Question

Grover's algorithm on a balanced function Let $ n $ be a positive integer and let $ f: \Sigma^{n} \rightarrow \Sigma $ be a balanced function. Thus, for \[ A_{0}=\left\{x \in \Sigma^{n}: f(x)=0\right\} \quad \text { and } \quad A_{1}=\left\{x \in \Sigma^{n}: f(x)=1\right\}, \] it is the case that $ \left|A_{0}\right|=\left|A_{1}\right|=2^{n-1} $. Which of the following statements best describes what happens when Grover's algorithm is run on the function $ f $ for some number of iterations, followed by a standard basis measurement of the $ n $ qubit register used in the algorithm? Answer options 1. After a single iteration, the resulting output string $ x $ is guaranteed to satisfy $ f(x)=1 $, with probability 1 . 2. After two iterations, the resulting output string $ x $ is guaranteed to satisfy $ f(x)=1 $, with probability 1 . 3. After any number of iterations, the resulting output string $ x $ satisfies $ f(x)=1 $ with probability $ 1 / 2 $. 4. After any number of iterations, the resulting output string $ x $ never satisfies $ f(x)=1 $; it is certain to satisfy $ f(x)=0 $.

          Grover's algorithm on a balanced function
Let \( n \) be a positive integer and let \( f: \Sigma^{n} \rightarrow \Sigma \) be a balanced function. Thus, for
\[
A_{0}=\left\{x \in \Sigma^{n}: f(x)=0\right\} \quad \text { and } \quad A_{1}=\left\{x \in \Sigma^{n}: f(x)=1\right\},
\]
it is the case that \( \left|A_{0}\right|=\left|A_{1}\right|=2^{n-1} \).
Which of the following statements best describes what happens when Grover's algorithm is run on the function \( f \) for some number of iterations, followed by a standard basis measurement of the \( n \) qubit register used in the algorithm?

Answer options
1. After a single iteration, the resulting output string \( x \) is guaranteed to satisfy \( f(x)=1 \), with probability 1 .
2. After two iterations, the resulting output string \( x \) is guaranteed to satisfy \( f(x)=1 \), with probability 1 .
3. After any number of iterations, the resulting output string \( x \) satisfies \( f(x)=1 \) with probability \( 1 / 2 \).
4. After any number of iterations, the resulting output string \( x \) never satisfies \( f(x)=1 \); it is certain to satisfy \( f(x)=0 \).

$Grover's algorithm on a balanced function Let n be a positive integer and let f: Σ^n→Σ be a balanced function. Thus, for A0={x ∈Σ^n: f(x)=0} and A1={x ∈Σ^n: f(x)=1}, it is the case that |A0|=|A1|=2^n-1. Which of the following statements best describes what happens when Grover's algorithm is run on the function f for some number of iterations, followed by a standard basis measurement of the n qubit register used in the algorithm? Answer options 1. After a single iteration, the resulting output string x is guaranteed to satisfy f(x)=1, with probability 1 . 2. After two iterations, the resulting output string x is guaranteed to satisfy f(x)=1, with probability 1 . 3. After any number of iterations, the resulting output string x satisfies f(x)=1 with probability 1 / 2. 4. After any number of iterations, the resulting output string x never satisfies f(x)=1; it is certain to satisfy f(x)=0.$

Added by Joel W.

Question

Please give Ace some feedback