Amplitude-Encoding VQC Overview

• Amplitude-encoding VQC is a paradigm that maps classical vectors into quantum state amplitudes using n = ⌈log₂ N⌉ qubits, offering exponential state space compression.
• Variational methods like AAE and ACAE employ hardware-efficient shallow circuits that optimize fidelity while mitigating the need for deep, exact state preparation.
• Comparative studies reveal significant accuracy gains with amplitude encoding over angle encoding, though increased circuit depth and noise sensitivity require careful resource management.

Amplitude-encoding variational quantum circuits (VQCs) constitute a powerful paradigm for compressing high-dimensional classical information into quantum amplitudes, enabling resource-efficient algorithms across quantum machine learning, quantum chemistry, and NISQ-era variational computation. At the core, these architectures seek to encode a classical vector $x \in \mathbb{C}^N$ (or $\mathbb{R}^N$) as the amplitudes of a quantum state $$\lvert\psi(x)\rangle = \sum_{i=0}^{N-1} x_i / \|x\| \, \lvert i\rangle$$ on $n = \lceil \log_2 N \rceil$ qubits, using exact or variationally-trained circuits. This approach leverages the exponential state space of quantum systems, offering both qubit reductions and (subject to expressivity) more powerful feature mappings than shallow, angle-based encodings, but at the expense of deeper state-preparation circuits and nontrivial optimization.

1. Mathematical Formulation of Amplitude Encoding

Given a classical data vector $x \in \mathbb{C}^N$, the objective is to encode it as a normalized quantum state

$$\lvert\psi(x)\rangle = \sum_{i=0}^{N-1} x_i \,|i\rangle, \qquad \|x\|^2 = \sum_{i=0}^{N-1} |x_i|^2 = 1$$

For real-valued data, the entries are real and normalization proceeds via the Euclidean norm. For complex vectors, phases are preserved. This mapping compresses an $N$-dimensional space into $n = \lceil \log_2 N \rceil$ qubits.

Exact state preparation, such as the Mottonen or Mottonen-Vartiainen algorithm, requires $O(2^n)$ rotations and CNOTs, while recursive decompositions yield a circuit of exponential depth in the number of features (Tudisco et al., 1 Aug 2025). For practical NISQ applications and high-dimensional data, approximate schemes based on variational quantum circuits are employed, setting parameters $\theta$ to maximize fidelity to a target amplitude-encoded state (Truger et al., 27 Feb 2024, Mitsuda et al., 2022, Nakaji et al., 2021).

2. Variational Amplitude-Encoding Algorithms

Amplitude-encoding VQCs often use a hardware-efficient ansatz. For encoding real vectors, approaches such as approximate amplitude encoding (AAE) variationally train a shallow circuit $U(\theta)$ to minimize distance between $U(\theta)\lvert 0\rangle$ and the desired state via a cost function—typically maximum mean discrepancy (MMD) between computational- and Hadamard-basis measurement distributions (Nakaji et al., 2021). For complex vectors, approximate complex amplitude encoding (ACAE) adopts a fidelity-based objective, utilizing the classical shadows protocol to efficiently estimate the fidelity $$F(\theta) = \text{Tr}[\rho_{\text{target}} \rho(\theta)]$$ (Truger et al., 27 Feb 2024, Mitsuda et al., 2022).

Both workflows employ a layered circuit design:

• Each layer applies parameterized single-qubit rotations (typically $R_X$, $R_Y$, $R_Z$ gates) on all qubits.
• Fixed entangling shemes (ring or chain topologies) deploy two-qubit gates such as CNOT or CZ for expressivity.
• Total variational parameters scale as $R = \text{(layers)} \times n$.

Training proceeds by gradient-based (e.g., Adam) or gradient-free (e.g., COBYLA, evolutionary) optimization of the cost function estimated via quantum circuit measurements (Truger et al., 27 Feb 2024, Nakaji et al., 2021).

3. Circuit Architectures and Resource Scaling

Exact Amplitude Encoding

• Qubit savings: $n = \lceil \log_2 N \rceil$ for $N$-dimensional vectors.
• Gate depth: $O(N)$; exponential in $n$ for Mottonen/Vartiainen state preparation (Morgan et al., 22 Aug 2025).
• Entanglement: Highly entangled states may result; explicit circuit depth amplifies noise vulnerability, particularly on NISQ hardware.

Approximate, Variational Amplitude Encoding

• Depth: $O(\text{layers} \times n)$ for shallow ansatz circuits, independent of $N$.
• Shots and measurements:
  • Fidelity estimation via classical shadows requires $O(\log(1/\delta)/\epsilon^2)$ single-shot Clifford measurements per estimator for additive error $\epsilon$ and error probability $\delta$ (Truger et al., 27 Feb 2024).
  • MMD objectives estimate moments using computational and Hadamard basis; measurement cost is $O(1/\sqrt{\text{shots}})$ per term (Nakaji et al., 2021).
• Parameter count: Polynomial in $n$ (e.g., 18 for a 3-qubit, 2-block SU(2) ansatz (Truger et al., 27 Feb 2024)).
• Scaling trade-off: Quasilogarithmic in input size, provided the target state is expressible by the chosen ansatz.

4. Methodological Landscape: Algorithms and Empirical Performance

ACAE and Warm-Start for VQE

The ACAE/VQE protocol proceeds in two phases (Truger et al., 27 Feb 2024):

1. ACAE Pretraining:
   • Optimize circuit parameters to maximize fidelity with a classical ground-state approximation using classical shadows.
   • Employ gradient-free optimization; typical iterations $N_{\text{pre}} \sim 50$.
2. VQE Refinement:
   • Initialize downstream VQE with pretrained parameters.
   • Continue minimizing the Hamiltonian expectation value via classical shot-based evaluation.

Empirically, warm started VQE achieves higher solution quality and faster convergence per quantum shot, avoiding local minima and barren plateaus often observed in random initialization. On random $8\times8$ Hermitian matrices, WS-VQE achieves median solution accuracy $r_{\rm appr}\approx0.88$ after 80 VQE-equivalent iterations, compared to $\approx0.70$ for standard VQE (Truger et al., 27 Feb 2024).

AAE/ACAE for Classification and QRNNs

For quantum classifiers, both real and complex amplitude encoding have been used. Empirical results confirm that amplitude/hybrid encoding schemes provide higher expressivity and accuracy than angle-only approaches, but at increased circuit depth (Biswas, 18 Mar 2025, Mitsuda et al., 2022, Morgan et al., 22 Aug 2025). In QRNNs, EnQode-style approximate amplitude encoding achieves depth-linear-in-qubit scaling and improves test MSE by 36% when enhanced with pre-normalized magnitude features, compared to angle encoding and classical RNN baselines (Morgan et al., 22 Aug 2025).

Hybrid Classical–Quantum Architectures

VQC-MLPNet demonstrates the use of amplitude-encoded quantum circuits as trainable generators for classical MLP parameters, with theoretical error bounds showing exponential improvement with circuit depth or number of qubits. Under noise, the hybrid architecture sustains significantly higher accuracy than standalone quantum or classical models (Qi et al., 12 Jun 2025).

5. Comparative Trade-offs: Amplitude vs. Angle Encoding

Amplitude encoding provides an exponential compression of classical data into quantum state space, reducing qubit requirements from $O(N)$ (e.g., angle encoding) to $O(\log N)$. However, this compression incurs exponential circuit depth for exact encoding, and increased vulnerability to noise due to deep multi-controlled-rotation circuits (Tudisco et al., 1 Aug 2025, Biswas, 18 Mar 2025, Morgan et al., 22 Aug 2025).

Empirical comparisons across data sets demonstrate:

• Higher Expressivity: Amplitude encoding can yield up to 40 percentage-point gains in VQC accuracy (as on the Wine dataset), provided the quantum device can implement the necessary depth (Tudisco et al., 1 Aug 2025).
• Resource trade-off: Circuit preparation is much slower than for angle encodings, which only require $O(N)$ single-qubit gates and minimal depth.
• Hybrid encoding: Combining amplitude, angle, and phase encoding (hybrid) achieves the highest observed classifier accuracy (0.90–0.95) in small-feature regimes (Biswas, 18 Mar 2025).

Amplitudes must be constructed from vectors of length $2^n$, typically requiring zero-padding or feature reduction (e.g., via PCA), and the depth of exact encoding targeting general quantum hardware is prohibitive at $n\gtrsim10$.

6. Practical Considerations and Applications

Practical VQC design using amplitude encoding requires careful choices:

• Circuit construction: For small $n$ ($n\lesssim10$), Qiskit’s Initialize implements exact Mottonen (Biswas, 18 Mar 2025). For larger $N$, variational ansatz and approximate methods (AAE/ACAE/EnQode) mitigate depth while preserving fidelity.
• Data pre-processing: Non-power-of-two dimensions require padding; sign/phase information in features implies the necessity of auxiliary rotations or ancilla qubits. To retain magnitude information (lost under standard amplitude encoding), augmenting with pre-normalized magnitude features is effective (Morgan et al., 22 Aug 2025).
• Optimization: Stochastic or gradient-free optimizers are used when gradients vanish (barren plateaus).
• Applications: Ground-state preparation (VQE), quantum classifiers, quantum reinforcement learning, QRNNs for time-series, and quantum hardware-efficient representations for classical neural nets (Truger et al., 27 Feb 2024, Mitsuda et al., 2022, Biswas, 18 Mar 2025, Chen et al., 2021, Qi et al., 12 Jun 2025, Morgan et al., 22 Aug 2025).

Empirical evidence supports amplitude-encoded VQCs as enabling deep representation with dramatically reduced parameter count and robust convergence for warm-start and hybrid applications (Truger et al., 27 Feb 2024, Qi et al., 12 Jun 2025, Biswas, 18 Mar 2025, Morgan et al., 22 Aug 2025). Nonetheless, circuit depth and noise impose limitations for current hardware, favoring variational and hybrid approaches in practice.

7. Open Challenges, Extensions, and Outlook

Research directions focus on increasing scalability, fidelity, and hardware efficiency:

• Ansatz and optimizer refinement: Structured ansätze can reduce barren-plateau risks; optimizer step-size adaptation mitigates differences between fidelity and energy landscapes (Truger et al., 27 Feb 2024).
• Efficient state preparation: Variational approximate encodings (AAE, ACAE, EnQode) enable amplitude encoding in shallow circuits. Increasing k-means clusters or ansatz depth improves fidelity at the cost of higher offline computation (Morgan et al., 22 Aug 2025).
• Hybrid/quasi-classical integration: Architectures generating classical network weights from quantum amplitudes achieve improved generalization and noise resilience, especially for tasks like genomics and semiconductor classification (Qi et al., 12 Jun 2025).
• Resource scaling: Alternating-register pipeline schemes, magnitude-augmented amplitude encoding, and measurement-reduction techniques (classical shadows, derandomized shadows) decrease wall-clock and quantum resource requirements (Morgan et al., 22 Aug 2025, Truger et al., 27 Feb 2024).
• Domain extensions: Warm-started amplitude-encoded VQCs generalize to QAOA, quantum PCA, quantum SVM, and quantum chemistry, as well as automation frameworks for hybrid quantum ML workflows (Truger et al., 27 Feb 2024, Biswas, 18 Mar 2025, Tudisco et al., 1 Aug 2025).

The challenge of exponential depth for exact amplitude encoding remains fundamental, but ongoing innovations in variational ansatz design, pre-processing, and hybridization strategies continue to expand the practical reach and utility of amplitude-encoding VQCs in NISQ- and post-NISQ quantum information processing.