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).

Question

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).

Count Iblis · Accepted Answer

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.