Quantum Data Structures and Encoding

Maria Schuld, Francesco Petruccione

Jincheng Zhang

This paper explores the application of topological information encoding as a novel approach to data compression and transmission, particularly within the context of complex quantum systems. Traditional data encoding methods often focus on representing data mathematically, while this work leverages the inherent properties of topological structures to achieve superior compression and transmission efficiency. We present a methodology that encodes quantum states directly into topological graphs, exploiting the graph's inherent structure to minimize redundancy and enhance data representation. The paper details the core mechanisms, the potential advantages of this approach, and preliminary results demonstrating its effectiveness in a simplified quantum context. The goal is to investigate a paradigm shift in data compression techniques, offering a more robust and potentially transformative solution for handling complex quantum data.

Andrew Vlasic, Anh T. Pham

Abstract Topological Data Analysis (TDA) is a well-established field derived to give insight into the geometric structure of real-world data. However, many methods in TDA are computationally intensive. The method that computes the respective Betti number has been shown to obtain a speed-up from translating the algorithm into a quantum circuit. The quantum circuit to calculate a particular Betti number requires a significant number of gates and, without a small record of data, is currently unable to be implemented on a NISQ-era processor. Given this NISQ-era restriction, a hybrid-method is proposed that calculates the Euclidean distance of the encoded data and computes the desired Betti number. This method is applied to a toy data set with different encoding techniques. The empirical results show the noise within the data is intensified with each encoding method as there is a clear change in the geometric structure of the original data, exhibiting information loss.

Adam Wesołowski, Dimitrios Thanos, Daniel Leykam +1 more

Many datasets encountered across a wide range of domains possess rich geometric and topological structure that is difficult to capture using conventional vector-based representations. Quantum machine learning offers the possibility of processing high-dimensional data in Hilbert spaces, but its practical success depends critically on how classical data is encoded into quantum states. We introduce \emph{quantum topological data encoding} (QTDE), a general framework for encoding topological information into quantum states via topology-driven quantum evolution. Our method generalises an existing topology-driven quantum encoding framework to higher-dimensional data. We test the proposed method on clique-complexes classification tasks, and provide preliminary evidence that topology-driven quantum representations can capture discriminative information beyond that available through direct comparisons of classical topological descriptors. The proposed quantum representations consistently outperform a baseline based on direct comparisons of the combinatorial Laplacians describing the underlying topological structure. We indicate several areas of application where the framework can be used to provide a more efficient and reliable data representation.

John A. Vaccaro, Yasuyoshi Mitsumori, Stephen Mark Barnett +4 more

Guang Lin, Toshihisa Tanaka, Qibin Zhao

Encoding classical data into quantum states is a central bottleneck in quantum machine learning: many widely used encodings are circuit-inefficient, requiring deep circuits and substantial quantum resources, which limits scalability on quantum hardware. In this work, we propose tensor network-based quantum data encoding (TNQE), a circuit-efficient framework built on structured unitary tensor network (TN) representations. TNQE first represents each classical input via a TN decomposition and then compiles the resulting tensor cores into an encoding circuit through two complementary core-to-circuit strategies. To make this compilation trainable while respecting the unitary nature of quantum operations, we introduce a unitary-aware constraint that parameterizes TN cores as learnable block unitaries, enabling them to be directly optimized and directly encoded as quantum operators. The proposed TNQE framework enables explicit control over circuit depth and qubit resources, allowing the construction of shallow, resource-efficient circuits. Across a range of benchmarks, TNQE achieves encoding circuits as shallow as $0.04\times$ the depth of amplitude encoding, while naturally scaling to high-resolution images ($256 \times 256$) and demonstrating practical feasibility on real quantum hardware.