Approximate Amplitude Encoding (AAE)

• Approximate Amplitude Encoding is a technique that employs optimization-driven, shallow variational circuits to embed classical data vectors into quantum state amplitudes with reduced circuit complexity.
• It leverages parameterized quantum circuits and measurement-based loss functions to approximate the target state while avoiding the exponential depth required by exact methods.
• AAE enables scalable quantum machine learning by achieving high fidelity (≈0.95–0.99) on NISQ devices and reducing resource requirements compared to traditional state preparation approaches.

Approximate Amplitude Encoding (AAE) is a family of techniques for embedding classical data vectors into the amplitudes of quantum states using shallow, variational circuits. These methods address the infeasibility of exact amplitude encoding for arbitrary real or complex vectors, which demands quantum circuits of exponential depth in the number of qubits. AAE instead employs optimization-driven quantum state preparation, leveraging parameterized architectures and measurement-based loss functions to approximate a target state efficiently. This paradigm is directly relevant to the design and scalability of quantum machine learning, particularly in near-term quantum devices (NISQ).

1. Mathematical Formulation and Problem Definition

The central objective in amplitude encoding is to produce an $n$-qubit quantum state such that its amplitudes represent a classical data vector %%%%1%%%% or $c \in \mathbb{C}^N$ for $N = 2^n$. The mapping is given by:

$$|x\rangle = \frac{1}{\|x\|_2} \sum_{i=0}^{N-1} x_i |i\rangle, \qquad |c\rangle = \sum_{i=0}^{N-1} c_i |i\rangle,$$

with normalization $\sum_i |x_i|^2 = 1$ or $\sum_i |c_i|^2 = 1$ (Tudisco et al., 1 Aug 2025, Mitsuda et al., 2022).

Exact state preparation using known algorithms—such as the Möttönen method—requires $O(2^n)$ depth and multi-qubit controlled rotations, rendering it impractical for $n > 6$ on NISQ hardware (Morgan et al., 22 Aug 2025). Approximate amplitude encoding replaces this with a variational quantum circuit (PQC) $U(\theta)$, trained so that:

$U(\theta) |0\rangle^{\otimes n} \approx |x\rangle,$

with $\theta$ optimized to minimize an overlap or distance-based objective (Nakaji et al., 2021, Mitsuda et al., 2022).

2. Variational Ansatz and Optimization Frameworks

AAE architectures universally leverage hardware-efficient variational ansätze. A typical circuit comprises $l = O(\text{poly}(n))$ layers, each consisting of single-qubit rotations (e.g., $R_y(\theta_{q,r})$, $R_z(\theta_{q,r})$), followed by entangling gates (CNOTs) in a prescribed connectivity pattern (Mitsuda et al., 2022, Nakaji et al., 2021).

Optimization proceeds by defining a cost function that quantifies the deviation between the circuit-prepared quantum state and the target. In the real-vector setting, the loss can aggregate discrepancies in computational ($Z$) and Hadamard-transformed ($X$) basis measurement probabilities, often via the maximum mean discrepancy (MMD):

$$\mathcal{L}(\theta) = \tfrac{1}{2} \mathrm{MMD}^2(q_\theta, p) + \tfrac{1}{2} \mathrm{MMD}^2(q^H_\theta, p^H),$$

where $q_\theta(i)$, $q^H_\theta(j)$ are sampled output distributions and $p(i), p^H(j)$ are theoretical expectations (Nakaji et al., 2021). For complex vectors, fidelity-based objectives are employed:

$$C(\theta) = 1 - |\langle \text{target} | \psi(\theta) \rangle|^2,$$

with fidelity estimates obtained through classical shadow tomography (Mitsuda et al., 2022, Truger et al., 2024).

Gradients are computed via the parameter-shift rule, which exploits the analyticity of $U(\theta)$ (Mitsuda et al., 2022, Nakaji et al., 2021).

3. Theoretical Properties: Controllability, Trainability, and Subspace Constraints

AAE workflows are underpinned by rigorous criteria for controllability and trainability. For Hamming-weight-preserving circuits, it is demonstrated that amplitude encoding in a fixed-weight subspace $\mathcal{B}_k^n$ (dimension $d_k = \binom{n}{k}$) is achievable if the Quantum Fisher Information Matrix (QFIM) attains maximal rank $d_k - 1$ almost everywhere in parameter space (Monbroussou et al., 2023):

$$\text{Full-subspace controllability: } \max_\theta \operatorname{rank} \text{QFIM}(\theta) = d_k - 1 \implies \forall~|x\rangle~\exists~\theta~\text{s.t.}~U(\theta)|e_s\rangle = |x\rangle.$$

Furthermore, trainability is certified: the variance of cost function gradients is polynomially bounded in $d_k$, avoiding exponentially vanishing gradients (barren plateaus) for fixed $k$ (Monbroussou et al., 2023). Analytic arguments confirm that QFIM rank is constant almost everywhere due to the analytic nature of circuit mappings (Monbroussou et al., 2023).

4. Measurement and Fidelity Estimation Techniques

Measurement-based evaluation in AAE extends beyond computational basis sampling. To encode both sign and magnitude, loss aggregation over $Z$ and $X$ basis outcomes is essential (Nakaji et al., 2021). For complex amplitude encoding, fidelity is efficiently estimated through classical shadows, utilizing random Clifford unitaries and the inverse shadow channel for robust observable estimation:

$$\hat f(\theta) = \frac{1}{N_\text{shot}} \sum_{i=1}^{N_\text{shot}} \text{Tr}(\rho_\text{target} \hat \rho_i(\theta)),$$

with sampling complexity $O(1/\epsilon^2)$ for error $\epsilon$, independent of the qubit count $n$ (Mitsuda et al., 2022, Truger et al., 2024).

5. Circuit Complexity, Scalability, and Practical Trade-offs

The essential advantage of amplitude encoding remains its qubit efficiency: $N$-dimensional data requires only $n = \lceil \log_2 N \rceil$ qubits (Tudisco et al., 1 Aug 2025, Biswas, 18 Mar 2025). However, practical state preparation incurs notable cost:

• Exact circuits (Möttönen, Shende–Bullock–Markov): depth $O(2^n)$, infeasible for $n \gg 6$ (Morgan et al., 22 Aug 2025, Pagni et al., 9 May 2025).
• Variational circuits (AAE): depth $O(\text{poly}(n))$, suitable for NISQ devices. For many tasks, fidelity $\gtrsim 0.94$ is achieved with moderate layers and classical optimizer iterations (Mitsuda et al., 2022, Nakaji et al., 2021, Morgan et al., 22 Aug 2025), though structured data may allow further depth reductions (Pagni et al., 9 May 2025).
• Resource efficiency: Empirical studies show that VQC classifiers using amplitude encoding require far fewer parameters than classical neural networks, e.g., 26 in quantum Cart-Pole RL vs. $> 700$ in comparable MLPs (Chen et al., 2021).

Approximate encoders such as EnQode combine clustering-based preprocessing with ansatz tuning, enabling rapid state preparation and practical deployment in sequential circuits (QRNNs) while maintaining robust performance (Morgan et al., 22 Aug 2025).

6. Application Domains and Empirical Outcomes

AAE methodologies have found utility in multiple quantum machine learning domains:

• Classification tasks: Compact Hadamard Classifier (CHC) uses ACAE to encode complex kernel vectors, achieving high accuracy in benchmarks such as Iris and credit card fraud detection, with PQC depth scaling as $O(\text{poly}(n, m, 1))$ (Mitsuda et al., 2022).
• Variational Quantum Eigensolver (VQE) warm starts: ACAE pretraining supplies optimized initial parameters, yielding 2–3× faster convergence (in shots) and higher approximation ratios compared to random initialization (Truger et al., 2024).
• Reinforcement learning: Amplitude encoding in VQC architectures yields >10$\times$ parameter compression over classical agents, with evolutionary optimization preserving sample efficiency (Chen et al., 2021).
• Financial indicators and quantum SVD: AAE enables variational estimation of the SVD entropy in stock-market datasets, tracking classical market indicators within 5% error even at moderate circuit depth (Nakaji et al., 2021).
• Time-series forecasting (QRNNs): Amplitude-encoded QRNNs, particularly with data normalization augmentation and circuit structure optimization, outperform angle-encoded models in MSE and resilience to NISQ noise (Morgan et al., 22 Aug 2025).

Empirical fidelity metrics and performance statistics consistently support the feasibility of AAE in near-term domains, with optimized PQCs reaching $F \approx 0.95–0.99$ for several datasets (Mitsuda et al., 2022, Nakaji et al., 2021, Morgan et al., 22 Aug 2025).

7. Limitations, Open Challenges, and Prospective Extensions

Principal limitations in AAE stem from circuit expressivity, ansatz design, and fidelity bottlenecks:

• Encoding fidelity: For highly irregular or large data vectors, approximate encoding may achieve only $F \approx 0.63$ (Truger et al., 2024). Tuning ansatz depths, optimizer schedules, and measurement resources can partially rectify this.
• Postselection and probabilistic success: Some modern encoding schemes (e.g., MCX-based sublinear-depth algorithms) rely on ancilla measurement and postselection, incurring a success probability $\rho$ dependent on data sparsity (Pagni et al., 9 May 2025).
• Hardware constraints: Encoding designs should exploit device-native gates (e.g., MCX on ion traps), minimize two-qubit depth, and leverage hybrid encoding for shallow circuit realizations (Biswas, 18 Mar 2025).

A plausible implication is that future AAE systems will incorporate hybrid quantum-classical compression (MPS preprocessing), advanced measurement strategies (shadow derandomization), and modular circuit architectures combining amplitude, angle, and phase encoding strategies for maximal efficiency on scaled quantum platforms (Chen et al., 2021, Biswas, 18 Mar 2025).

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References: For rigorous implementation details and extended results, see (Monbroussou et al., 2023, Mitsuda et al., 2022, Truger et al., 2024, Nakaji et al., 2021, Tudisco et al., 1 Aug 2025, Morgan et al., 22 Aug 2025, Chen et al., 2021, Biswas, 18 Mar 2025, Pagni et al., 9 May 2025).