Switching_Bs

 Went on a bit of an exploration this afternoon, trying to understand the notion

of changing the basis of quantum computation. Thesimplest example is going from

the Z basis to the X basis with a Hadamard gate. Below:

                                                                 

Here is how Perplexity AI explains it:

The Hadamard gate plays a crucial role in creating superposition states in qubits, which is a fundamental concept in quantum computing. Here's how the Hadamard gate creates superposition:

1. The computational basis states |0⟩ and |1⟩ represent the definite states of a qubit, where it is either in the state 0 or 1.

2. When the Hadamard gate is applied to the |0⟩ state, it creates an equal superposition of |0⟩ and |1⟩ states:

H|0⟩ = (|0⟩ + |1⟩) / √2 [1][4]

3. Similarly, when the Hadamard gate is applied to the |1⟩ state, it creates an equal superposition with a relative phase difference:

H|1⟩ = (|0⟩ - |1⟩) / √2 [1][4]

4. The resulting states (|0⟩ + |1⟩) / √2 and (|0⟩ - |1⟩) / √2 are superposition states, where the qubit is in a linear combination of the |0⟩ and |1⟩ states with equal probability amplitudes.[3][5]

5. The Hadamard gate is its own inverse, meaning that if you apply it twice to a state, you get back the original state. This property is useful for creating and manipulating superposition states in quantum algorithms.[1][3]

The ability of the Hadamard gate to create superposition states is crucial for exploiting quantum parallelism, where a quantum computer can perform operations on multiple states simultaneously. This is a key advantage of quantum computing over classical computing and enables various quantum algorithms and protocols.[4][5]

It's important to note that the superposition state created by the Hadamard gate is a linear combination of the basis states, and the qubit is not in a definite state of 0 or 1 until it is measured. The measurement process causes the superposition to collapse into one of the basis states randomly, with probabilities determined by the probability amplitudes.[3][4]

Citations:

[1] https://quantumcomputing.stackexchange.com/questions/7041/how-do-2-hadamard-gates-act-on-a-single-qubit

[2] https://www.quantum-inspire.com/kbase/hadamard/ https://www.quantum-inspire.com/kbase/hadamard/

[3] https://medium.datadriveninvestor.com/the-power-of-hadamard-in-quantum-computing-simplify-calculations-9c1e9ba948ba?gi=bd18f7996548null

[4] https://softwaredominos.com/home/science-technology-and-other-fascinating-topics/hadamard-gates-introducing-superposition-into-quantum-circuits/

[5] https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Quantum_Tutorials_%28Rioux%29/08:_Quantum_Teleportation/8.77:_Using_Quantum_Gates_to_Create_Superpositions_and_Entangled_States

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