GROVER's Algorithm

Grover’s Algorithm for $(2^x\equiv 1(\mathrm{mod}4))$

Problem Statement

We need to find ( x ) such that:

$2^x\equiv 1(\mathrm{mod}4)$

Classical Solution

Classically, we can solve this by checking values of ( x ):

• $2^0\equiv 1(\mathrm{mod}4)$
• $2^0\equiv 1(\mathrm{mod}4)$
• $2^0\equiv 1(\mathrm{mod}4)$
• $2^0\equiv 1(\mathrm{mod}4)$
• $2^4\equiv 0(\mathrm{mod}4)$
• $...$

From this, we observe that ( 2^x \equiv 1 \pmod{4} ) only holds for ( x = 0 ).

Using Grover’s Algorithm

Grover’s algorithm provides a quadratic speedup for searching through unsorted databases or solving unstructured search problems. To apply Grover’s algorithm to find $(x)$ such that $(2^x\equiv 1(\mathrm{mod}4))$, we need to construct a quantum oracle that marks the correct solution.

Steps to Apply Grover’s Algorithm

1. Oracle Construction:

   • Create an oracle ( O ) that flips the sign of the amplitude of the state $(|x⟩)$ if $2^x\equiv 1(\mathrm{mod}4)$. This can be represented as: $$O|x⟩=\{\begin{array}{ll}-|x⟩& \text{if }{2}^x\equiv 1(\mathrm{mod}4)\\ |x⟩& \text{otherwise}\end{array}$$

2. Initialization:

   • Initialize a quantum register in a superposition of all possible values of $x$ using Hadamard gates:
   $$H^{\otimes n}|0⟩=\frac{1}{\sqrt{N}}\sum _{x=0}^{N-1}|x⟩$$

   where ( N ) is the number of possible values of ( $x$ ).

3. Grover Iterations:

   • Apply the Grover diffusion operator, which inverts the amplitude of each state about the average amplitude. This step amplifies the amplitude of the solution state(s).

4. Measurement:

   • Measure the quantum register. The result will be the value of ( x ) that satisfies $2^x\equiv 1(\mathrm{mod}4)$ with high probability.

Practical Consideration

In practice, $2^x\equiv 1(\mathrm{mod}4)$ has a trivial solution $x=0$ However, if we consider a more complex modular exponentiation problem, such as finding $x$ such that $2^x\equiv y(\mathrm{mod}N)$ for some arbitrary $y$ and $N$, Grover’s algorithm becomes more applicable. The steps would involve constructing a suitable oracle for the given problem.