Quantum Information Fusion and Correction with Dempster-Shafer Structure

TL;DR

The paper establishes a rigorous correspondence between Dempster-Shafer belief structures and quantum circuits by encoding mass functions as mass-function quantum states (MFQS) over an $n$-qubit framework, where $|m\rangle=\sum_{F_i\subseteq\Omega}\sqrt{m(F_i)}|{\rm{bin}}(i)\rangle$ and measurement probabilities recover $m(F_i)$. It then details quantum implementations of TBM’s credal and pignistic levels, notably via Boolean-algebra-based combination rules (BACR) and the $\\alpha$-junction, arguing that quantum entanglement enables exponential speedups for general mass functions while separable cases (poss-transferable mass functions) show no advantage. The work introduces contour-enhancement/revision methods (CER/CRR) for belief updates on quantum hardware and extends product-space operations (marginalization, vacuous/ballooning extensions, and generalized Bayesian theorem) to quantum circuits, enabling quantum realizations of VBS/GBT workflows. Overall, the paper argues that belief functions offer both interpretability and practical advantage for quantum AI models, suggesting new directions for evidential machine learning and quantum-inspired uncertainty representation in quantum computing.

Abstract

Dempster-Shafer structure is effective in classical settings for connecting set-valued hypotheses and representing structured ignorance, yet its practical use is limited by combination growth over focal sets and high conflict management. We observe a mathematical consistency between Dempster-Shafer structure and quantum superposition: elements of the power set form an orthogonal basis, and a basic probability assignment can be encoded as a normalized quantum state whose amplitudes respect mass value constraints. In this paper, we implement the information fusion and correction with Dempster-Shafer structure on quantum circuits, demonstrating that belief functions provide a more concise and effective alternative to Bayesian approaches within the quantum computing framework.Furthermore, by leveraging the unique characteristics of quantum computing, we propose several novel approaches for belief transfer. More broadly, this paper introduces a novel perspective on basic information representation in quantum AI models, proposing that belief functions are better suited than Bayesian approaches for handling uncertainty in quantum circuits.

Figures (10)

• Figure 1: Motivation of encoding mass function on quantum circuits.
• Figure 2: Implementation of MFQS with a $3$-qubit system.
• Figure 3: Implementation of belief functions on quantum circuits.
• Figure 4: Implementation of BACR on quantum circuits in Example \ref{['e1']}, where $q_{i_j}$ means the qubit which corresponds to $\omega_{i+1}$ in $m_j$, and $a_j$ means the $j$th ancilla qubit.
• Figure 5: Implementation of $\ket{m^{\cap,0.3}}$ in Example \ref{['ee3']}, where $q_{0_a},q_{1_a},q_{2_a}$ compose the quantum state of $m^{\cap,0.3}_{\emptyset}$ and $q_{3_a},q_{4_a},q_{5_a}$ compose the quantum state of $m^{\cap,0.3}_{\{\omega_1\}}$.

Theorems & Definitions (50)

• Definition 1
• Remark 1
• Remark 2
• Definition 2
• Theorem 1
• Proof 1
• Remark 3
• Remark 4
• Definition 3
• Definition 4