Question

1 point 1 5. We have ran the Shor's algorithm (n = 8, N = 28 = 256) and measured the value y 165. Suppose we are lucky and on the 512, 265 + 512 there is a rational number with denominator r < VN interval 165 256 16. Find this rational number and write down its denominator. If there's no such number write 0. Enter answer here

Answer 1

the rational number with denominator r < VN interval 165 256 16 is 256/1683.

Explanation:

To find the rational number with denominator r < VN interval 165 256 16, we can perform the following steps:

1. Calculate the continued fraction expansion of the measured value y = 165:

165 = 0 + 1/(6 + 1/(1 + 1/(1 + 1/13)))

2. Write down the continued fraction expansion:

165 = [0; 6, 1, 1, 1, 13]

3. Determine the convergents of the continued fraction:

The convergents of the continued fraction [0; 6, 1, 1, 1, 13] are:

- Convergent 1: 0/1

- Convergent 2: 1/6

- Convergent 3: 1/7

- Convergent 4: 2/13

- Convergent 5: 15/98

- Convergent 6: 197/1291

- Convergent 7: 256/1683

4. Check if any of the convergents have a denominator less than the VN interval [165, 256, 16]:

Among the convergents, the denominator of the rational number 256/1683 falls within the VN interval.

5. Write down the denominator of the rational number:

The denominator of the rational number is 1683.

Therefore, the rational number with denominator r < VN interval 165 256 16 is 256/1683.