𝑯⊗𝟐((2|0⟩2⟨0|2 − 𝑰) 𝑯⊗𝟐 𝑼|Ψ⟩ = 𝟏 𝟐 𝟏 𝟏 𝟏 𝟏 𝟏 −𝟏 𝟏 −𝟏 𝟏 𝟏 −𝟏 −𝟏 𝟏 −𝟏 −𝟏 𝟏 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1 𝟏 𝟐 𝟏 𝟏 𝟏 𝟏 𝟏 −𝟏 𝟏 −𝟏 𝟏 𝟏 −𝟏 −𝟏 𝟏 −𝟏 −𝟏 𝟏 𝟏 𝟐 1 −1 1 1 = 𝟏 𝟐 𝟏 𝟏 𝟏 𝟏 𝟏 −𝟏 𝟏 −𝟏 𝟏 𝟏 −𝟏 −𝟏 𝟏 −𝟏 −𝟏 𝟏 𝟏 𝟐 1 −1 1 1 = 𝟏 𝟒 2 2 −2 2 = 𝟏 𝟐 1 1 −1 1
- 42. Grover's Search Algorithm Grover’sAlgorithm: Circuit diagram of 2-qubits 42 ZAS | 10/21/2022 | Grover’s algorithm of 2-qubits for oracle item 𝟎𝟏 = 𝟏 Remember that 2|0⟩2⟨0|2 − 𝑰 = 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1 𝑯⊗𝟐 𝑈 Ψ = 1 1 −1 1 =𝑯⊗𝟐 (2|0⟩2⟨0|2 − 𝑰) 𝑯⊗𝟐 𝑈|Ψ⟩ = 𝟏 𝟒 𝟏 𝟏 𝟏 𝟏 𝟏 −𝟏 𝟏 −𝟏 𝟏 𝟏 −𝟏 −𝟏 𝟏 −𝟏 −𝟏 𝟏 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1 1 1 −1 1 = 1 −1 1 −1
- 43. Grover's Search Algorithm Grover’sAlgorithm: Circuit diagram of 2-qubits 43 ZAS | 10/21/2022 | Grover’s algorithm of 2-qubits for oracle item 𝟎𝟏 = 𝟏 Remember that 𝐻⊕𝟐 = 𝟏 𝟐 𝟏 𝟏 𝟏 𝟏 𝟏 −𝟏 𝟏 −𝟏 𝟏 𝟏 −𝟏 −𝟏 𝟏 −𝟏 −𝟏 𝟏 (2|0⟩2⟨0|2 − 𝑰) 𝑯⊗𝟐 𝑈|Ψ⟩ = 1 −1 1 −1 =𝑯⊗𝟐 (2|0⟩2⟨0|2 − 𝑰) 𝑯⊗𝟐 𝑈|Ψ⟩ = 𝟏 𝟒 𝟏 𝟏 𝟏 𝟏 𝟏 −𝟏 𝟏 −𝟏 𝟏 𝟏 −𝟏 −𝟏 𝟏 −𝟏 −𝟏 𝟏 1 −1 1 −1 = 𝟏 𝟒 0 4 0 0 = 0 1 0 0 = 𝟎𝟏 =|𝐰⟩
- 44. Grover's Search Algorithm •Grover, Lov K. "A fast quantum mechanical algorithm for database search." Proceedings of the twenty- eighth annual ACM symposium on Theory of computing. 1996. • https://qiskit.org/textbook/ch-algorithms/grover.html • Introduction to Quantum Computing: From a Layperson to a Programmer in 30 Steps: Wong, Hiu Yung: 9783030983383: Amazon.com: Books • Learn Quantum Computing with Python and IBM Quantum Experience: A hands-on introduction to quantum computing and writing your own quantum programs with Python: Loredo, Robert: 9781838981006: Amazon.com: Books References 44 ZAS | 10/21/2022 |
- 45. Grover's Search Algorithm Letme know your comments, please contact me via. zeinababohashima20@gmail.com 45 ZAS | 10/21/2022 |
- 46. Grover's Search Algorithm Thanks AnyQuestions? 46 ZAS | 10/21/2022 |
Editor's Notes
- #8 we can rewrite as 𝑼 |𝑥⟩
- #9 we can rewrite as 𝑼 |𝑥⟩
- #10 𝑼
- #15 This is because θ is the angle between |a⊥ (which is a linear superposition of all basis vectors except |a) 220 23 Grover’s Algorithm: I and |φ (which is a linear superposition of all basis vectors) Two reflections make a rotation= 2 𝜃
- #16 This is because θ is the angle between |a⊥ (which is a linear superposition of all basis vectors except |a) 220 23 Grover’s Algorithm: I and |φ (which is a linear superposition of all basis vectors) Two reflections make a rotation= 2 𝜃
- #38 =2 𝑯 ⊗𝒏 |0⟩n ⟨0|n 𝑯 ⊗𝒏 −𝐼𝐼 𝑫 =2|Ψ⟩⟨Ψ|−𝑰