Angle-Encoding VQC with Circular Entanglement

• Angle-encoding VQC circuits with circular entanglement are parameterized quantum circuits that encode classical data via single-qubit rotations and cyclic CNOT gates.
• They balance expressivity, entangling power, and noise resilience by leveraging optimized architectures for near-term quantum hardware.
• Empirical benchmarks demonstrate their effectiveness in tasks like fraud detection and tumor classification, highlighting superior connectivity and gradient flow.

Angle-encoding variational quantum circuits (VQCs) with circular entanglement are a distinct class of parametrized quantum circuits used in quantum machine learning and reinforcement learning, characterized by the encoding of classical features into quantum states via single-qubit rotations and the use of cyclically interconnected CNOT gates to generate high expressivity and connectivity in the circuit ansatz. These architectures are optimized for near-term quantum hardware by balancing the requirements of entangling power, gradient transport, and noise resilience. Empirical benchmarks indicate their effectiveness in a range of supervised and reinforcement learning tasks, providing a robust trade-off between expressivity, generalization, and trainability across different applications (Abbou et al., 26 Sep 2025, Yurtseven, 29 Nov 2025, Kruse et al., 2023).

1. Formulation of Angle-Encoding and Input Mapping

In angle-encoding VQCs, each input feature is mapped to the angle of a single-qubit rotation. For an $n$-dimensional classical feature vector $x = (x_1, \ldots, x_n)$, the input encoding is constructed as

$$U_{\text{encode}}(x) = \bigotimes_{i=1}^n R_Y(\theta_i), \quad \theta_i = x_i \pi,$$

where $R_Y(\theta)$ denotes the rotation about the $Y$ axis:

$R_Y(\theta) = \exp(-i \theta Y / 2).$

For raw features, preprocessing steps linearly normalize $x_i$ into $[-1,1]$ so that $x_i \pi \in [-\pi, \pi]$, maximizing the representational coverage of the Bloch sphere (Yurtseven, 29 Nov 2025). In settings with variable feature scaling or continuous state spaces, an additional trainable scaling parameter $\lambda_i$ may be introduced, so that each normalized feature is encoded as $R_Y(\lambda_i \hat{s}_i)R_Z(\lambda_i \hat{s}_i)$, where $\hat{s}_i$ is the normalized input (Kruse et al., 2023).

2. Circuit Architecture and Variational Ansatz

A typical angle-encoding VQC circuit consists of:

• An initial encoding layer of $R_Y$ (and possibly $R_Z$) gates with angles determined by classical inputs.
• $L$ variational layers, each comprising:
  • For each qubit: two trainable single-qubit rotations, commonly $R_X(\theta)$ and $R_Y(\phi)$.
  • A block of $n$ CNOT gates arranged in a circular entanglement (ring) topology.

The total depth is $L$ repetitions of the variational+entangler block. For $n=4$, with $L=2$, and a minimal ansatz ($R_Y$ only), the number of trainable parameters is $n \times L$, but more general setups employ both $R_X$ and $R_Y$ per qubit per layer, yielding $2nL$ parameters (Abbou et al., 26 Sep 2025, Yurtseven, 29 Nov 2025).

For example, a four-qubit angle-encoding VQC with $L=2$ layers and circular entanglement is defined by:

• Initial $R_Y(x_j \pi)$ gates for $j = 0,1,2,3$.
• Each layer: $R_X(\theta_{l, j}) \rightarrow R_Y(\phi_{l, j})$ for each qubit $q_j$, followed by $\text{CNOT}_{j\rightarrow (j+1)\bmod 4}$ for $j=0,1,2,3$ (Yurtseven, 29 Nov 2025).

3. Circular (Ring) Entanglement Topology

Circular entanglement refers to the CNOT block where each qubit is both a control and target once per layer, connecting $q_j$ to $q_{(j+1) \bmod n}$ for all $j \in \{0, ..., n-1\}$:

$$\mathcal{U}_{\text{ent}} = \prod_{j=0}^{n-1} \text{CNOT}_{j\rightarrow (j+1) \bmod n}$$

This ring topology ensures full connectivity after two layers, avoids the presence of "dark" parameters that never affect the measurement, and enables more uniform gradient propagation compared to chain or partial-connectivity layouts (Kruse et al., 2023). Empirical studies indicate that alternative entanglement strategies—such as linear chains or random pairs—yield either reduced convergence rates or barren plateaus (Kruse et al., 2023, Abbou et al., 26 Sep 2025).

4. Parameterization, Training, and Regularization

Angle-encoding VQCs with circular entanglement are trained using gradient-based optimizers, frequently Adam or AdamW, with gradients estimated via the parameter-shift rule. In benchmarked architectures, initial parameter values are either drawn uniformly from $[0,2\pi)$ or as $\mathcal{N}(0, 0.1)$-distributed variables (Abbou et al., 26 Sep 2025, Yurtseven, 29 Nov 2025, Kruse et al., 2023).

Training strategies include:

• Binary cross-entropy as the primary loss function:

$$\mathcal{L}(\phi) = -\frac{1}{N} \sum_{k=1}^N \left[ y^{(k)} \log p_0(x^{(k)};\phi) + (1-y^{(k)}) \log (1-p_0(x^{(k)};\phi)) \right],$$

where $p_0$ is the probability of measuring $|0\rangle$ in the readout qubit (Abbou et al., 26 Sep 2025, Yurtseven, 29 Nov 2025).

• $L_2$-regularization to mitigate overfitting and promote generalization (Abbou et al., 26 Sep 2025).
• Label smoothing and gradient clipping to enhance stability (Yurtseven, 29 Nov 2025).
• Early stopping and learning rate scheduling to accelerate convergence (Yurtseven, 29 Nov 2025).

Empirical evidence suggests optimal convergence for shallow circuits ($L \leq 2$) on present-day NISQ devices, and deeper circuits when the number of qubits is small (Kruse et al., 2023).

5. Empirical Performance and Application Benchmarks

Angle-encoding VQCs with circular entanglement have been empirically validated in several models:

• Financial fraud detection: For $n=4$, $L=2$ angle-encoded VQC with circular entanglement achieved 91.1% test accuracy, outperforming linear connectivity (90.7%) and closely matching full connectivity schemes (92.0%) (Abbou et al., 26 Sep 2025).
• Image-based tumor classification: In hybrid quantum-classical CNNs, a parallel angle-encoding VQC with circular entanglement increased the test accuracy baseline from 84.17% (classical) to 86.54%. Improvements were also observed in precision (0.8670 vs. 0.8397) and F1-score (0.9131 vs. 0.8967), confirmed as statistically significant via a one-sided Wilcoxon signed rank test at $p = 0.03125$ with Cohen’s $d = 2.14$ (Yurtseven, 29 Nov 2025).
• Reinforcement learning for continuous action environments: Circular entanglement constructs were critical for solving OpenAI Gym benchmarks. Alternative layouts led to poorly conditioned gradients or unused parameters, while the circular CNOT ring enabled solutions with fewer training episodes and order-of-magnitude lower parameter counts compared to classical networks (Kruse et al., 2023).

Across studies, circular entanglement provides the strongest trade-off between quantum circuit expressivity and classical trainability, producing superior or competitive performance relative to alternative entanglement topologies.

6. Implementation Considerations and Deployment Guidelines

Angle-encoding VQC circuits with circular entanglement are tailored for NISQ-era practicalities

• Qubit count: Typically one per feature; four-qubit circuits are a standard benchmark (Abbou et al., 26 Sep 2025, Yurtseven, 29 Nov 2025).
• Single-qubit gates per layer: $n$ for encoding + $n$ (or $2n$) for variational gates.
• Entangling gates per layer: $n$ CNOTs arranged circularly.
• Total circuit depth: For $n=4$, $L=2$, this yields $24$ gates plus measurements (Abbou et al., 26 Sep 2025).
• Hardware mapping: Circuits should align with the physical qubit layout to minimize SWAP overheads. On ring or heavy-hex topologies, the circular pattern can often be mapped with minimal connectivity cost (Yurtseven, 29 Nov 2025).
• Noise mitigation: Employ zero-noise extrapolation and readout error correction. Limiting $L$ to $\leq2$ is advised on current platforms.

Recommended workflow:

1. Prototype on a noiseless statevector simulator for circuit verification.
2. Transition to noisy, shot-based simulators for hardware-relevant performance, using on the order of $1024$ shots and calibration runs (Abbou et al., 26 Sep 2025).
3. Average across circuit batches and apply practical tweaks (see Table below).

Element	Recommendation (n=4, L=2)	Source
Qubit count	4	(Abbou et al., 26 Sep 2025)
Entanglement per layer	4 CNOTs, circular topology	(Abbou et al., 26 Sep 2025)
Total circuit depth	24 single- and two-qubit gates	(Abbou et al., 26 Sep 2025)
Parameter init	Uniform [0,2π) or N(0,0.1)	(Yurtseven, 29 Nov 2025)
Optimizer	Adam or AdamW, LR 0.001 – 0.01	(Yurtseven, 29 Nov 2025)
Loss	Cross-entropy + L2 regularization	(Abbou et al., 26 Sep 2025)
Regularization	Early stop, label smooth, grad clip	(Yurtseven, 29 Nov 2025)

7. Advantages and Limitations of Circular Entanglement

Circular entanglement endows angle-encoding VQCs with several advantages:

• Expressivity: Rapidly achieves all-to-all effective connectivity with minimal circuit depth.
• Trainability: Empirically avoids vanishing-gradient “barren plateau” issues seen in alternative layouts; all variational parameters impact output via the entanglement structure (Kruse et al., 2023).
• Hardware alignment: Naturally reflects many physical device topologies, reducing transpilation overhead.

Limitations and challenges include:

• Expressivity Limit: While effective, angle encoding in Y (or YZ) rotations may underperform nonlinear maps like ZZ features for certain tasks, as angle encoding may be less expressive for high-order feature correlations (Abbou et al., 26 Sep 2025).
• Depth-Noise Trade-off: Increased circuit depth for larger $L$ or $n$ can exacerbate noise-induced degradation on hardware, motivating careful regularization and shallow architectures (Yurtseven, 29 Nov 2025).
• Overparameterization: Excessive parameter counts or layers can result in overfitting or barren plateaus, necessitating empirical tuning per dataset and application (Kruse et al., 2023).

A plausible implication is that the circular entanglement pattern represents a near-optimal ansatz for NISQ VQC models across tasks where a balance of trainability and expressivity is required, especially for datasets with $\leq 6$ features.

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References:

(Abbou et al., 26 Sep 2025, Yurtseven, 29 Nov 2025, Kruse et al., 2023)