Hadamard gate in Grover algorithm

Hadamard gate in Grover algorithm Ask Question Asked 3 years, 9 months ago Modified 2 days ago Viewed 390 times 1 What is the need to apply the Hadamard gate as the first step while designing the diffuser circuit in the implementation of Grover's algorithm? I know what the gate does but I cannot understand what specific purpose it serves in this case. circuit-constructiongrovers-algorithmhadamard Share Improve this question Follow edited Jun 2, 2022 at 6:01 Martin Vesely 15.6k4 4 gold badges 34 34 silver badges 76 76 bronze badges asked Jun 1, 2022 at 19:09 Chanchal Sharma 111 1 bronze badge Add a comment 3 Answers Sorted by: Highest score (default) Date modified (newest first) Date created (oldest first) 0 The H gate is used to create an equal superposition of all computational states, each of which represents an item in the search region. Then the following circuits are applied to rotate the vector to the target. Share Improve this answer Follow answered Jun 2, 2022 at 0:20 刘环宇 893 3 bronze badges Add a comment 0 The Hadamard gate is used (in most descriptions) as a simplification to generate a superposition of the database. Say we want to find a marked element in a database D 𝐷 . The simplification is based on the assumption that the database has a number of entries that's a power of 2 2 . Formally, one would use a operator U D 𝑈 𝐷 that generates a superposition of all the elements in the database D 𝐷 . For Example: Say the database consists the integers from {0,1,...,7} { 0 , 1 , . . . , 7 } . Then the Hadamard transform H ⊗3 𝐻 ⊗ 3 performs the map U D :|0⟩↦ ∑ 7 i=0 |i⟩ 𝑈 𝐷 : | 0 ⟩ ↦ ∑ 𝑖 = 0 7 | 𝑖 ⟩ . Circuit with 0,1, ..., 7 If the database would instead be the integers {0,1,...,6} { 0 , 1 , . . . , 6 } , then one would use a different operator U D :|0⟩↦ ∑ 6 i=0 |i⟩ 𝑈 𝐷 : | 0 ⟩ ↦ ∑ 𝑖 = 0 6 | 𝑖 ⟩ . Circuit with 0,1, ..., 6, where U D 𝑈 𝐷 is called "{0,...,6}". (You can verify that the operators generate the databases by looking at the probability distribution after the operator) Share Improve this answer Follow edited Jun 3, 2022 at 9:10 answered Jun 3, 2022 at 9:04 Fleeep 3741 1 silver badge 5 5 bronze badges Add a comment 0 In our Grover’s algorithm, we: Add the initial Hadamard gates ie put our circuit in equal superposition |s> Apply the oracle Apply reflection about |s> For the reflection (diffuser), we need to implement U = 2|s><s| - 1 As we want to add negative phase to every state orthogonal to |s>, we implement as follows: Transform |s> to |0> by applying Hadamard gates (this is why we use it!) Apply a circuit to add negative phase to states orthogonal to |0> Transform |0> back to |s> using Hadamard gates Without the Hadamards, we cannot add the negative phase to states orthogonal to |0> as it needs to be converted first. This link may help: https://qiskit.org/textbook/ch-algorithms/grover.html#2.-Example:-2-Qubits- Share Improve this answer Follow answered Jun 29, 2022 at 13:23 Shraddha 262 2 bronze badges Add a comment Your Answer Post Your Answer By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions circuit-constructiongrovers-algorithmhadamard See similar questions with these tags. The Overflow Blog Open source for awkward robots Domain expertise still wanted: the latest trends in AI-assisted knowledge for... 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