How is a quantum gate or unitary transformation on a qubit state performed using the Bloch sphere?
A quantum gate or unitary transformation on a qubit state can be performed using the Bloch sphere representation, which provides a geometric visualization of the qubit's state space. The Bloch sphere is a useful tool for understanding and manipulating spin systems, such as the Larmor precession of a qubit.
To begin, let's consider a qubit in a superposition state represented by the vector |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex probability amplitudes and |0⟩ and |1⟩ are the basis states. The Bloch sphere provides a way to represent this qubit state geometrically.
The Bloch sphere is a unit sphere with the north pole representing the state |0⟩ and the south pole representing the state |1⟩. Any point on the surface of the sphere corresponds to a pure state of the qubit, while points inside the sphere represent mixed states. The state vector |ψ⟩ can be represented by a point on the surface of the Bloch sphere.
Now, let's consider how a quantum gate or unitary transformation can be applied to the qubit state using the Bloch sphere. A quantum gate is a mathematical operation that transforms the qubit state according to certain rules. In the Bloch sphere representation, a quantum gate corresponds to a rotation of the state vector |ψ⟩ around an axis on the surface of the sphere.
The axis of rotation is determined by the gate's action on the basis states |0⟩ and |1⟩. For example, if we consider the Pauli-X gate, which flips the qubit state, it corresponds to a rotation of π radians around the x-axis of the Bloch sphere. This means that the state vector |ψ⟩ is rotated by π radians around the x-axis, resulting in a new state vector |ψ'⟩.
To perform the rotation, we can use the following formula:
|ψ'⟩ = U|ψ⟩,
where U is the unitary transformation corresponding to the desired gate. In the case of the Pauli-X gate, the unitary transformation U is given by:
U = |0⟩⟨1| + |1⟩⟨0|.
Applying this transformation to the state vector |ψ⟩, we get:
|ψ'⟩ = (|0⟩⟨1| + |1⟩⟨0|)(α|0⟩ + β|1⟩)
= α|1⟩ + β|0⟩.
This means that the state vector |ψ⟩ = α|0⟩ + β|1⟩ is transformed into the state vector |ψ'⟩ = α|1⟩ + β|0⟩, which corresponds to a rotation of π radians around the x-axis of the Bloch sphere.
Similarly, other quantum gates can be represented as rotations around different axes on the Bloch sphere. For example, the Pauli-Y gate corresponds to a rotation of π radians around the y-axis, while the Pauli-Z gate corresponds to a rotation of π radians around the z-axis.
A quantum gate or unitary transformation on a qubit state can be performed using the Bloch sphere representation by rotating the state vector around an axis on the surface of the sphere. The axis of rotation is determined by the gate's action on the basis states. The Bloch sphere provides a visual and intuitive way to understand and manipulate qubit states.
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