The first quantum algorithm we will look at is the Bernstein-Vazirani algorithm.

Given a secret s $\in \{0,1\}^n$ and the function f : $\{0,1\}^n \rightarrow \{0,1\}$ defined as f(x) = x * s
\* denotes the inner product of two bitstrings here
For Bitstrings x and y of length n the inner product x * y is $x_1y_1 + ... + x_ny_n$ mod 2
The goal is to find the secret s using as few queries of f as possible.
So as few evaluations of f as possible.

We will look at a quantum algorithm that will find s with only one evaluation of f.

![[Pasted image 20260730144713.png]]

The top wire consists of n qubits in state 0.   $\ket{0}^n = \ket{0} \otimes ... \otimes \ket{0}$ 
The bottom wire is in state 1.
Both wires together are in state $\ket{0^n1} = \ket{0}^n \otimes \ket{1}$ 

First, we apply the Hadamard gate on all qubits.
The resulting state is calculated as follows:

![[Pasted image 20260730145119.png]]