# Exact Equivalence between Quantum Adiabatic Algorithm and Quantum Circuit Algorithm

## TL;DR

This paper proves a precise equivalence between quantum circuit algorithms and quantum adiabatic algorithms, showing they can be transformed into each other with the same complexity and that adiabatic algorithms can be exponentially sped up through path optimization.

## Contribution

It provides a rigorous proof of the exact transformation between quantum circuit and adiabatic algorithms, establishing their equivalence and potential for exponential speedup.

## Key findings

- Quantum circuit algorithms can be transformed into adiabatic algorithms with the same time complexity.
- Quantum adiabatic algorithms can be exponentially sped up by choosing optimal evolution paths.
- The transformation preserves the computational complexity, confirming their fundamental equivalence.

## Abstract

We present a rigorous proof that quantum circuit algorithm can be transformed into quantum adiabatic algorithm with the exact same time complexity. This means that from a quantum circuit algorithm of $L$ gates we can construct a quantum adiabatic algorithm with time complexity of $O(L)$. Additionally, our construction shows that one may exponentially speed up some quantum adiabatic algorithms by properly choosing an evolution path.