Consider the observable M that corresponds to a measurement in the sign basis where the measurement value is 3 if the outcome is + and −2 if the outcome is −.

Write M in the standard basis.
If M is the Hamiltonian acting on a qubit which is in the state |ψ(0)⟩=|0⟩ at time 0, what is the state of the qubit at time t? Assume that you are working in units such that ℏ=1. You may use e and t in your answer.
Now you perform an X=(0110) measurement on the qubit at time 0. What is the expected value of your measurement?
What if you perform the above measurement at time t=4π3 instead? What is the expected value of your measurement?

We have:

|+> = 1/sqrt(2)[|0> + |1>]

|-> = 1/sqrt(2)[|0> - |1>]

M = 3|+><+| - 2 |-><-|

Insert the above expressions and expand, or just evaluate the matrix elements. E.g.:

<0|M|0> =

3<0|+><+|0> - 2 <0|-><-|0> =

3/2 - 1 = 1/2

I was able to do part 1 and part 2. I am stuck on

Now you perform an X=(0110) measurement on the qubit at time 0. What is the expected value of your measurement?
What if you perform the above measurement at time t=4π3 instead? What is the expected value of your measurement?

To write the observable M in the standard basis, we need to express it as a linear combination of the Pauli matrices. In the sign basis, the measurement value corresponds to + for outcome + and -2 for outcome -.

In the standard basis, the Pauli matrices are defined as follows:
σ_x = |0><1| + |1><0|
σ_y = -i |0><1| + i |1><0|
σ_z = |0><0| - |1><1|

The observable M can be written in terms of these Pauli matrices as:
M = 3 * |0><0| + (-2) * |1><1|

Now, let's determine the state of the qubit at time t given that M is the Hamiltonian acting on the qubit initialized in the state |ψ(0)⟩=|0⟩ at time 0. The time evolution of a quantum state can be obtained using the Schrödinger equation.

The Schrödinger equation is given by: i𝑑𝑑𝑡|ψ(t)⟩ = ℋ|ψ(t)⟩

Since our Hamiltonian ℋ is M, we have:
i𝑑𝑑𝑡|ψ(t)⟩ = M|ψ(t)⟩

Substituting M and the initial state |ψ(0)⟩=|0⟩, we have:
i𝑑𝑑𝑡|ψ(t)⟩ = (3 * |0><0| + (-2) * |1><1|)|ψ(t)⟩

We can solve this differential equation to find the time evolution of |ψ(t)⟩. To do this, we can express the state |ψ(t)⟩ in the standard basis as:
|ψ(t)⟩ = α(t)|0⟩ + β(t)|1⟩

Substituting this into the differential equation, we get two coupled equations:
i𝑑𝑑𝑡α(t) = 3α(t)
i𝑑𝑑𝑡β(t) = -2β(t)

Solving these equations, we find:
α(t) = α(0) * e^(3t)
β(t) = β(0) * e^(-2t)

Given that the initial state |ψ(0)⟩=|0⟩, we have α(0) = 1 and β(0) = 0. Therefore, the state of the qubit at time t is:
|ψ(t)⟩ = e^(3t)|0⟩

Now, let's move on to the measurement part. Performing an X=(0110) measurement on the qubit at time 0 means we measure the observable X in the standard basis. The expected value of the measurement is given by the expectation value of the observable with respect to the state |ψ(0)⟩.

The expectation value is calculated as:
⟨X⟩ = ⟨ψ(0)|X|ψ(0)⟩

Substituting X=(0110) and |ψ(0)⟩=|0⟩, we get:
⟨X⟩ = ⟨0|(0110)|0⟩

Evaluating this expression, we find:
⟨X⟩ = 0 * 0 + 1 * 1 + 1 * 1 + 0 * 0 = 2

Therefore, the expected value of the X measurement at time 0 is 2.

Now, let's consider the measurement at time t=4π/3. We can follow the same steps as above but using the state |ψ(t)⟩ = e^(3t)|0⟩ = e^(3(4π/3))|0⟩ = e^(4π)|0⟩.

Substituting this into the expression for the expectation value, we get:
⟨X⟩ = ⟨e^(4π)|0|(0110)|e^(4π)|0⟩

Evaluating this expression, we find:
⟨X⟩ = e^(8π) * 0 + e^(8π) * 1 + e^(8π) * 1 + e^(8π) * 0 = 2 * e^(8π)

Therefore, the expected value of the X measurement at time t=4π/3 is 2 * e^(8π).