GAPs for Shallow Implementation of Quantum Finite Automata
Mansur Ziatdinov, Aliya Khadieva, Abuzer Yakaryılmaz · Electronic Proceedings in Theoretical Computer Science · 2023
Quantum fingerprinting is a technique that maps classical input word to a quantum state.The obtained quantum state is much shorter than the original word, and its processing uses less resources, making it useful in quantum algorithms, communication, and cryptography.One of the examples of quantum fingerprinting is quantum automata algorithms for MOD p = {a i•p | i ≥ 0} languages, where p is a prime number.However, implementing such an automaton on the current quantum hardware is not efficient.Quantum fingerprinting maps a word x ∈ {0, 1} n of length n to a state |ψ(x) of O(log n) qubits, and uses O(n) unitary operations.Computing quantum fingerprint using all available qubits of the current quantum computers is infeasible due to a large number of quantum operations.To make quantum fingerprinting practical, we should optimize the circuit for depth instead of width in contrast to the previous works.We propose explicit methods of quantum fingerprinting based on tools from additive combinatorics, such as generalized arithmetic progressions (GAPs), and prove that these methods provide circuit depth comparable to a probabilistic method.We also compare our method to prior work on explicit quantum fingerprinting methods.