Question1. If there is a state $|\phi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+e^{i\theta}|1\rangle)$, and I want to know the angle $\theta$. What kind of measurement should I do? Could somebody give me the quantum circuit?
Question2. How to perform a measurement with base $M{{({{\theta }_{k}})}_{\pm }}=\left\{ 1/\sqrt{2}\left( |0\rangle \pm {{e}^{-i{{\theta }_{k}}}}|1\rangle \right) \right\}$ on IBMQ?
Answer to the first question:
As mentioned in the comments of the question I assume that we can prepare $|\phi \rangle$ as many as we want. Let's calculate the relative phase for this one qubit pure state:
$$|\psi \rangle = \frac{1}{\sqrt{2}} \left( |0\rangle + e^{i\theta}|1\rangle\right)$$
We are going to execute $2$ different experiments in order to estimate $\theta$. In the first experiment we apply this circuit:
circuit_experiment_1.h(q[0])
circuit_experiment_1.measure(q[0], c[0])
The state after Hadamard gate:
$$H \frac{1}{\sqrt{2}} \left( |0\rangle + e^{i\theta}|1\rangle\right) = \frac{1}{2}\left[(1 + e^{i\theta})| 0 \rangle + (1 - e^{i\theta})| 1 \rangle \right]$$
The probabilities of $|0\rangle$ and $|1\rangle$ states:
\begin{align*} P(0) = \frac{1}{4}\left| 1 + e^{i\theta} \right|^2 = \frac{1}{2}(1 + \cos(\theta)) \\ P(1) = \frac{1}{4}\left| 1 - e^{i\theta} \right|^2 = \frac{1}{2}(1 - \cos(\theta)) \end{align*}
From here we can see that:
$$\theta = \pm \arccos\big(P(0) - P(1)\big)$$
because the range of usual principal value arccosine function is equal to $[0, \pi]$. So we will need the second experiment in order to estimate the $sign(\theta)$. But, before that, how to find $P(0)$ and $P(1)$ with the described experiment? We will need to execute the circuit $N$ times (bigger $N$ gives better precision) and take into accout these relations between measurement outcomes and probabilities:
\begin{align*} P(0) = \lim_{N \rightarrow \infty} \frac{N_{0}}{N} \qquad P(1) = \lim_{N \rightarrow \infty} \frac{N_{1}}{N} \end{align*}
where $N_{0}$ is the number of $|0\rangle$ measurement outcomes and $N_{1}$ is the number of $|1\rangle$ measurement outcomes. Also, note that:
$$\langle X \rangle = \langle \psi | X | \psi \rangle = \langle \psi |H Z H| \psi \rangle = P(0) - P(1)$$
So, the formula can be written in this way:
$$\theta = \pm \arccos \big( \langle X \rangle \big)$$
The sign of the $\theta$
Now we should determine the $sign(\theta)$ with this circuit:
circuit_experiment_2.sdg(q[0])
circuit_experiment_2.h(q[0])
circuit_experiment_2.measure(q[0], c[0])
The state after applying $S^{\dagger}$ and $H$ gates:
$$H S^{\dagger} \frac{1}{\sqrt{2}} \left( |0\rangle + e^{i\theta}|1\rangle\right) = \frac{1}{2}\left[(1 - i e^{i\theta})| 0 \rangle + (1 + i e^{i\theta})| 1 \rangle \right]$$
with the same logic:
\begin{align*} P'(0) = \frac{1}{4}\left| 1 - ie^{i\theta} \right|^2 = \frac{1}{2}(1 + \sin(\theta)) \\ P'(1) = \frac{1}{4}\left| 1 + ie^{i\theta} \right|^2 = \frac{1}{2}(1 - \sin(\theta)) \end{align*}
So after determining the $P'(0)$ and $P'(1)$ from the second experiment we will find the sign of the $\theta$:
$$sign(\theta) = sign(\arcsin\left(P'(0) - P'(1)\right)) = sign(P'(0) - P'(1))$$
because the range of usual principal value of arcsine function is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
Also, note that for the expectation value of the $Y$ operator (as can be seen from this answer) we have this relation:
$$\langle Y \rangle = \langle \psi| Y | \psi\rangle = \langle \psi| S H Z H S^{\dagger} | \psi\rangle = P'(0) - P'(1)$$
By taking this into account and combining two results:
\begin{align*} \theta = sign \big(\langle Y \rangle \big) \arccos \big(\langle X \rangle \big) \end{align*}
An approach for finding the relative phase of an arbitrary pure state is described in this answer.
Answer to the second question:
Here is the circuit for measuring in $M{{({{\theta }_{k}})}_{\pm }}=\left\{ 1/\sqrt{2}\left( |0\rangle \pm {{e}^{-i{{\theta }_{k}}}}|1\rangle \right) \right\}$ basis. I assume here that $\theta_k$ is given:
circuit.u1(theta_k, q[0]) # q[0] is one of the qubits
circuit.h(q[0])
circuit.measure(q[0], c[0]) #c[0] is a classical bit
If the state was $M(\theta _k)_+= 1/\sqrt{2}\left( |0\rangle + e^{-i\theta _k}|1\rangle \right)$, then the outcome of the circuit will be $|0\rangle$, and if it was $M(\theta _k)_-= 1/\sqrt{2}\left( |0\rangle - e^{-i\theta _k}|1\rangle \right)$, then the outcome of the circuit will be $|1\rangle$. So this way we will be able to measure in $M{{({{\theta }_{k}})}_{\pm }}$ basis.
I would just like to share a code for testing a phase measurement on IBM Q:
OPENQASM 2.0;
include "qelib1.inc";
qreg q[1];
creg c[1];
//measuring theta in
//(|0> + |1>exp(itheta))
h q[0]; //(|0> + |1>)
t q[0]; //(|0> + |1>exp(ipi/4))
//s q[0]; //(|0> + |1>exp(ipi/2))
//u1 (pi/8) q[0]; //(|0> + |1>exp(ipi/8))
h q[0]; //measurment in Hadamard basis
measure q[0] -> c[0];
Tested on IBM Q Armonk (1 qubit processor).
EDIT (based on Davit comment): To infer a sign of the phase, a measurement in circular basis (i.e. adding $S^\dagger$ gate before Hadamard gate) has to be done as well. Combining results from measurement in Hadamard basis and circular basis gives complete knowledge about the phase.
