This paper presents an exponential speedup for solving an oracular problem using a quantum walk, as opposed to classical algorithms. The quantum algorithm is based on a continuous-time quantum walk, which differs from previous quantum algorithms that rely on quantum Fourier transforms. The problem involves finding the EXIT vertex in a graph given an oracle that provides the graph structure. The quantum walk efficiently traverses the graph, while no classical algorithm can solve the problem in subexponential time with high probability. The problem is defined on a graph $ G_n' $, which is a modification of a previously studied graph. The graph consists of two balanced binary trees connected by a random cycle. The quantum walk on this graph can reach the EXIT in polynomial time, while classical algorithms require exponential time. The quantum walk is implemented using an oracle that provides the graph structure and a consistent coloring of the edges. The algorithm uses a sequence of unitary operations to simulate the quantum walk, which is shown to traverse the graph in linear time. The paper proves that the quantum walk can find the EXIT in polynomial time, while classical algorithms cannot do so in subexponential time. The quantum walk is shown to propagate through the graph in linear time, with the quantum walk on a line with a defect at the center being equivalent to the quantum walk on the graph $ G_n' $. The analysis of the quantum walk on a line shows that it can traverse the graph in linear time, with the quantum walk on a finite line without a defect also being equivalent to the quantum walk on the graph $ G_n' $. The paper concludes that the quantum walk provides an exponential speedup for solving the oracular problem, as no classical algorithm can solve it in subexponential time. The quantum walk algorithm is shown to find the EXIT with high probability using a polynomial number of oracle calls. The results demonstrate that quantum walks can provide a significant advantage over classical algorithms for certain problems.This paper presents an exponential speedup for solving an oracular problem using a quantum walk, as opposed to classical algorithms. The quantum algorithm is based on a continuous-time quantum walk, which differs from previous quantum algorithms that rely on quantum Fourier transforms. The problem involves finding the EXIT vertex in a graph given an oracle that provides the graph structure. The quantum walk efficiently traverses the graph, while no classical algorithm can solve the problem in subexponential time with high probability. The problem is defined on a graph $ G_n' $, which is a modification of a previously studied graph. The graph consists of two balanced binary trees connected by a random cycle. The quantum walk on this graph can reach the EXIT in polynomial time, while classical algorithms require exponential time. The quantum walk is implemented using an oracle that provides the graph structure and a consistent coloring of the edges. The algorithm uses a sequence of unitary operations to simulate the quantum walk, which is shown to traverse the graph in linear time. The paper proves that the quantum walk can find the EXIT in polynomial time, while classical algorithms cannot do so in subexponential time. The quantum walk is shown to propagate through the graph in linear time, with the quantum walk on a line with a defect at the center being equivalent to the quantum walk on the graph $ G_n' $. The analysis of the quantum walk on a line shows that it can traverse the graph in linear time, with the quantum walk on a finite line without a defect also being equivalent to the quantum walk on the graph $ G_n' $. The paper concludes that the quantum walk provides an exponential speedup for solving the oracular problem, as no classical algorithm can solve it in subexponential time. The quantum walk algorithm is shown to find the EXIT with high probability using a polynomial number of oracle calls. The results demonstrate that quantum walks can provide a significant advantage over classical algorithms for certain problems.