Information and quantum information of deterministic data with applications to systems analysis

Farhad Farokhi

An alternative measure of information leakage for quantum encoding of classical data is defined. An adversary can access a single copy of the state of a quantum system that encodes some classical data and is interested in correctly guessing a general randomized or deterministic function of the data (e.g., a specific feature or attribute of the data in quantum machine learning) that is unknown to the security analyst. The resulting measure of information leakage, referred to as maximal quantum leakage, is the multiplicative increase of the probability of correctly guessing any function of the classical data upon observing measurements of the quantum state. Maximal quantum leakage is shown to satisfy the postprocessing inequality (i.e., applying a quantum channel reduces information leakage) and independence property (i.e., leakage is zero if the quantum state is independent of the classical data), which are fundamental properties required for privacy and security analysis. It also bounds accessible information. Effects of global and local depolarizing noise models on the maximal quantum leakage are established.

Lev B. Levitin

Fang‐Fang Du, Ling‐Hui Li, Chen Huang +2 more

Entanglement in high-dimensional systems plays a vital role in quantum information processing (QIP), as it extends the available Hilbert space, supports the manipulation of more intricate information, and broadens the scope of implementable quantum operations. In the paper, we present a deterministic scheme for the four-dimensional two-qudit Bell entangled state analysis for the four-photon system in the polarization degree of freedom, resorting to the weak cross-Kerr nonlinearity and some linear optical components. In our scheme, the analysis of four-dimensional Bell entangled state is constructed by three-photon Toffoli gate and quantum nondemolition detectors (QND 1 and QND 2 ). By phase shifts of the coherent beams, the 16 orthogonal four-dimensional two-qudit Bell states can be completely distinguished from each other. The proposal can be extended to the four-dimensional three-qudit entangled state analysis and n -qudit entangled state analysis. In addition, our deterministic scheme for high-dimensional entangled state analysis can be applied to high-dimensional quantum teleportation, including both uncontrolled and controlled quantum teleportation protocols. Moreover, our scheme can achieve high fidelity and efficiency, further broadening its applicability in high-dimensional QIP.

Igor Vasil'evich Volovich

Jincheng Zhang

Quantum computing offers unique computational capabilities, particularly in the realm of topological quantum information. This paper explores the application of adaptive geometry to analyze topological quantum information, investigating how quantum computation can enhance the efficiency of geometric topology. We develop a framework that leverages the inherent properties of quantum systems to model and understand topological quantum data structures. The core claim is that utilizing quantum computation allows for a significant improvement in the analysis and manipulation of geometric topology, ultimately leading to enhanced computational efficiency. The research will focus on establishing a methodology for quantifying and optimizing the impact of quantum algorithms on topological data analysis.

Jason Iaconis, Sayonee Ray, Samwel Sekwao +2 more

Topological data analysis (TDA) provides a powerful framework for extracting information about the shape of complex, unstructured data, but the classical cost of computing high dimensional topological features limits its application. Quantum algorithms for TDA offer a route around this bottleneck, yet existing approaches typically focus on exact or high precision Betti number estimation, making the regime for practical quantum advantage appear narrow. Here, we instead frame quantum TDA as a feature-extraction method for downstream data analysis by extracting low-order spectral information from the combinatorial Laplacian as a proxy for high-dimensional topology. We support this perspective from both the application and algorithmic sides. First, we show that higher-order TDA features improve predictive performance in two time-series applications: functional MRI analysis for neurodegenerative disease classification and financial time-series analysis for identifying market instability. Second, we develop a moment-based quantum algorithm and show that low-order moments, including the relative trace, are strongly correlated with high-dimensional Betti information, even when the relative Betti number is small. Finally, we present circuit constructions, resource estimates, quantum-classical crossover projections, and experimental results from a Barium development system similar to the forthcoming IonQ Tempo line, extracting Laplacian-derived observables from graph instances and quantitatively comparing them with exact Betti information. Together, these results establish quantum TDA as a practical approach for extracting topological features from classically challenging data