Quantum Data Encoding and Representations

• Quantum data encoding and representations are methods that map classical data—such as scalars, vectors, and images—into quantum states, enabling quantum information processing.
• These schemes, including basis, angle, amplitude, and continuous-variable encodings, balance resource efficiency, expressivity, and circuit complexity in quantum algorithms.
• By dictating feature map geometry and noise sensitivity, these techniques are critical for developing practical quantum machine learning, simulation, and hybrid classical-quantum workflows.

Quantum data encoding and representations refer to the suite of mathematical, algorithmic, and physical techniques by which classical data—scalars, vectors, high-dimensional arrays, images, or structured objects—are mapped into the state space of quantum systems for the purpose of quantum information processing, machine learning, and simulation. These encoding schemes determine the feature expressivity, information capacity, and resource requirements of quantum algorithms, with direct implications for quantum algorithmic performance, classical-quantum integration, and the viability of quantum machine learning and hybrid workflows.

1. Foundations and Theoretical Landscape

Quantum data encoding serves as the bridge between classical feature spaces (e.g., ℝⁿ, binary strings, structured tensors) and the complex projective Hilbert spaces utilized by quantum circuits. The encoding process is formalized as a map $E: x \mapsto |\psi(x)\rangle$ (pure-state) or $x \mapsto \rho(x)$ (mixed-state) that translates classical data $x$ into a quantum state suitable for further quantum processing. This abstraction layer is distinct from the actual data loading circuit, allowing the same mathematical encoding to be realized via multiple physical implementations or approximations (Agliardi et al., 2024).

Encoding choices directly impact the size and geometry of the induced feature manifold in Hilbert space, the separability of data classes, sensitivity to noise, and resource scaling in both qubit and gate count.

2. Core Encoding Schemes and Mathematical Formalisms

2.1 Basis (Computational) Encoding

All input bits (or small integers) are encoded as individual qubits in computational basis states: $|x\rangle = |x_{n-1}\ldots x_0\rangle$ Preparation requires a width equal to the data size and merely a layer of parallel $X$ gates; it provides no superposition or compression (Rath et al., 2023, Lang et al., 2024).

2.2 Angle (Rotation/Phase) Encoding

Real-valued data components are mapped onto single-qubit rotation gates (e.g., $R_y(2x_i)$) so that each qubit encodes a local geometric parameter: $$|\psi(x)\rangle = \bigotimes_{i=1}^{n} \left(\cos x_i |0\rangle + \sin x_i |1\rangle\right)$$ This encoding is resource efficient (one qubit per feature, circuit depth $O(1)$, if parallelized) and suitable for variational circuits, but expressivity is limited by its product state nature (Rath et al., 2023, Vlasic et al., 2022, Schuld et al., 2020).

2.3 Amplitude Encoding

A real or complex vector $x \in \mathbb{R}^N$ is encoded in the amplitudes of an $n$-qubit superposition: $$|\psi(x)\rangle = \sum_{i=0}^{N-1} x_i |i\rangle, \quad N = 2^n$$ Achieves exponential compression in qubits, but is costly to prepare (requires $O(N)$ to $O(N/\sqrt{\log N})$ gates or more, depending on method) and sensitive to noise (Pagni et al., 9 May 2025, Ashhab, 2021, Brunet et al., 2023, Mitarai et al., 2018).

2.4 Continuous-Variable Encodings: Displacement & Squeezing

Continuous-variable quantum computing (CVQC) admits infinite-dimensional encodings:

• Displacement encoding: $|\psi_{\mathrm{disp}}(x)\rangle = D(c x)|0\rangle$, where $D$ is a bosonic displacement.
• Squeezing encoding: $|\psi_{\mathrm{sq}}(x)\rangle = S(d x)|0\rangle$, where $S$ is a single-mode squeeze. Squeezing encodings increase feature manifold curvature, offering high expressivity with tunable trade-offs in learnability and resource costs, especially for high-dimensional problems (Rath et al., 9 Apr 2025).

2.5 Structured and Hybrid Encodings

• Matrix Product State (MPS): Data vector is factorized into local tensors, capturing area-law entanglement:

$$|\psi\rangle = \sum_{s_1,...,s_N} \mathrm{Tr}[A_1^{s_1}\cdots A_N^{s_N}] |s_1\cdots s_N\rangle$$

Efficiently realized in $O(n\,\chi)$ circuit depth, with bond dimension $\chi$ controlling truncation error. Optimal qubit-feature mapping strongly reduces truncation error and enhances downstream learning accuracy (Jeon et al., 2024, Green et al., 23 Feb 2025).

• Quantum-inspired Encodings: Instance-Level (ILS), Global Discrete (GDS), and Class Conditional Value (CCVS) strategies mimic quantum logic in classical preprocessing for efficient integration with classical ML pipelines (Rath et al., 15 Jun 2025).

3. Resource Analysis and Circuit Complexity

Preparation cost and scalability are explicit bottlenecks:

• Standard amplitude encoding circuits scale with $O(N)$ controlled rotations or $O(N/\sqrt{\log N})$ MCX gates in optimized schemes, using only $n+2$ qubits and well-suited to hardware with multi-controlled gate support (Pagni et al., 9 May 2025, Ashhab, 2021, Kim et al., 2024).
• Methods using dynamic-range enhanced angle encoding (significand/exponent separation) permit greater precision at reduced qubit overheads for large datasets (Sinha et al., 2022).
• MPS-based methods compress high-dimensional encodings into shallow linear-depth circuits, assuming favorable entanglement structure (Green et al., 23 Feb 2025, Jeon et al., 2024).
• In analog physical quantum processors, e.g., neutral-atom arrays, sparse geometric dot representations enable fully quantum-native, qubit-frugal data mapping bypassing digital circuit overhead altogether (Sharma et al., 20 Dec 2025).

4. Expressivity, Topology, and Learnability

The feature map induced by quantum encodings realizes nonlinear kernels in exponentially large Hilbert spaces. Metrics to characterize expressivity include:

• Manifold dimension (rank of the Jacobian $\partial|\psi(x)\rangle/\partial x$)
• Minimum pairwise Hilbert distances between encoded classes
• Topological invariants, e.g., persistent Betti numbers associated with the simplicial complex formed by the feature map images. Angle, amplitude, and IQP encodings induce notably different topological signatures, which impact downstream learning tasks such as clustering and classification (Vlasic et al., 2022).

Expressivity can be increased by circuit repetition or entangling gates (raising accessible Fourier ranks), but at increased cost and potential loss of generalization (sharp eigenvalue decay in the induced kernel operator) (Schuld et al., 2020, Rath et al., 9 Apr 2025).

5. Image and Structured Data Encodings

Several specialized schemes exist for multidimensional or structured classical data:

• Quantum Lattice & Phase Encoding: Use $N^2$ qubits for $N \times N$ images, high-fidelity and minimal depth but non-scalable in width (Kulkarni et al., 31 Jan 2025, Brunet et al., 2023).
• FRQI (Flexible Representation of Quantum Images): Maps pixel brightness into color qubit rotations, position encoded in a logarithmic number of qubits but incurs heavy gate depth for preparation (Lang et al., 2024, Kulkarni et al., 31 Jan 2025).
• QPIE/Amplitude Encoding: Direct amplitude map of normalized image tensors, qubit-efficient but challenging state-preparation and probabilistic readout (Brunet et al., 2023, Sharma et al., 20 Dec 2025).
• Analog sparse-dot representations: For Rydberg-atom platforms, images compressed to edge-following point clouds mapped into spatial quantum registers, with pruning optimizing both qubit count and matching fidelity (Sharma et al., 20 Dec 2025).

6. Applications, Empirical Performance, and Trade-offs

Quantum encodings serve as feature maps for hybrid quantum-classical machine learning, kernel-based classifiers, generative modeling, quantum principal component analysis, and structured data simulation. Empirical benchmarking reveals:

• For linear models (Logistic Regression, SVM) and kNN, classical or classical PCA features retain or exceed quantum-embedded accuracy.
• Quantum encodings improve classification metrics in ensemble/tree-based models where separability in Hilbert space is leveraged, subject to a trade-off with embedding and kernel evaluation costs (Rath et al., 9 Apr 2025, Rath et al., 2023, Rath et al., 15 Jun 2025).
• Squeezing encoding, at optimal parameter scaling, offers a practical compromise between expressivity and runtime.
• Fidelity-preserving, data-driven compression (FPQE) with amplitude-encoding substantially improves quantum neural network accuracy and structural similarity metrics for complex image data compared to classical dimensionality reduction (Lu et al., 19 Nov 2025).

7. Limitations, Robustness, and Future Directions

Critical challenges remain in quantum data encoding:

• Preparation costs impose strong limits on achievable quantum advantage for arbitrary data; sublinear-depth algorithms, graph-theoretic decompositions, and variational compression are active research frontiers (Pagni et al., 9 May 2025, Ashhab, 2021, Jeon et al., 2024).
• Robustness to device noise depends intimately on the encoded state’s location relative to noise-invariant subspaces. Only some encoding/noise combinations yield full or partial robustness, with systematic bounds available via fidelity analysis (LaRose et al., 2020).
• Quantum analog-digital conversions enable nonlinearity, amplitude thresholding, and hybrid algorithm design but often incur probabilistic preparation costs (Mitarai et al., 2018).
• As hardware matures, integration of exact and approximate data representations, tensor-network-based compressions, and quantum-inspired classically tractable surrogates will shape the landscape of practical quantum data encoding (Agliardi et al., 2024, Sharma et al., 20 Dec 2025, Rath et al., 15 Jun 2025).

Quantum data encoding and representation thus form a technically rich and rapidly evolving foundation within quantum information science, critically dictating the feasibility and scope of quantum advantage across scientific and industrial domains.