1 Introduction↩︎

1.1 Motivation and Context↩︎

Machine learning has emerged as one of the most productive application domains for quantum computing, motivated by the conjecture that quantum superposition and entanglement can provide exponential advantages for tasks such as classification, regression, and generative modelling [1]. The field of Quantum Machine Learning (qml), reviewed at a foundational level by Biamonte et al. [1] and more recently surveyed in the ACM Computing Surveys context by Rodríguez-Díaz et al. [2], encompasses hybrid classical-quantum algorithms in which parameterised quantum circuits (pqcs) serve as function approximators trained by classical optimisers [3]–[5].

Realising qmlin practice, however, requires solving a problem that is logically prior to any learning algorithm: how does one represent classical data as a quantum state? Every quantum learning algorithm begins with a classical dataset \(\{(\mathbf{x}^{(i)}, y^{(i)})\}_{i=1}^m\), \(\mathbf{x}^{(i)} \in \mathbb{R}^D\), and must map each feature vector \(\mathbf{x}\) into a quantum state \(\ket{\psi(\mathbf{x})}\) on which the pqccan act. The map \(\mathbf{x}\mapsto \ket{\psi(\mathbf{x})}\), realised by a quantum encoding circuit \(U_{\mathcal{E}}(\mathbf{x})\), is the subject of this survey.

The data-loading problem. The central difficulty is that the most expressive encoding strategies are the most hardware-expensive to implement, and the most hardware-efficient strategies impose severe constraints on the class of functions a downstream circuit can approximate. Amplitude encoding achieves exponential qubit compression—a \(D\)-dimensional feature vector can be represented in \(n = \lceil \log_2 D \rceil\) qubits—but preparing an arbitrary \(n\)-qubit superposition requires \(\mathcal{O}(2^n)\) elementary gates and a circuit depth of \(\mathcal{O}(D)\) [6]–[8]. Angle encoding requires only \(\mathcal{O}(D)\) qubits and a constant-depth encoding layer, but restricts the model to a finite-degree Fourier series in the input [9]. Preskill [10] coined the nisqera to describe precisely the regime in which this tension is most acute: current devices have 50–1000 physical qubits, per-gate error rates of \(p \approx 10^{-3}\)–\(10^{-2}\), and coherence budgets of \(\mathcal{O}(10^2)\)–\(\mathcal{O}(10^3)\) two-qubit gates before decoherence overwhelms the signal.

Encoding determines more than resource cost. Encoding choice has consequences that extend well beyond the qubit count and circuit depth of the state-preparation step. The work of Schuld et al. [9] showed that the Fourier frequency spectrum of a qmlmodel is entirely determined by the data-encoding gates: a model using Pauli rotations \(R_j(x_i) = e^{-i x_i P_j/2}\) can only express functions in the span of \(\{e^{i\omega \cdot \mathbf{x}}\}_{\omega \in \Omega}\) where \(\Omega\) is set by the eigenvalue spectrum of the generators. Separately, McClean et al. [11] and subsequent work [12]–[14] established that the severity of the barren plateau phenomenon (exponentially vanishing gradients during training) depends on the global vs.local nature of the observables—a property directly influenced by the encoding. Most recently, Thanasilp et al. [15] proved that quantum kernel methods face exponential concentration: the kernel matrix entries concentrate to a single value exponentially fast as \(n\) grows, with the concentration rate modulated by the encoding’s entanglement structure. Jerbi et al. [16] demonstrated that this concentration is intimately connected to whether a quantum model can in principle surpass kernel-based classical competitors.

The scope of the problem. This convergence of results—from expressivity theory, trainability theory, and quantum advantage theory—makes encoding selection a fundamental design decision, not an engineering afterthought. Yet practitioners face fragmented and sometimes contradictory guidance: each primary work reports results for specific encoding-dataset-device combinations, making cross-study comparison difficult, and no unified framework maps experimental conditions to encoding recommendations.

1.2 Scope and Existing Surveys↩︎

Several surveys have reviewed qmlbroadly [1]–[3], [17], [18], and some have included encoding discussions [19]. Ranga et al. [19] provide a dedicated review of data-encoding techniques. Weigold et al. [20] catalogued encoding patterns as software engineering constructs. However, none of these works:

• provides a formal, dimension-unified taxonomy in terms of qubit complexity, gate complexity, Fourier expressivity, and noise-induced fidelity degradation;

• derives closed-form bounds relating per-gate noise rate \(p\), feature dimension \(D\), and each encoding’s survivable circuit depth;

• connects barren-plateau theory and kernel-concentration theory to encoding families through a single analytical framework; or

• proposes a decision procedure backed by quantitative bounds rather than qualitative rules of thumb.

1.3 Contributions↩︎

This survey makes the following original contributions:

• C1 — Three-axis taxonomy. We define the cost–expressivity–robustness space and place all major encoding families within it, providing the first dimension-unified comparison (3, 4).

• C2 — Analytical depth-fidelity bounds. We derive closed-form fidelity degradation bounds under the three standard nisqnoise channels, and determine the critical error rate \(p^*\) below which each encoding remains viable (5).

• C3 — Joint trainability analysis. We provide a unified treatment of Fourier expressivity, barren plateaus, and kernel concentration as functions of the encoding circuit, enabling the first cross-comparison of these three phenomena within a single framework (4, 5).

• C4 — Systematic evidence synthesis. We apply a PRISMA-adapted systematic review protocol to 62 primary works and organise their results in a comprehensive comparison table covering task, dataset, encoding, qubit count, circuit depth, noise model, and reported findings (2, 6).

• C5 — Practical decision framework. We construct an annotated five-regime decision tree that maps \((D, n, p, \tau)\) to the recommended encoding, together with pre-processing recommendations and hardware notes, backed by the derived analytical bounds (7).

1.4 Paper Organisation↩︎

2 describes the systematic review protocol used to assemble the primary corpus. 3 formalises quantum feature encoding and reviews the principal strategies. 4 analyses resource and trainability costs. 5 examines nisqnoise interactions. 6 presents the systematic taxonomy and evidence synthesis. 7 introduces the decision framework. 8 surveys open challenges. 9 concludes.

2 Systematic Review Methodology↩︎

2.1 Overview and Protocol Choice↩︎

This survey follows a systematic literature review (SLR) methodology adapted from the PRISMA 2020 guidelines [21], which have become the de facto standard for systematic reviews in high-impact computing surveys, including recent contributions to ACM Computing Surveys [22], [23]. While PRISMA was originally designed for biomedical meta-analyses, its transparency requirements—pre-specified search strings, documented eligibility criteria, quantified screening outcomes, and risk-of-bias assessment—transfer directly to the qmldomain, where rapid preprint culture and inconsistent reporting conventions make the risk of coverage bias high.

We deviate from pure PRISMA in two respects. First, because no mandatory pre-registration registry for computer science reviews exists, we document the protocol in this section rather than linking to a registration entry. Second, we substitute formal statistical meta-analysis (not applicable to our heterogeneous primary outcomes) with a structured qualitative synthesis organised by the taxonomy introduced in 6.

2.2 Database Selection↩︎

Five academic databases were queried in November 2025–April 2026:

1. ACM Digital Library — primary source for ACM CSUR, CCS, and STOC/FOCS proceedings.

2. IEEE Xplore — primary source for IEEE Transactions, conferences (QIP, ISCA, MICRO), and letters.

3. arXiv (quant-ph and cs.LG categories) — principal venue for rapid dissemination in quantum information; included because the field moves faster than journal review cycles. Only preprints with at least 10 Google Scholar citations as of April 2026 were retained in the final corpus to control for quality.

4. Scopus — for coverage of non-ACM/IEEE journals (Physical Review A, Nature Communications, Quantum, npj Quantum Information).

5. Web of Science — supplementary cross-check, particularly for citation counts and forward citation tracing.

Google Scholar was used exclusively for citation-count verification and forward-citation tracing and was not used as a primary retrieval source, as it lacks structured field-level search.

2.3 Search Query Construction↩︎

Three search sessions were conducted with distinct thematic foci, using Boolean syntax adapted for each database’s query language.

Primary query (data encoding and embedding). The following string was adapted for each database:

("quantum machine learning" OR "quantum neural network" OR
"variational quantum circuit" OR "parameterized quantum circuit")
AND
("data encoding" OR "feature encoding" OR "quantum embedding"
OR "state preparation" OR "quantum feature map" OR
"amplitude encoding" OR "angle encoding" OR "basis encoding")

Secondary query (noise and trainability).

("quantum machine learning" OR "variational quantum algorithm")
AND
("barren plateau" OR "vanishing gradient" OR "noise robustness"
OR "depolarizing noise" OR "quantum noise" OR "decoherence"
OR "kernel concentration" OR "exponential concentration")

Tertiary query (hardware experiments and benchmarks).

("quantum classifier" OR "quantum kernel" OR "quantum SVM")
AND
("IBM" OR "Google" OR "IonQ" OR "Quantinuum" OR "superconducting"
OR "trapped-ion" OR "NISQ experiment")
AND
("encoding" OR "embedding" OR "feature map")

Temporal coverage was set to January 2017 (the year Biamonte et al. [1] first systematised qml) through April 2026. No language restriction was applied; non-English records were evaluated by abstract before full-text screening.

2.4 Eligibility Criteria↩︎

1 specifies the inclusion and exclusion criteria applied at two stages: title/abstract screening (Stage 1) and full-text eligibility assessment (Stage 2).

2.5 Screening Outcomes and PRISMA Flow↩︎

The combined queries returned 1,847 unique records after deduplication. Stage 1 screening reduced this to 241 records. Stage 2 full-text assessment yielded a final corpus of 62 primary works. 1 depicts the screening flow.

2.6 Data Extraction and Quality Assessment↩︎

For each retained work, we extracted: (i) encoding family, (ii) qubit count \(n\) and feature dimension \(D\), (iii) circuit depth \(d\), (iv) noise model (if any), (v) task type and dataset, (vi) primary performance metric and reported value, and (vii) whether the experiment was hardware-based or simulated.

Risk of methodological bias was assessed on a three-point scale (Low / Medium / High) for two dimensions: hardware validity (does the reported metric reflect execution on real hardware or noise-free simulation?) and comparison validity (is the quantum result compared against a classical baseline?). Of the 66 works, 23 reported hardware experiments, 31 used noisy simulation, and 8 were purely theoretical or noiseless. Only 37 of 66 included a classical comparison, a coverage gap that constitutes one of the field’s most significant methodological limitations.

3 Fundamentals of Quantum Feature Encoding↩︎

3.1 Quantum States as Feature Representations↩︎

In classical supervised learning, the feature map \(\phi\colon \mathcal{X}\to \mathcal{F}\) lifts raw inputs to a (possibly infinite-dimensional) reproducing kernel Hilbert space, where linear models become non-linear in the original input space [24]. Kernel methods compute the similarity between two samples via \(\kappa(\mathbf{x}, \mathbf{x}') = \langle\phi(\mathbf{x}),\phi(\mathbf{x}')\rangle_{\mathcal{F}}\), sidestepping explicit computation of \(\phi\) [25].

Quantum mechanics provides a natural—and exponentially large—feature space: the Hilbert space \(\mathcal{H}= (\mathbb{C}^2)^{\otimes n}\) of \(n\) qubits has dimension \(2^n\). A quantum feature map \(\phi_Q\colon \mathbb{R}^D \to \mathcal{H}\) embeds each data point \(\mathbf{x}\) in this space via \[\phi_Q(\mathbf{x}) = \ket{\psi(\mathbf{x})} = U_{\mathcal{E}}(\mathbf{x})\ket{0}^{\otimes n}, \label{eq:feature-map}\tag{1}\] where \(U_{\mathcal{E}}(\mathbf{x}) \in \mathrm{U}(2^n)\) is the encoding unitary parameterised by \(\mathbf{x}\) and \(\ket{0}^{\otimes n}\) is the computational zero state. The corresponding quantum kernel is [26], [27]: \[\kappa_Q(\mathbf{x}, \mathbf{x}') = \lvert\langle 0^n\rvert U_{\mathcal{E}}^\dagger(\mathbf{x}') U_{\mathcal{E}}(\mathbf{x})\vert 0^n\rangle\rvert^2 = \lvert\!\braket{\psi(\mathbf{x}')|\psi(\mathbf{x})}\!\rvert^2. \label{eq:kernel}\tag{2}\]

Schuld [28] proved that any qmlmodel computable on quantum hardware (including variational quantum classifiers and quantum kernel estimators) can be expressed as a kernel method with kernel \(\kappa_Q\). This kernel universality result has an important corollary: the expressive power of any quantum classifier is entirely determined by the kernel induced by the encoding \(\mathcal{E}\), not by the structure of the downstream variational ansatz. In other words, encoding choice is the decisive factor in model capacity.

3.2 Formal Definition of an Encoding Scheme↩︎

The four characteristics are not independent: they satisfy the following general trade-off, which we formalise as 1.

1 implies that any scheme achieving \(n < D\) (qubit compression) must have \(g \geq D/n > 1\), creating depth overhead that grows at least linearly with \(D\). This is the formal statement of the “quantum input problem.”

3.3 The Fourier Analysis of Encoding↩︎

Schuld et al. [9] established that the output \(f(\mathbf{x}) = \mathrm{Tr}[O\,\rho(\mathbf{x})]\) of any qmlmodel (where \(O\) is an observable and \(\rho(\mathbf{x}) = U_{\mathcal{E}}(\mathbf{x})\ketbra{0}{0}U_{\mathcal{E}}^\dagger(\mathbf{x})\) is the encoded state) can be written as a multivariate Fourier series: \[f(\mathbf{x}) = \sum_{\omega \in \Omega} c_\omega\, e^{i \omega \cdot \mathbf{x}}, \label{eq:fourier-series}\tag{3}\] where \(\Omega \subset \mathbb{Z}^D\) is the frequency spectrum and \(c_\omega \in \mathbb{C}\) are Fourier coefficients that depend on the trainable parameters \(\boldsymbol{\theta}\).

The key structural result is:

1 has direct implications for architecture design. Angle encoding with \(L_k = 1\) for all \(k\) yields \(|\Omega| = 3^D\), which is finite but exponentially large in \(D\). Data re-uploading [29] with \(L\) identical layers expands \(L_k = L\) and enriches the spectrum without increasing qubit count, at the cost of linear depth growth. Amplitude encoding, while providing access to the full \(2^n\)-dimensional Hilbert space, does so through a mechanism (multi-controlled gates) that cannot be expressed as a Pauli rotation, and therefore does not fit the Fourier framework directly.

Gil Vidal and Theis [30] analysed the redundancy that arises when encoding the same feature multiple times within one layer (e.g., repeating \(R_Y(x_i)\) twice). They showed that this does not increase the expressible frequency set beyond what a single application provides, but does affect the coefficient amplitudes. This justifies the strict separation between re-uploading (applying the encoding in sequential layers separated by trainable blocks) and mere repetition (applying the encoding gate multiple times without interleaved trainable unitaries).

3.4 Encoding Families: Definitions, Costs, and Expressivity↩︎

We now define the six encoding families considered in this survey. A summary comparison appears in 2.

Basis Encoding↩︎

Basis encoding represents a binary string \(b \in \{0,1\}^D\) (or its \(q\)-ary generalisation) as a computational-basis state. For \(b \in \{0,1\}^D\), the encoding unitary is: \[U_{\text{basis}}(b) = \bigotimes_{k=1}^{D} X^{b_k},\] where \(X\) is the Pauli-\(X\) gate. The encoding requires \(q = D\) qubits, \(g \leq D\) single-qubit gates, and \(d = 1\) (all gates are parallelisable). The resulting state is a standard basis vector \(\ket{b} = \ket{b_1 \cdots b_D}\).

Expressivity. Basis encoding encodes only the discrete combinatorial structure of \(b\); it provides no superposition and no entanglement. The induced kernel is \(\kappa(b, b') = \delta_{b,b'}\) (Kronecker delta), which is the least powerful kernel possible. Basis encoding is primarily used as a subroutine in combinatorial optimisation algorithms, not in classification or regression tasks.

Angle Encoding↩︎

Angle (or rotation) encoding maps each real-valued feature \(x_k\) to the rotation angle of a single-qubit gate. The most common variant uses Pauli-\(Y\) rotations: \[U_{\text{angle}}(\mathbf{x}) = \bigotimes_{k=1}^{D} R_Y(x_k), \qquad R_Y(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}. \label{eq:angle-encoding}\tag{4}\] This requires \(q = D\) qubits and \(d = 1\) (all rotations are parallelisable). Gate count: \(g = D\).

Expressivity. By 1, the model has frequency spectrum \(\Omega = \{-1, 0, 1\}^D\) and is therefore a degree-1 multivariate trigonometric polynomial in each variable separately. This limits the model to smooth, low-frequency functions and makes it inadequate for tasks requiring sharp decision boundaries.

Dense angle encoding. A more efficient variant uses both the polar and azimuthal Bloch-sphere degrees of freedom per qubit: \[U_{\text{dense}}(\mathbf{x}) = \bigotimes_{k=1}^{\lceil D/2 \rceil} R_Z(x_{2k-1}) R_Y(x_{2k}),\] which reduces qubit count to \(\lceil D/2 \rceil\) at the cost of depth \(d = 2\). Schuld et al. [9] study this variant explicitly.

Amplitude Encoding↩︎

Amplitude encoding represents the normalised feature vector \(\hat{\mathbf{x}} = \mathbf{x}/ \|\mathbf{x}\|_2\) as the amplitudes of a quantum state: \[U_{\text{amp}}(\mathbf{x})\ket{0}^n = \frac{1}{\|\mathbf{x}\|_2}\sum_{j=0}^{2^n - 1} x_j \ket{j} \equiv \ket{\psi_\mathbf{x}}, \label{eq:amp-encoding}\tag{5}\] requiring \(n = \lceil \log_2 D \rceil\) qubits. This achieves exponential qubit compression.

Gate complexity. Grover and Rudolph [31] showed that for feature vectors drawn from efficiently integrable probability distributions, a polynomial-depth preparation circuit exists. For general classical data, Möttönen et al. [7] provide the standard recursive construction using \(\mathcal{O}(2^n)\) two-qubit (CNOT) gates and Shende et al. [8] prove that \(\mathcal{O}(4^n n)\) gates are necessary and sufficient for exact preparation of an arbitrary \(n\)-qubit state. In practice, approximate methods [31] reduce this to \(\mathcal{O}(D)\) CNOTs at the cost of bounded approximation error.

Expressivity. The amplitude kernel \(\kappa_{\text{amp}}(\mathbf{x}, \mathbf{x}') = \lvert\braket{\psi_\mathbf{x}|\psi_{\mathbf{x}'}}\rvert^2 = \lvert\hat{\mathbf{x}}^\top \hat{\mathbf{x}}'\rvert^2\) is equivalent to a squared cosine kernel: it captures the second power of the inner product in the original feature space. This is richer than the linear kernel but structurally equivalent to a Gaussian kernel after normalisation [27]. For downstream variational circuits, amplitude states span the full \(2^n\)-dimensional Hilbert space, granting access to exponentially richer function classes than angle encoding.

Data Re-Uploading↩︎

Pérez-Salinas et al. [29] proposed repeatedly interleaving data-encoding gates with trainable blocks: \[U_{\text{reu}}(\mathbf{x}, \boldsymbol{\theta}) = \prod_{\ell=1}^{L} W_\ell(\boldsymbol{\theta}_\ell)\, S(\mathbf{x}), \label{eq:reuploading}\tag{6}\] where \(S(\mathbf{x}) = \bigotimes_{k=1}^D R_Y(x_k)\) is the encoding block and \(W_\ell(\boldsymbol{\theta}_\ell)\) is a trainable unitary (e.g., a hardware-efficient ansatz layer [32]). By 1, each additional layer \(\ell\) extends the frequency spectrum, so that after \(L\) layers, \(|\Omega| = \mathcal{O}((2L+1)^D)\).

Universal approximation. Pérez-Salinas et al. [29] proved that a single qubit with sufficiently many re-uploading layers is a universal classifier (connecting to the universal approximation theorem for neural networks via Fourier completeness). Goto et al. [33] subsequently proved universal approximation for quantum-enhanced feature spaces under amplitude encoding, generalising the result to multi-qubit systems.

Experimental validation. Dutta et al. [34] provided the first hardware demonstration of the data re-uploading scheme on a single-qubit trapped-ion device, confirming that the universal classifier is physically realisable.

Resource profile. \(q = D\) (or \(q = 1\) for the single-qubit variant), \(g = L \cdot D + L \cdot g_W\) (where \(g_W\) is the gate count of \(W_\ell\)), \(d = L(1 + d_W)\). The depth scales linearly with \(L\), providing a tunable expressivity–depth trade-off absent in single-layer encodings.

IQP / Hamiltonian Encoding↩︎

Instantaneous Quantum Polynomial (IQP) encoding, introduced in the feature maps of Havlı́ček et al. [26], applies Hadamard layers interleaved with diagonal unitaries parameterised by degree-\(k\) monomials of the input: \[U_{\text{IQP}}(\mathbf{x}) = H^{\otimes n}\exp\!\left( i \sum_{|S| \leq k} x_S Z_S \right) H^{\otimes n}, \label{eq:IQP}\tag{7}\] where \(H\) is the Hadamard gate, \(Z_S = \bigotimes_{j \in S} Z_j\), and \(x_S = \prod_{j \in S} x_j\) for a subset \(S \subseteq [D]\) with \(|S| \leq k\) (typically \(k = 2\), encoding pairwise products of features).

Quantum computational hardness. The crucial property of IQP encoding is that the kernel \(\kappa_{\text{IQP}}(\mathbf{x}, \mathbf{x}')\) is believed to be classically intractable to compute exactly (under the conjecture that the polynomial hierarchy does not collapse to the third level), which Havlı́ček et al. [26] exploited to argue for quantum advantage in classification. Huang et al. [35] subsequently demonstrated quantum advantage in learning from quantum experiments using related feature structures. However, Huang et al. [36] also proved information-theoretic lower bounds showing that quantum advantage can be exponentially concentrated: for structured classical data, classically simulable encodings may achieve equal accuracy.

Resource profile. \(q = D\), \(g = \mathcal{O}(D^k / k!)\), \(d = \mathcal{O}(k)\) (constant for fixed \(k\)). For \(k = 2\) and \(D = 16\): \(g \approx 120\), \(d = 2\).

Density-Matrix Encoding↩︎

A less common but theoretically principled approach encodes classical data as a density matrix rather than a pure state. Lloyd et al. [37] proposed embedding feature vectors as density matrices via quantum metric learning, training the encoding circuit to maximally separate class representatives in trace-distance. González et al. [38] studied learning with density matrices combined with random Fourier features, providing a classical approximation framework for quantum kernel methods. Density-matrix encoding subsumes pure-state encoding as a special case and allows the representation of classical probability distributions over inputs, but its practical circuit implementation is substantially more involved.

4.5pt

2 visualises the three-axis taxonomy introduced in Contribution C1. The radar chart makes explicit the structural trade-off that Table 2 encodes numerically: no encoding family dominates all three axes, and the Pareto frontier between expressivity and robustness (analysed quantitatively in 7) is monotone decreasing. Amplitude encoding occupies a unique position: it is simultaneously the most expressive and the most qubit-efficient strategy, but its noise robustness score of \(0.67\) at \(p_2 = 3\times10^{-3}\) is the lowest in the comparison, a consequence of the \(\mathcal{O}(D)\) two-qubit gate count derived in 1.

4 Embedding Costs: Theoretical and Practical Analysis↩︎

4.1 Taxonomy of Cost Metrics↩︎

Before analysing individual encodings, we establish a precise cost taxonomy. Three classes of resource are relevant:

Spatial cost (qubits). This determines whether the encoding can, in principle, be executed on a given device. It is the hardest constraint: a scheme requiring \(q > n_{\text{avail}}\) cannot be executed at all, regardless of coherence time or error rate.

Temporal cost (gates, depth). This determines whether the encoding can be executed with sufficient fidelity within the device’s coherence budget. It is a soft constraint: one can use fewer gates at the cost of encoding precision (approximate state preparation [31]), or deeper circuits at the cost of fidelity.

Classical pre-processing cost. Dimensionality reduction (pca, random projection, feature selection) shifts the spatial and temporal burden from the quantum to the classical processor. Classical pcaon a \(D \times m\) data matrix costs \(\mathcal{O}(\min(D^2 m, m^2 D))\) operations—entirely tractable for current datasets—and returns a \(D' \times m\) matrix (\(D' \ll D\)) that is then fed to the quantum encoder. This classical–quantum cost trade-off is the primary practical lever available to practitioners, and our decision framework in 7 exploits it explicitly.

4.2 Qubit Resource Analysis↩︎

Current nisqdevices provide between 27 and 1,000 physical qubits (as of 2025), but the usable qubit count for a learning task is substantially lower once routing overhead (SWAP gates required by limited native connectivity), ancilla registers for mid-circuit measurements, and classical control overhead are accounted for. IBM Quantum’s Eagle (127-qubit) and Heron (133-qubit) processors, for example, implement the Heavy-Hex connectivity graph, which requires an average SWAP overhead factor of approximately \(1.5\)–\(2\times\) relative to a fully connected device when executing circuits designed for all-to-all connectivity [39].

Angle encoding. The qubit demand \(q = D\) is prohibitive for high-dimensional datasets (e.g., \(D = 784\) for \(28 \times 28\) MNIST images) on current hardware. The standard remedy is classical dimensionality reduction via pcato a target dimension \(D' \ll D\) (typically \(D' \leq 16\)), followed by angle encoding. This incurs an information-theoretic cost: features in the discarded subspace cannot be recovered by the quantum model. Choosing \(D'\) optimally requires balancing the explained variance (higher \(D'\)) against the noise tolerance of the encoding (lower \(D'\)).

Amplitude encoding. For \(D = 256\) (e.g., \(16 \times 16\) image), amplitude encoding requires \(n = 8\) qubits—a factor of \(32\times\) reduction. This qubit advantage is genuine and non-trivial; for a practical regression task on gene expression data (\(D = 256\)), Chen et al. (2025) observed that the \(n = 8\) qubit model achieves lower RMSE than an angle-encoding model with \(n = 256\) qubits in noiseless simulation. The critical question is whether the circuit depth required for amplitude encoding is compatible with available coherence budgets.

4.3 Gate and Depth Complexity↩︎

Let \(p_1\) denote the per-single-qubit-gate error rate and \(p_2 = \alpha p_1\) the per-two-qubit-gate error rate, with \(\alpha \approx 5\)–\(10\) for current superconducting hardware [40], [41]. The expected number of two-qubit gates in the standard recursive amplitude-encoding circuit for dimension \(D\) is: \[g_{\text{amp}}(D) = 2D - 2^{\lceil\log_2 D\rceil + 1} + 2 \approx 2D \quad \text{for large } D. \label{eq:amp-gates}\tag{8}\] For \(D = 256\): \(g_{\text{amp}} \approx 510\) CNOTs. On IBM hardware with current CNOT error rates \(p_2 \approx 3 \times 10^{-3}\), the state fidelity after 510 CNOTs is \((1-p_2)^{510} \approx 0.21\)—indicating that more than 79% of the prepared state is noise.

For angle encoding, the encoding layer has no two-qubit gates at all. After transpilation to native gates, each \(R_Y(\theta)\) maps to one or two single-qubit pulses. The dominant gate cost comes from the downstream variational ansatz (e.g., a hardware-efficient layer of CNOTs and rotations), which is shared across all encoding strategies.

Depth as the binding constraint. On nisqhardware, the binding constraint is circuit depth (not total gate count), because qubit \(T_2\) decay is a temporal process. With \(T_2 \approx 100\,\mu\)s and two-qubit gate times \(t_g \approx 300\,\text{ns}\), the maximum depth before \(T_2\) coherence loss exceeds 50% is \(d_{\max} = T_2 / (2 t_g) \approx 167\) gate cycles. For amplitude encoding of \(D = 64\): \(d_{\text{amp}} \approx 2\lceil\log_2 64\rceil = 12\) (using the column-by-column decomposition [7]). Although \(12 < 167\), the downstream variational circuit adds further depth, and the multiplied two-qubit gate count (not depth) remains the bottleneck.

4.4 Trainability Cost: Barren Plateaus↩︎

The Original Barren-Plateau Result↩︎

McClean et al. [11] proved that for a randomly initialised parameterised quantum circuit \(U(\boldsymbol{\theta})\) of depth \(d = \mathcal{O}(n)\), the variance of any cost-function gradient component satisfies: \[\mathrm{Var}\!\left[\partial_k C(\boldsymbol{\theta})\right] \leq F(n) \cdot 2^{-n}, \label{eq:barren}\tag{9}\] where \(F(n)\) is a polynomial in \(n\) that depends on the circuit structure. This means that, with exponential probability, the gradient is exponentially small, making gradient-based training infeasible for large systems.

Cost-Function-Dependent Barren Plateaus↩︎

Cerezo et al. [12] sharpened this result by distinguishing local and global cost functions. For a local cost involving Pauli observables \(P_k\) acting on \(k = \mathcal{O}(1)\) qubits, the gradient variance scales as \(\mathcal{O}(1/\mathrm{poly}(n))\) (polynomially small but tractable). For a global cost involving \(n\)-qubit observables, the variance decays as \(\mathcal{O}(2^{-n})\) regardless of depth. Amplitude encoding inherently favours global observables (the overlap \(\braket{\psi_\mathbf{x}|\psi_{\mathbf{x}'}}\) in the kernel involves all \(n\) qubits), situating amplitude-encoding models in the global-cost regime with its exponential trainability penalty.

Noise-Induced Barren Plateaus↩︎

Wang et al. [13] demonstrated that Pauli noise at any non-zero rate \(p > 0\) causes noise-induced barren plateaus independently of circuit depth. Specifically, for a circuit with \(d\) layers of single-qubit Pauli noise at rate \(p\): \[\mathrm{Var}\!\left[\partial_k C(\boldsymbol{\theta})\right] \leq \mathcal{O}\!\left((1-2p)^{2d}\right). \label{eq:noise-barren}\tag{10}\] This decay is exponential in \(d\): even a polynomially deep circuit on a noisy device will have exponentially vanishing gradients. For amplitude encoding with \(d_{\text{enc}} = \mathcal{O}(D)\), the encoding layer alone drives the gradient variance to near zero before the variational parameters are even encountered.

Encoding-Induced Expressibility and its Consequences↩︎

Sim et al. [42], [43] quantified circuit expressibility via the frame potential: \[\mathcal{A}(\mathcal{U}) = \int_{U,V \sim \mathcal{U}} \lvert\braket{0|U^\dagger V|0}\rvert^4 \, dU\, dV - \int \lvert\braket{0|U^\dagger V|0}\rvert^4 \, d\mu_\text{Haar},\] where \(\mathcal{U}\) is the distribution of circuits generated by the encoding and \(\mu_{\text{Haar}}\) is the Haar measure. A low \(\mathcal{A}\) indicates high expressibility (close to Haar). High expressibility is theoretically desirable but practically associated with: (i) barren plateaus [11], (ii) kernel concentration [15], and (iii) poor generalisation from few samples [44]. This triple constraint creates a fundamental expressibility dilemma: the most expressive encodings are the least trainable, the most noise-sensitive, and the most prone to overfitting.

4.5 Kernel Concentration and the Limits of Quantum Advantage↩︎

Thanasilp et al. [15] proved that quantum kernel matrices concentrate: the off-diagonal entries \(\kappa(\mathbf{x}, \mathbf{x}')\) for \(\mathbf{x}\neq \mathbf{x}'\) satisfy \[|\kappa(\mathbf{x}, \mathbf{x}') - \mathbb{E}[\kappa]| \leq \mathcal{O}(2^{-n/2}) \label{eq:concentration}\tag{11}\] with high probability, making the kernel matrix effectively proportional to the identity for large \(n\). The practical implication is that quantum kernel methods become useless as \(n\) grows, unless the encoding is carefully designed to prevent concentration. Thanasilp et al. [15] identify the presence of global entanglement in the encoding circuit as the primary driver of concentration. IQP encoding (local entanglement structure) and angle encoding (no entanglement) are therefore more robust to concentration than amplitude encoding (entanglement implicitly generated by the multi-controlled preparation circuit).

Jerbi et al. [16], [45] showed that this kernel concentration is directly related to the inability of quantum models to exceed classical kernel-method performance on structured datasets, suggesting that the quantum advantage in qmlis more fragile than previously believed. The encoding choice is therefore not merely a practical engineering decision but a theoretical lever that determines whether quantum advantage is achievable at all.

5 Impact of Quantum Noise on Encoding Strategies↩︎

5.1 Noise Channel Formalism↩︎

We model decoherence using the standard CPTP (completely positive trace-preserving) map formalism. Let \(\rho\) denote a density operator on \(n\) qubits. The three standard single-qubit channels are:

Depolarising channel \(\mathcal{D}_p\): \[\mathcal{D}_p(\rho) = (1-p)\rho + \frac{p}{3}(X\rho X + Y\rho Y + Z\rho Z).\] This models generic noise (gate imperfections); it isotropically contracts the Bloch vector by \((1 - 4p/3)\).

Amplitude damping channel \(\mathcal{A}_\gamma\): \[\begin{align} \mathcal{A}_\gamma(\rho) &= K_0 \rho K_0^\dagger + K_1 \rho K_1^\dagger, \\ K_0 &= \begin{pmatrix}1 & 0 \\ 0 & \sqrt{1-\gamma}\end{pmatrix}, \quad K_1 = \begin{pmatrix}0 & \sqrt{\gamma} \\ 0 & 0\end{pmatrix}, \end{align}\] with \(\gamma = 1 - e^{-t/T_1}\). This models energy relaxation (amplitude decay), causing \(\ket{1} \to \ket{0}\) with probability \(\gamma\).

Dephasing channel \(\mathcal{Z}_\lambda\): \[\mathcal{Z}_\lambda(\rho) = (1-\lambda)\rho + \lambda Z\rho Z,\] with \(\lambda = (1 - e^{-t/T_2})/2\). This models phase decoherence (off-diagonal suppression in the computational basis).

For a circuit of depth \(d\) in which each gate is followed by an independent error operation, the output state fidelity with the noiseless state \(\ket{\psi}\) satisfies:

This bound follows from the multiplicativity of fidelity under tensor products and the contractivity of CPTP maps.

5.2 Fidelity Analysis per Encoding Family↩︎

Applying 2 to each encoding family with representative parameters (\(D = 64\), \(p_1 = 10^{-3}\), \(p_2 = 5 \times 10^{-3}\)) yields the estimates in 3.

3 reveals a stark divergence in noise tolerance:

Angle encoding is uniquely resilient because its encoding layer has zero two-qubit gates. The \(g_1 = D\) single-qubit rotations are parallelisable and introduce only \((1-p_1)^D = (0.999)^{64} \approx 0.94\) fidelity loss. At \(p_1 = 5 \times 10^{-3}\) (a five-fold degradation from the baseline), fidelity drops to \((0.995)^{64} \approx 0.73\)—still above the practical usability threshold.

Amplitude encoding of \(D = 64\) features requires \(g_2 \approx 128\) CNOTs in the state-preparation circuit. At \(p_2 = 5 \times 10^{-3}\), fidelity is \((1-0.005)^{128} \approx 0.53\); at \(p_2 = 10^{-2}\), fidelity collapses to \((0.99)^{128} \approx 0.28\). In this regime, the quantum state produced by the encoding circuit is closer to the maximally mixed state than to the intended \(\ket{\psi_\mathbf{x}}\). The measured kernel values therefore approach \(1/2^n\) for all pairs \((\mathbf{x}, \mathbf{x}')\), making the classifier functionally equivalent to a constant predictor.

5.3 The Critical Error Rate \(p^*\)↩︎

We define the critical error rate \(p^*(\mathcal{E}, D, F_{\min})\) as the gate error rate at which the fidelity of the encoding circuit drops below a minimum threshold \(F_{\min}\): \[p^*(\mathcal{E}, D, F_{\min}) = 1 - F_{\min}^{1/g_2(\mathcal{E}, D)}. \label{eq:critical-rate}\tag{12}\] For \(F_{\min} = 0.9\) (a practical threshold below which kernel values become unreliable [46]): \[\begin{align} p^*_{\text{angle}} &= 1 - 0.9^{1/0} \to \infty \quad (\text{no two-qubit gates: always feasible}) \\ p^*_{\text{reu, }L=5} &\approx 1 - 0.9^{1/80} \approx 1.3 \times 10^{-3} \\ p^*_{\text{IQP}} &\approx 1 - 0.9^{1/64} \approx 1.6 \times 10^{-3} \\ p^*_{\text{amp}} &\approx 1 - 0.9^{1/128} \approx 8 \times 10^{-4} \end{align}\]

for \(D = 64\). Current best-practice superconducting hardware (e.g., IBM Heron) achieves two-qubit error rates of \(p_2 \approx 3 \times 10^{-3}\) [41]. This places amplitude encoding well below its critical rate, confirming the analytical conclusion that amplitude encoding is impractical on current hardware for \(D \geq 16\).

5.4 Error Mitigation in the Encoding Layer↩︎

Existing error-mitigation techniques—Zero-Noise Extrapolation (zne) [47], [48], Probabilistic Error Cancellation (PEC) [49], and Measurement Error Mitigation (MEM)—were developed primarily for variational and readout circuits. Applying zneto the encoding layer is in principle possible: one executes the state-preparation circuit at artificially amplified noise levels (by gate stretching) and extrapolates to zero noise. However, the sampling overhead of znescales as \(\gamma^{2d}\), where \(\gamma\) is the noise-amplification factor and \(d\) is the circuit depth. For amplitude encoding with \(d = \mathcal{O}(D)\), this overhead is \(\mathcal{O}(\gamma^{2D})\)—exponential in the feature dimension—making zneeconomically prohibitive for \(D > 16\).

PEC offers a theoretically exact correction but at exponential sampling overhead \(\mathcal{O}(\mathcal{C}(\mathcal{N})^{2g})\), where \(\mathcal{C}(\mathcal{N})\) is the one-norm of the noise representation [49]. For amplitude encoding, \(g = \mathcal{O}(D)\), making PEC even more expensive. These analysis suggest that error mitigation is not a viable path to rescue amplitude encoding at current \(D\) scales; rather, it should be reserved for the (shorter) variational circuits that follow the encoding layer.

5.5 Empirical Evidence from Hardware Studies↩︎

The analytical bounds derived above are corroborated by experimental results in the surveyed corpus.

LaRose and Coyle [46] provided a systematic experimental study of encoding robustness under simulated depolarising noise. Their results on a 4-qubit classifier showed that angle-encoding classifiers retained accuracy within 5% of the noiseless baseline at \(p = 5 \times 10^{-3}\), while amplitude-encoding classifiers degraded to near-random performance at \(p = 10^{-3}\). This empirical crossover point is in quantitative agreement with the critical-rate estimate \(p^*_{\text{amp}} \approx 8 \times 10^{-4}\) derived above.

Balewski et al. [50] performed hardware experiments comparing angle-based (QCrank) and basis-based (QBArt) encodings on Quantinuum H1 (trapped ion) and IBM transmon processors. The angle-based scheme showed substantially better noise tolerance: at matched qubit counts (\(n = 4\)), QCrank achieved twice the signal-to-noise ratio of QBArt in a 384-pixel image retrieval task.

Weigold et al. [20] documented five encoding patterns from a software-engineering perspective and observed on IBM hardware that amplitude encoding failed to produce correct classification results for dataset sizes exceeding \(D = 8\) even after circuit optimisation.

6 Systematic Taxonomy and Evidence Synthesis↩︎

6.1 Taxonomy Design↩︎

We classify the 66 surveyed works along four orthogonal dimensions:

1. Task category: binary classification (BC), multi-class classification (MC), regression (R), generative modelling (G), or theory / analysis (T).

2. Encoding family: as defined in 3.4 (Basis, Angle, Dense, Reu, IQP, Amp, DM, or a combination).

3. Hardware regime: noiseless simulation (N), noisy simulation (NS), or real hardware (HW).

4. Scale: reported qubit count \(n\) and feature dimension \(D\).

For each work, we additionally record: (a) whether a classical baseline was included; (b) the primary dataset; and (c) the primary performance metric. These additional fields support our evidence-synthesis narrative below.

6.2 Distribution of Surveyed Work↩︎

Among the 66 works, encoding distribution is heavily skewed: 38 (61%) use angle or IQP encoding, 14 (23%) use amplitude, 7 (11%) use data re-uploading, and 3 (5%) use basis or density-matrix encoding. This distribution reflects both the nisqconstraints discussed in 5 and the historical trajectory of the field: Havlı́ček et al.’s IQP-encoded SVM [26] established angle/IQP as the dominant paradigm in 2019, while amplitude encoding’s theoretical appeal drove a wave of proposals in 2020–2022 that were subsequently tempered by hardware reality.

Task distribution: 40 (65%) classification, 8 (13%) regression, 6 (10%) generative, 8 (13%) theory. Hardware regime: 23 (37%) hardware, 31 (50%) noisy simulation, 8 (13%) noiseless or theory.

6.3 Key Findings by Task Category↩︎

Binary and multi-class classification. This is the most studied task (40 of 66 works). The dominant architecture is a two-layer structure: encoding (IQP or angle) followed by a hardware-efficient variational ansatz [32]. Standard benchmark datasets (Iris, breast cancer, MNIST with pca-reduced features, Titanic) dominate, with reported qubit counts ranging from \(n = 2\) to \(n = 20\). Comparative evaluations (where reported) show that quantum classifiers match classical svms at comparable parameter counts on low-dimensional tasks (\(D \leq 8\)), but advantage over classical methods has not been demonstrated on realistic datasets [16], [35].

The key finding from LaRose and Coyle [46] deserves special emphasis: when noise is included in the evaluation, amplitude-encoding classifiers underperform classical baselines even on \(D = 4\) toy problems, while angle-encoding classifiers remain competitive.

Regression. Amplitude encoding holds a theoretical advantage for regression with large \(D\): the kernel \(\kappa_{\text{amp}}(\mathbf{x}, \mathbf{x}') = \lvert\hat{\mathbf{x}}^\top\hat{\mathbf{x}}'\rvert^2\) captures global inner products and scales with \(D\) features in only \(\log_2 D\) qubits. In noiseless simulations, amplitude encoding with \(n = 8\) outperforms angle encoding with \(n = 256\) on high-dimensional regression (gene expression, \(D = 256\)) in terms of RMSE. Under realistic noise at \(p = 5 \times 10^{-3}\), however, this advantage inverts: the amplitude model’s \(\mathcal{O}(D)\) encoding depth destroys signal before the variational circuit begins. This crossover is consistent with the \(p^*\) estimate derived in 5.

Generative modelling. Quantum generative adversarial networks (QGANs) represent the most amplitude-intensive application: the generator network must produce arbitrary probability distributions over \(D\)-dimensional feature space, which is most naturally represented as an amplitude state. Zoufal et al. [51] demonstrated learning of 1D financial distributions with \(n = 3\)–\(6\) qubits on IBM hardware; amplitude encoding was used but restricted to \(D \leq 8\). Lloyd and Weedbrook [52] provided theoretical frameworks for QGANs but did not address the encoding overhead. The emerging consensus is that QGANs are the most promising near-term application for amplitude encoding, provided \(D\) is kept small enough for current coherence budgets (\(D \leq 16\) for \(p \approx 3 \times 10^{-3}\)).

Image classification. Encoding pixel data presents a distinct sub-problem: images have spatial structure (local correlations) that should ideally be exploited by the encoding. The frqischeme [53] and NEQR [54] were designed for this purpose, using \(2n+1\) qubits to represent a \(2^n \times 2^n\) image. Henderson et al. [55] introduced quanvolutional layers, which apply random quantum circuits as feature extractors before classical processing. The approach uses angle encoding on \(4\)-qubit circuits applied in sliding windows, remaining well within nisqconstraints.

6.4 Comprehensive Literature Table↩︎

4 presents the 25 most representative works from the 66-work corpus. The full 66-entry table is available as Electronic Supplementary Material.

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6.5 Cross-Cutting Themes↩︎

Four cross-cutting themes emerge from the systematic analysis:

Theme 1: Noise tolerance trumps theoretical expressivity. Of the 23 hardware experiments in the corpus, 19 (83%) use angle or IQP encoding despite amplitude encoding’s theoretical expressivity advantage. This strongly suggests that practitioners have implicitly converged on noise-tolerant strategies even when not explicitly motivated by noise analysis.

Theme 2: Classical comparison is inconsistently reported. Only 37 of 66 works (60%) include a classical baseline. Among these, 28 show quantum classifiers matching but not exceeding classical baselines, 7 show the quantum model outperforming on a specially constructed problem, and 2 show genuine advantage on a realistic dataset. This suggests that the field’s progress metrics are insufficiently standardised.

Theme 3: Feature dimension is rarely pushed above \(D = 64\). Only 8 of 66 works use \(D > 64\); of these, 7 apply pcafirst. This implies that the high-\(D\) regime—where amplitude encoding’s qubit compression would be most impactful—remains largely unexplored experimentally.

Theme 4: Re-uploading is underexplored relative to its potential. Despite the strong theoretical guarantees of universal approximation [29], [33] and the hardware validation by Dutta et al. [34], only 7 of 66 works use data re-uploading. The likely reason is the lack of standardised software frameworks for re-uploading circuits at the time of publication of many surveyed works.

7 A Practical Decision Framework for Encoding Selection↩︎

7.1 Problem Formalisation↩︎

We formalise the encoding-selection problem as follows.

Since \(\mathrm{Acc}(\mathcal{E}, \mathcal{I})\) is not analytically tractable in general, we approximate the optimal selection by a decision procedure that eliminates hardware-infeasible encodings and ranks the remaining candidates by expressivity (1) and noise-adjusted fidelity (3).

7.2 Pre-Processing Decisions↩︎

Before applying the decision tree, two pre-processing decisions must be resolved:

Pre-processing Decision P1: Dimensionality reduction. If \(D > n_{\text{enc}} \cdot \kappa\) where \(n_{\text{enc}}\) is the qubit budget for encoding and \(\kappa\) is the encoding-specific bits-per-qubit ratio (\(\kappa = 1\) for standard angle, \(\kappa = 2\) for dense angle, \(\kappa = 2^{n}/D\) for amplitude), dimensionality reduction is mandatory. Standard options: pca(linear, analytically justifiable for continuous features), random projections (preserves inner products by Johnson–Lindenstrauss), or feature selection (preserves interpretability). pcais the default recommendation; select \(D'\) components explaining \(\geq 95\%\) of variance and check feasibility with the reduced dimension \(D'\) in place of \(D\).

Pre-processing Decision P2: Feature scaling. Angle encoding requires features in \([0, 2\pi]\) to avoid wrapping artefacts in the rotation. Amplitude encoding requires \(\|\mathbf{x}\|_2 \neq 0\) for normalisation. Standard min-max scaling to \([0, \pi]\) is recommended for angle encoding; \(\ell_2\) normalisation for amplitude encoding.

7.3 The Five-Regime Decision Framework↩︎

Based on the analytical bounds of [sec:costs] [sec:noise] and the empirical evidence of 6, we propose the five-regime framework formalised in 3 and visualised in 4.

Regime 1 — Expressive encoding, low-noise device↩︎

Conditions: \(D \leq n\), \(p \leq p^*_{\text{IQP}}\), \(\tau = \texttt{classification}\).

Recommendation: IQP encoding with \(k=2\). The circuit depth is constant (\(d = 2\)), within the coherence budget even for \(n = 20\), and the induced kernel is believed to be classically intractable [26]. Use a two-layer IQP feature map (repeating the IQP block twice provides more expressivity without significant depth increase) [58].

Trainability note: Kernel concentration may arise for \(n > 12\) [15]; monitor kernel matrix condition number and switch to Regime 5 if the matrix becomes near-singular.

Regime 2 — Standard regime: angle encoding↩︎

Conditions: \(D \leq n\), \(p > p^*_{\text{IQP}}\) (noise too high for IQP), or (any \(p\), \(\tau \neq \texttt{classification}\) but \(D \leq n\)).

Recommendation: Angle encoding (standard or dense). Pre-scale features to \([0, \pi]\). Use standard angle for \(D = n\); use dense angle if the qubit budget is tight (\(D' \leq 2n\) after pca). Downstream variational circuit depth: limit to \(d_{\text{VQC}} \leq 6 \cdot \lceil\log_2 n\rceil\) to stay within the noise-barren-plateau limit of 10 .

Regime 3 — Qubit-constrained, generative task↩︎

Conditions: \(D > n\), \(\tau = \texttt{generative}\).

Recommendation: Reduce \(D \to D' = 2n\) via pca (maximising variance retention within the qubit budget), then apply dense angle encoding. Generative tasks benefit from encoding that spans the full Bloch-sphere parameter space per qubit; dense angle encoding achieves this with depth 2. Amplitude encoding is not recommended here despite its qubit efficiency, because QGANs require many encoding passes during training, and the accumulated noise from \(\mathcal{O}(D)\) depth encoding circuits is prohibitive.

Regime 4 — Extreme compression: amplitude encoding with mitigation↩︎

Conditions: \(D \gg n\) (specifically \(\log_2 D \leq n\)), \(p \leq p^*_{\text{amp}}(D)\), \(\tau = \texttt{regression}\).

Recommendation: Amplitude encoding with zneapplied to the state-preparation circuit. This regime is currently accessible only for \(D \leq 16\) and \(n \leq 4\) on best-available hardware (\(p \approx 10^{-3}\)). Mandatory feasibility check: verify \(F_{\text{amp}} \geq 0.90\) using 13 before committing to this regime. Fallback: if the check fails, reduce \(D\) via pcaand apply Regime 2.

Regime 5 — Expressivity-critical: data re-uploading↩︎

Conditions: \(D \leq n\), \(p\) moderate (\(p_{\text{angle}}^* \leq p \leq p_{\text{reu}}^*(L)\) for achievable \(L\)), depth budget \(d_{\text{budget}} \geq L(1 + d_W)\).

Recommendation: Data re-uploading with \(L\) layers, where \(L\) is chosen as the largest integer satisfying the feasibility constraint. Use layer-by-layer initialisation [59] to avoid barren plateaus at the initial layers. Sweke et al.’s stochastic gradient analysis [56] justifies using mini-batch gradient updates, which are preferable to full-batch for re-uploading circuits on hardware.

7.4 Pareto Analysis↩︎

5 plots the Pareto frontier in the two-dimensional space of expressivity (Fourier degree, as a proxy for model capacity) vs.noise robustness (state-preparation fidelity at \(p_2 = 3 \times 10^{-3}\), \(D = 64\)). No encoding dominates all others simultaneously: the frontier is monotone decreasing, confirming that expressivity and robustness are fundamentally at odds.

The data re-uploading family traces a controllable curve parameterised by \(L\): each additional layer adds expressivity at a fidelity cost determined by the depth-fidelity bound of 2. For the specified hardware parameters, the curve crosses the feasibility threshold \(F_{\min} = 0.90\) at \(L \approx 3\) (corresponding to a Fourier degree of \(3\) per feature dimension). This gives a concrete recommendation: for a system with \(p_2 = 3 \times 10^{-3}\), re-uploading should not exceed \(L = 3\) layers if state-preparation fidelity is to remain above 90%.

7.5 Worked Examples↩︎

Example A — MNIST digit classification, IBM Heron. \(D = 784\) (raw), \(n = 10\) (encoding register), \(p_2 = 3 \times 10^{-3}\), \(\tau = \texttt{classification}\). Step 1: \(D = 784 > n = 10\), apply pcato \(D' = 10\). Step 2: check \(p^*_{\text{IQP}} = 1 - 0.90^{1/50} \approx 2.1 \times 10^{-3}\); since \(p = 3 \times 10^{-3} > 2.1 \times 10^{-3}\), IQP fails feasibility. Step 3: check depth budget; for \(L = 3\), \(d_W = 2\): total depth \(= 3(1+2) = 9 < d_{\max}\). Fidelity: \((1-3\times10^{-3})^{30} \approx 0.914 > 0.90\). Decision: Regime 5 (re-uploading, \(L = 3\)).

Example B — Gene expression regression, IonQ Aria. \(D = 256\), \(n = 8\) (logical), \(p_2 = 10^{-3}\), \(\tau = \texttt{regression}\). \(p^*_{\text{amp}}(D=256) = 1 - 0.90^{1/512} \approx 2.1 \times 10^{-4}\). Since \(p = 10^{-3} > 2.1 \times 10^{-4}\), amplitude encoding fails. Fallback: pcato \(D' = 8\), then Regime 2 (angle). Decision: Regime 2 with pcapre-processing.

Example C — Financial distribution learning, IBM Falcon. \(D = 8\), \(n = 3\), \(p_2 = 5 \times 10^{-3}\), \(\tau = \texttt{generative}\). \(p^*_{\text{amp}}(D=8) = 1 - 0.90^{1/16} \approx 6.6 \times 10^{-3}\). Since \(p = 5 \times 10^{-3} < 6.6 \times 10^{-3}\), amplitude encoding is feasible. However, \(\tau = \texttt{generative}\) routes through Regime 3; checking: \(D = 8 \leq n = 3\)? No, \(D > n\), but \(\log_2 8 = 3 = n\), so amplitude uses exactly \(n\) qubits. Treat as a special case of Regime 4 with \(\tau = \texttt{generative}\): amplitude with zne. Decision: Regime 4 (amplitude + zne), consistent with Zoufal et al.’s approach [51].

8 Open Challenges and Future Directions↩︎

8.1 Trainable and Adaptive Encodings↩︎

Data re-uploading [29] treats the encoding circuit as a fixed function of the data, with trainable parameters only in the interleaved ansatz blocks. A natural extension is to make the encoding generators themselves trainable: replacing \(R_Y(x_k) = e^{-ix_k Y/2}\) with \(R_Y(\alpha_k x_k + \beta_k)\) where \((\alpha_k, \beta_k)\) are learnable. This meta-encoding approach allows the model to learn which Fourier frequencies are most useful for the given dataset. The parameter-shift rule [57] extends to arbitrary encoding generators, making gradient-based learning of \((\alpha_k, \beta_k)\) straightforward in principle.

The open question is whether learning the encoding parameters jointly with the ansatz parameters introduces a meta-level barren plateau. Current evidence from the expressibility analysis of Schuld et al. [9] and the universal approximation results of Goto et al. [33] suggests that the encoding parameter landscape is benign for shallow circuits, but no rigorous result exists for the joint-optimisation case.

8.2 Hardware-Native Encoding Design↩︎

Current encoding strategies are designed for idealised, fully connected quantum processors. In practice, native connectivity constraints (e.g., Heavy-Hex graph for IBM, linear chain for early IonQ) require SWAP-gate insertion whenever a gate acts on non-adjacent qubits, increasing both depth and gate count. For amplitude encoding, whose preparation circuit involves multi-controlled Toffoli gates across distant qubits, the SWAP overhead can multiply the gate count by a factor of \(2\)–\(3\times\) [39].

Hardware-native encoding design would construct the preparation circuit to respect native connectivity from the outset. Preliminary work on connectivity-aware state preparation exists for specific topologies, but no general framework has been developed. The problem is equivalent to a quantum circuit routing problem [60] with the additional constraint that the prepared state must encode the full feature vector.

8.3 Error Mitigation Integrated into State Preparation↩︎

zneand PEC have been applied extensively to variational circuits but not systematically to encoding circuits. The challenge is that encoding circuits are data-dependent: every new input \(\mathbf{x}\) requires re-execution of the state-preparation circuit, and the noise characterisation (required for PEC) changes with the circuit. A promising direction is data-aware error mitigation, where the noise model is parameterised as a function of \(\mathbf{x}\) and learned from a calibration dataset. This would allow PEC to be applied to encoding circuits with overhead that scales polynomially with \(D\) rather than exponentially.

A related direction is developing noise-robust state preparation algorithms: circuits that are by construction less sensitive to Pauli noise. Recent work on symmetric derivatives [57] suggests that encoding circuits with higher-order symmetry properties may have inherently lower noise sensitivity, although this connection has not been formally established.

8.4 Encoding for Structured and Non-Euclidean Data↩︎

The encoding strategies reviewed above assume \(\mathbf{x}\in \mathbb{R}^D\): dense, Euclidean feature vectors. Real-world datasets are frequently structured: graphs (molecular property prediction, social networks, knowledge bases), sequences (protein structures, time series), and hierarchical or set-valued data (parse trees, point clouds). Classical machine learning has developed architectures—graph neural networks, transformers, and recurrent networks—that exploit structural inductive biases, but the quantum analogues are at an early stage.

Quantum walks as structural encodings. Quantum walks on graphs provide a principled route to encoding adjacency structure. A continuous-time quantum walk on a graph \(G=(V,E)\) with adjacency matrix \(A\) evolves as \(\ket{\psi(t)} = e^{-iAt}\ket{v_0}\), producing a superposition over graph vertices weighted by the walk amplitude. This constitutes a natural structural feature map: \(\phi_Q(G) = e^{-iAt}\ket{v_0}\), where the quantum state encodes the graph topology through the generator \(A\). The walk can be approximated to depth \(\mathcal{O}(\|A\|_{\max} t)\) using standard Hamiltonian simulation techniques [61].

Quantum Graph Neural Networks (QGNNs). Verdon et al. [62] proposed the first QGNN architecture, in which a parameterised quantum circuit is constructed from the graph structure: nodes and edges correspond to qubits and two-qubit gates respectively, and the circuit depth scales with the graph diameter rather than the number of nodes. The graph topology is thus encoded directly into the circuit connectivity, circumventing the need for an explicit \(\mathbb{R}^D\) feature vector. This approach naturally avoids the compression–depth trade-off of 1 for Euclidean data, since the encoding circuit and the variational ansatz coincide.

Recent work on equivariant quantum circuits has introduced encoding strategies that respect graph symmetries (e.g., permutation invariance) [62], [63], which is a necessary condition for meaningful learning on graphs with unlabelled nodes. These circuits achieve encoding-layer gate counts of \(\mathcal{O}(|E|)\)—linear in the number of edges—at constant depth if the graph has bounded degree, making them well-suited to NISQ constraints.

Open problems. Three encoding-specific questions for structured data remain unresolved. First, the kernel concentration problem of Thanasilp et al. [15] has not been analysed for graph-structured encodings: it is unknown whether the walk-based kernel \(\kappa_G(G,G') = |\langle 0|e^{iAt} e^{-iA't}|0\rangle|^2\) concentrates exponentially as \(|V|\) grows. Second, no systematic analysis exists of the Fourier expressivity of walk-based encodings in the sense of 1: the generator of the walk is not a Pauli rotation, so the frequency spectrum result does not apply directly. Third, for molecular data, the most natural structural encoding—mapping atomic orbital basis functions to qubit registers—is equivalent to the first-quantisation representation used in quantum chemistry [64], whose resource costs are substantially different from the encodings surveyed here. A unified theory of structured quantum encoding—encompassing walk-based, equivariant, and chemistry-motivated schemes within the cost–expressivity–robustness framework of 2—is the most underexplored direction at the intersection of quantum hardware and learning theory.

8.5 Towards Fault-Tolerant Encoding↩︎

As quantum hardware approaches fault-tolerant operation (projected for the late 2020s with surface codes and \(>10^4\) physical qubits [39]), the encoding landscape will shift. With logical error rates \(p_L \approx 10^{-10}\), amplitude encoding of \(D = 10^6\)-dimensional vectors becomes, in principle, feasible from a fidelity perspective. The binding constraint shifts to the number of logical operations per encoding circuit: with \(\mathcal{O}(D)\) CNOTs compiled to surface-code logical gates, the time overhead per encoding pass becomes the practical limit.

Quantifying this overhead requires understanding the logical gate time for cnotunder surface codes (estimated at \(\sim 1\)–\(10\,\mu\)s with cycle times of \(\sim 1\,\mu\)s), the parallelisability of the state-preparation circuit, and the available classical-data throughput. A preliminary analysis suggests that encoding \(D = 10^3\) features with a fault-tolerant amplitude circuit would require \(\mathcal{O}(10^4)\) surface-code cycles and \(\mathcal{O}(10^6)\) physical qubits [64]—far beyond near-term hardware but relevant for the 2030–2035 time horizon.

8.6 Benchmarking Standards for Encoding Evaluation↩︎

One of the most pressing methodological gaps identified in 6 is the absence of standardised benchmarking protocols for encoding evaluation. A standardised benchmark suite would include: (i) a fixed set of classical datasets at multiple scales (\(D \in \{4, 16, 64, 256\}\)), (ii) a standardised noise model (parameterised by \(p\)) to be used in simulation when hardware is unavailable, (iii) a mandatory classical baseline (kernel svmwith the same feature dimension), and (iv) a standardised reporting format (encoding name, \(n\), \(g_2\), \(d\), \(p\), accuracy, compared with classical baseline).

The Quantum Computing Benchmarking Consortium (cf. [65]) has made progress on hardware-level benchmarks but has not addressed encoding-specific metrics. Establishing such standards would dramatically increase the comparability of results across research groups and hardware platforms, and would make the kind of meta-analysis performed in 6 substantially more rigorous. We recommend that the quantum computing community adopt the format described above as a reporting standard for all experimental qmlpapers, analogous to the role that ImageNet and GLUE benchmarks play in classical deep learning.

To operationalise this recommendation, Table 5 proposes a concrete minimal reporting schema. Any experimental QML paper using a specific encoding strategy should populate all fields; omission of a field requires explicit justification. We strongly encourage QML journals and conferences to adopt this schema as a mandatory supplementary checklist for paper submissions, akin to the reproducibility checklists pioneered by classical machine learning venues such as NeurIPS. Adopting this schema uniformly would make the kind of systematic meta-analysis conducted in Section 6 substantially more rigorous in future surveys.

8.7 Quantum Advantage Conditions for Encoding-Sensitive Tasks↩︎

The results of Jerbi et al. [16] and Huang et al. [35] establish that quantum advantage in learning depends critically on whether the data itself has quantum structure (i.e., it was generated by a quantum process) or is purely classical. For classical data, the information-theoretic bounds of Huang et al. [36] suggest that the advantage of quantum encodings over classical kernel methods is at most polynomial for most practically relevant function classes.

A fundamental open question is: for which specific encoding strategies and task distributions does a provable quantum advantage exist? The IQP encoding of Havlı́ček et al. [26] achieves advantage only for a purpose-built problem; no natural dataset is known for which amplitude or angle encoding provably outperforms all classical baselines. Resolving this question requires combining the Fourier analysis of 3.3 with complexity-theoretic arguments (relating the hardness of computing \(\kappa_Q\) to well-studied computational problems) and statistical learning theory (bounding the sample complexity of learning with \(\kappa_Q\)). The work of Thanasilp et al. [15] and the out-of-distribution generalisation bounds of Caro et al. [66] provide useful building blocks, but a unified theory remains absent.

9 Conclusion↩︎

This survey has provided a unified, analytical treatment of quantum feature encoding strategies from three complementary perspectives—resource cost, expressivity, and noise robustness—synthesising 66 primary works assembled via a PRISMA-adapted systematic review protocol.

Core analytical result. We derived the critical error rate \(p^*(\mathcal{E}, D, F_{\min})\) below which each encoding strategy is hardware-feasible. For current nisqdevices (\(p_2 \approx 3 \times 10^{-3}\)) and \(D = 64\), amplitude encoding is infeasible (\(p^*_{\text{amp}} \approx 8 \times 10^{-4} < p_{\text{current}}\)), while angle encoding and data re-uploading (with \(L \leq 3\)) remain viable. This result, corroborated by the empirical evidence of LaRose and Coyle [46] and Balewski et al. [50], provides a principled, hardware-grounded criterion for encoding selection that supersedes qualitative rules of thumb.

Joint trainability framework. By connecting the Fourier analysis of Schuld et al. [9], the barren-plateau theory of McClean et al. [11] and Wang et al. [13], and the kernel-concentration results of Thanasilp et al. [15], we identified a fundamental expressibility dilemma: the most expressive encodings are simultaneously the least trainable, the most noise-sensitive, and the most prone to kernel concentration. Resolving this dilemma requires encoding strategies that are both structured (to avoid concentration) and of controllable depth (to avoid barren plateaus). Data re-uploading with layer-by-layer initialisation is currently the most promising candidate.

Practical impact. The five-regime decision framework of 7 translates the theoretical results into operationally actionable recommendations. The framework handles the full range of practical settings—from low-dimensional classification on noisy superconducting hardware (Regime 2) to high-dimensional regression with error mitigation (Regime 4)—and provides concrete fallback paths when primary encodings fail feasibility checks.

Outlook. As hardware error rates approach \(p_2 \approx 10^{-4}\) (projected for early fault-tolerant devices in the 2028–2032 window), the critical depth threshold for amplitude encoding will rise from \(g_2 = 2D\) at \(p_2 = 8 \times 10^{-4}\) to \(g_2 = 2D \cdot k\) for \(k \approx 4\)–\(5\). This will open a window for amplitude encoding of \(D \leq 64\)–\(128\) dimensional data, shifting the optimal regime from pure angle encoding toward a hybrid architecture in which amplitude encoding handles global feature structure and angle encoding handles local feature refinement. Designing encoding strategies for this intermediate regime is, in our assessment, the highest-priority research problem at the intersection of quantum hardware and machine learning.

AI Disclosure↩︎

During the preparation of this work, the authors used Claude 4.6 Sonnet for the purpose of proofreading the manuscript and improving the English grammatical structure. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the technical accuracy and the final content of the publication.

References↩︎