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Quantum computation

  1. Oct 18, 2013 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant 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?

    3. 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?
     
  2. jcsd
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