What is the final state of the first qubit after applying the Hadamard gate and the CNOT gate to the initial state |0⟩|0⟩?
The final state of the first qubit after applying the Hadamard gate and the CNOT gate to the initial state |0⟩|0⟩ can be determined by considering the step-by-step transformation of the state vector.
Let's start with the initial state |0⟩|0⟩, which represents two qubits in the state |0⟩. The first qubit is denoted as qubit 1, and the second qubit is denoted as qubit 2.
The Hadamard gate (H) is a single-qubit gate that transforms the state of a qubit. When applied to qubit 1, the Hadamard gate takes the state |0⟩ to the superposition state (|0⟩+|1⟩)/√2. Therefore, the state after applying the Hadamard gate to qubit 1 is ((|0⟩+|1⟩)/√2)|0⟩.
Next, we apply the CNOT gate to qubit 1 and qubit 2. The CNOT gate is a two-qubit gate that performs a conditional operation based on the state of the control qubit (qubit 1) and the target qubit (qubit 2). In this case, qubit 1 is the control qubit, and qubit 2 is the target qubit.
The CNOT gate flips the state of the target qubit (qubit 2) if the control qubit (qubit 1) is in the state |1⟩. Since the control qubit (qubit 1) is in the state ((|0⟩+|1⟩)/√2), we need to consider the two cases separately.
Case 1: Control qubit (qubit 1) is in the state |0⟩:
In this case, the CNOT gate does not flip the state of the target qubit (qubit 2). Therefore, the state after applying the CNOT gate in this case is ((|0⟩+|1⟩)/√2)|0⟩.
Case 2: Control qubit (qubit 1) is in the state |1⟩:
In this case, the CNOT gate flips the state of the target qubit (qubit 2). Therefore, the state after applying the CNOT gate in this case is ((|0⟩+|1⟩)/√2)|1⟩.
To determine the final state, we need to consider both cases and combine the results. We can express the final state as a superposition of the two cases:
Final state = (1/√2)((|0⟩+|1⟩)/√2)|0⟩ + (1/√2)((|0⟩+|1⟩)/√2)|1⟩
Simplifying this expression, we get:
Final state = (1/2)(|00⟩+|01⟩+|10⟩+|11⟩)
So, the final state of the first qubit after applying the Hadamard gate and the CNOT gate to the initial state |0⟩|0⟩ is (1/2)(|0⟩+|1⟩).
The final state of the first qubit is a superposition of the states |0⟩ and |1⟩, each with a coefficient of 1/2.
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