Shallow Implementation of Quantum Fingerprinting with Application to Quantum Finite Automata

TL;DR

This paper presents a shallow-depth quantum fingerprinting implementation using additive combinatorics, making it more practical for current quantum hardware, especially for automata related to prime-modulus languages.

Contribution

It introduces explicit quantum fingerprinting methods based on additive combinatorics, optimizing circuit depth for current quantum devices.

Findings

Circuit depth comparable to probabilistic methods

Explicit methods outperform previous approaches in depth

Applicable to quantum automata for prime-modulus languages

Abstract

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 algorithm for \(MOD_{p}=\{a^{i\cdot p} \mid i \geq 0\}\) 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 \(x \in \{0,1\}^{n}\) of length \(n\) to a state \(\ket{\psi(x)}\) 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.