Question

When applying the Hadamard gate to a qubit in the |0⟩ state, what is the resulting state vector?

Accepted Answer

Applying the Hadamard gate to a qubit in the |0⟩ state results in the state vector 1/sqrt(2)|0⟩ + 1/sqrt(2)|1⟩. The Hadamard gate is a fundamental quantum logic gate that creates a state of superposition, where the qubit exists as both 0 and 1 simultaneously. The state |0⟩ represents the qubit&#x27;s ground state, often visualized as a vector [1, 0] in linear algebra. The Hadamard gate is represented by a matrix with values 1/sqrt(2) for its entries, except for the bottom right entry which is -1/sqrt(2). When you multiply this matrix by the column vector [1, 0], you perform a calculation where the first component becomes 1/sqrt(2) times 1 plus 1/sqrt(2) times 0, resulting in 1/sqrt(2), and the second component becomes 1/sqrt(2) times 1 plus -1/sqrt(2) times 0, resulting in 1/sqrt(2). This creates the output state, which is commonly denoted as |+⟩. This result means that if you measure the qubit in the computational basis after the operation, there is a 50 percent probability of observing the state as |0⟩ and a 50 percent probability of observing it as |1⟩.

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When applying the Hadamard gate to a qubit in the |0⟩ state, what is the resulting state vector?