Shallow Implementation of Quantum Fingerprinting with Application to Quantum Finite Automata

TL;DR

The paper addresses the practical realization of quantum fingerprinting-based finite automata for the MOD_p language on near-term quantum hardware by shifting to shallow-depth circuit designs. It introduces GAP-based coefficient sets from additive combinatorics to construct large, structured K with depth $\lceil \log p - 2 \log \varepsilon \rceil + 2$, achieving depth comparable to probabilistic methods while increasing width only polynomially. Through theoretical analysis and extensive numerical experiments, including noisy-device simulations, the authors show that the shallow approach maintains competitive error behavior in ideal devices and significantly outperforms standard circuits under realistic noise, enabling MOD_p recognition with a small number of qubits. These results advance the practicality of quantum fingerprinting in current hardware and highlight open questions about fundamental depth lower bounds for such automata.

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 \(\ket{ψ(x)}\) 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.

Figures (9)

• Figure 1: Deep fingerprinting circuit example. Gate $U_{j}$ is a rotation $R_{y}(4\pi k_{j}x / p)$. Controls in controlled gates run over all binary strings of length $s$
• Figure 2: Shallow fingerprinting circuit example. Gate $R_{j}$ is a rotation $R_{y}(4\pi t_{j}x / p)$
• Figure 3: Circuit $\mathcal{C}_j$ for AIKPS subsequence. Gate $R_{j}$ is a rotation $R_{y}(4\pi (r_{j}^{-1}) / p)$. Gate $R_{j,k}$ is a rotation $R_{y}(2^{k-1} \cdot 4\pi (r_{j}^{-1}) / p)$
• Figure 4: Circuit for $U_{a}(T)$. Gate $R_{j}$ is a rotation $R_{y}(4\pi (r_{j}^{-1}) / p)$. Gate $R_{j,k}$ is a rotation $R_{y}(2^{k-1} \cdot 4\pi (r_{j}^{-1}) / p)$
• Figure 5: Computational errors for $m=3,4,5$ of original and shallow automata

Theorems & Definitions (8)

• definition 1: TaoVu2006
• theorem 1: TaoVu2006
• definition 2: TaoVu2006
• theorem 2
• proof
• corollary 1
• theorem 3: Circuit depth for AIKPS sequences
• proof