 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 3

The <quantum Fourier transform> measurement supplies a rational approximation
$$
\frac{c}{2^m}\simeq\frac{s}{r},
$$
where $r$ is the desired <multiplicative order>. Here
$$
\frac{c}{2^m}=\frac{256}{512}=\frac12.
$$
Its reduced <continued fraction convergent> has denominator two, so <continued-fraction recovery in quantum order finding> proposes $r=2$. Direct <order verification from divisors> confirms it:
$$
8^2=64\equiv1\pmod{21},
\qquad
8\not\equiv1\pmod{21}.
$$
Hence
$$
\boxed{\operatorname{ord}_{21}(8)=2.}
$$
As a check, the classical final step of the <Shor algorithm> gives
$$
\boxed{\gcd(8-1,21)=7,
\qquad
\gcd(8+1,21)=3.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 10D / d / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 10D / d / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 10D / d / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)