Generalized Deutsch-Jozsa Algorithm for Applications in Data Classification, Logistic Regression, and Quantum Key Distribution

TL;DR

The paper introduces the Generalized Deutsch-Jozsa (GDJ) algorithm, extending the original Deutsch-Jozsa framework to a two-register oracle with a Bell-state ancilla to retrieve both the global function type (constant or balanced) and explicit output values in a single query. This richer information per query is leveraged for practical tasks across data classification, quantum machine learning ensembles, and quantum cryptography, including a GDJ-based QKD protocol with a higher key rate and improved eavesdropper detection. The authors detail the Generalized Deutsch Algorithm (GDA) and the GDJ circuit, analyze resource and query implications, and illustrate how GDJ supports ensemble encoding, feature/parameter encoding, and interference-driven decision making. They also discuss robustness to noise, information-theoretic advantages, and provide simulations and protocol-level considerations for real hardware implementations. Overall, GDJ enhances interpretability and efficiency in quantum information processing by simultaneously exposing function type and values, with broad applicability to high-impact quantum technologies.

Abstract

We present a generalized Deutsch-Jozsa (DJ) quantum algorithm that not only determines both the global type of an unknown Boolean function (constant or balanced) but also determines explicit output values of the function in a single oracle query. Unlike the original DJ algorithm, which identifies only whether a function is constant or balanced, our generalization retrieves actual function output values at the same time with using a Bell state as ancilla. This makes a richer function characterization with minimal queries to have practical quantum advantages, e.g. data classification, logistic regression, and quantum cryptography.

Figures (10)

• Figure 1: Quantum circuit for the Generalized Deutsch Algorithm (GDA), where $\Phi^-$ denotes a Bell state, and $\Phi^-_1$ and $\Phi^-_2$ refer to the first and second qubits (or parts) of the Bell state $\Phi^-$, respectively.
• Figure 2: Quantum Circuit of Generalized Deutsch-Jozsa Algorithm (GDJA).
• Figure 3: Query complexity for classical and quantum algorithms in computations based on determining a constant or balanced function.
• Figure 4: Simulation of the oracles used in (a) the Deutsch algorithm (DA), (b) the generalized Deutsch algorithm (GDA), and (c) the generalized Deutsch–Jozsa algorithm (GDJA) through a “Life–Death choice game.” In this analogy, different input pathways lead to two possible outcome rooms: “Life” or “Death”, representing the binary outputs of the oracle. The objective of the game is to identify which paths lead to which rooms, i.e., to uncover the true nature of the output states. (d) Four possible configurations exist for the two outcome rooms: (1) Life–Death, (2) Life–Life, (3) Death–Life, and (4) Death–Death. When viewed as a game, the generalized algorithms guarantee a winning probability of 1, since they can fully determine both the nature of the doors (outputs) and the mapping of the input pathways.
• Figure 5: Standard deviation of accuracy versus dimension for the DJ and GDJ algorithms, shown for dimensions up to 50 (left) and 500 (right). The red dashed line represents the Berry-Esseen bound as a theoretical reference for convergence.