Quantum Computation Homework: Bloch Sphere Coordinates

In summary, the Bloch sphere coordinates for the qubit at time t with Hamiltonian H=BX and initial condition |ψ(0)⟩=|0⟩ are θ=2Bt and ϕ=π/2, with the ranges 0<θ<π and 0<ϕ<2π.
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Homework Statement



assume that we are working in units such that ℏ=1. When providing Bloch sphere coordinates, please ensure that 0<θ<π and 0<ϕ<2π whenever applicable.

Homework Equations



we start with the qubit |Ψ(0)⟩=|0⟩ and we apply a magnetic field in the x^ direction at t=0. This corresponds to the Hamiltonian H=BX, where B is a constant and X is the usual bit flip gate. What are the coordinates (θ,ϕ) of this qubit on the Bloch sphere at time t, as a function of B and t?

The Attempt at a Solution



We have the evolution equation : i∂t|ψ(t)>=H|ψ(t)>, with H constant, solved as |ψ(t)>=e−iHt|ψ(0)>. Here |ψ(t)⟩ is a 2-dimensional complex vector. so to find an expression for e−iHt. Thinking at the definition of the exponential as a Taylor serie, and that X2=Id, and get the result, then the expression for |ψ(t).. Then |ψ(t)>=cosθ/2|0>+e^(iϕ)*sinθ/2|1> (up to a global phase).
i got Θ = 2*B*t and phi = pi/2
but do not know is it correct?
 
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it's important to always double check your work and make sure your equations are correct. In this case, your solution is not quite correct. The correct solution is θ=2Bt and ϕ=π/2. This can be derived by solving the Schrodinger equation for the given Hamiltonian and using the initial condition of |ψ(0)⟩=|0⟩. It's also important to note that the Bloch sphere coordinates should be in the ranges specified in the homework statement, so the correct solution would be 0<θ<π and 0<ϕ<2π.
 

1. What is a Bloch sphere and how is it used in quantum computation?

A Bloch sphere is a visual representation of the quantum states of a qubit, which is the basic unit of information in quantum computation. It is a sphere with the north and south poles representing the qubit states 0 and 1, and all other points on the sphere representing superpositions of these states. The Bloch sphere is used to visualize and manipulate the complex amplitudes and phases of qubits, making it a useful tool for quantum computation.

2. What are Bloch sphere coordinates and how are they related to quantum states?

Bloch sphere coordinates are used to describe the position of a point on the Bloch sphere. The x, y, and z coordinates represent the amplitudes of the qubit states 0, 1, and a superposition, respectively. The length of the vector from the center of the sphere to the point on the surface represents the probability of measuring that state. Therefore, Bloch sphere coordinates can be used to represent and manipulate the quantum states of a qubit.

3. How do you convert between Bloch sphere coordinates and the state vector representation of a qubit?

To convert between Bloch sphere coordinates and the state vector representation of a qubit, you can use the following equations:

x = Re(a), y = Im(a), z = |b| where a is the complex amplitude of the state 0, b is the complex amplitude of the state 1, and |b| represents the magnitude of b. Conversely, you can convert from Bloch sphere coordinates to the state vector representation using the following equations: a = (x + iy)/sqrt(2), b = z.

4. How can Bloch sphere coordinates be used to perform quantum operations on qubits?

Bloch sphere coordinates can be used to perform quantum operations on qubits by manipulating the position of the point on the sphere. For example, applying a Hadamard gate to a qubit would rotate the point on the Bloch sphere by 90 degrees around the x-axis, representing a superposition of the qubit states 0 and 1. By manipulating the Bloch sphere coordinates, various quantum gates and operations can be performed on qubits, allowing for complex quantum computations.

5. Are there any limitations to using Bloch sphere coordinates in quantum computation?

While Bloch sphere coordinates are a useful tool for visualizing and manipulating quantum states, they have some limitations. They can only represent single qubits and cannot be used for multi-qubit systems. Additionally, they do not account for decoherence, which is the loss of quantum information due to interactions with the environment. Therefore, while Bloch sphere coordinates are a valuable tool, they may not accurately reflect the behavior of qubits in real-world systems.

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