Suppose you are given an endless supply of qubit pairs in the state 1/sqrt(2)|00> + e^iphi/sqrt(2)|10>. To estimate the phase angle phi, you run Fourier sampling (i.e. Hadamard on each qubit followed by a standard basis measurement) on this state repeatedly. After 100,000,000 measurements, you find that the outcome 01 never occurred. What is phi ? Please provide your answer in the range phi belongs to [0,2pi).

Hadamard transform is defined as:

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

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

The state is:

1/sqrt(2)|00> + e^iphi/sqrt(2)|10>

We then have that:

<01|U|s> = 0

You can evaluate the l.h.s. by letting U act on the bra vector. Since U equals its own inverse, U-dagger = U. We thus have:

U-dagger|01> = U|01> =

1/2 [|0> + |1>][|0> - |1>] =

1/2 [|00> - |01> +|10> - |11>]

We thus have:

<01|U|s> =

1/2 [<00|s> - <01|s> + <10|s> - <11|s>]

=

1/(2 sqrt(2)) [1 + exp(i phi)]

This is zero for phi = pi. You can also be more precise by considering the squared norm of the amplitude and saying that this must be less than some small number, where you choose that small number based on the fact that in 100,000,000 measurements you didn't observe 01. You then get a small interval around pi.

To estimate the phase angle phi in the given scenario, we can use the concept of quantum interference and the properties of Fourier sampling.

Let's start by understanding the state of the qubit pair. The given state can be written as:

1/sqrt(2)|00> + e^(i*phi)/sqrt(2)|10>

Next, we consider the effect of Hadamard (H) gate on the first qubit. The Hadamard gate transforms the basis states as follows:

H|0> = 1/sqrt(2)(|0> + |1>)
H|1> = 1/sqrt(2)(|0> - |1>)

Applying the H gate to the first qubit yields the following state:

H(1/sqrt(2)|00> + e^(i*phi)/sqrt(2)|10>)
= 1/2 (|0> + |1>) [1/sqrt(2)(|0> + |1>)] + 1/2 (|0> - |1>) [e^(i*phi)/sqrt(2)(|0> - |1>)]
= 1/2 [1/sqrt(2)(|0> + |1>)(|0> + e^(i*phi)|1>) + 1/sqrt(2)(|0> - |1>)(|0> - e^(i*phi)|1>)]

Simplifying this expression, we get:

1/2 [1/sqrt(2)(|00> + e^(i*phi)|10>) + 1/sqrt(2)(|00> - e^(i*phi)|10>)]

= 1/2[|0>|0> + e^(i*phi/2)|1>|0>] + 1/2[|0>|0> - e^(i*phi/2)|1>|0>]
= 1/2[|0>|0> + e^(i*phi/2)|1>|0> + |0>|0> - e^(i*phi/2)|1>|0>]
= 1/2[2|0>|0>]

We can see that applying the Hadamard gate on the first qubit causes the state to collapse to |0>|0> with certainty. Therefore, the measurement outcome will always be 00.

Since we are given that the outcome 01 never occurs even after 100,000,000 measurements, this rules out the possibility of any phase angle phi. In other words, phi must be equal to 0.

Therefore, the value of phi in the range [0, 2pi) is 0.