Modified Amplitude Encoding Overview

• Modified amplitude encoding is a set of techniques that generalize standard amplitude encoding, mitigating exponential gate-depth and improving hardware compatibility.
• It leverages methods like block-parallel encoding, polynomial MPS representations, and neural amplitude mapping to achieve efficient and adaptable quantum state preparation.
• These approaches enable practical quantum machine learning, robust signal processing, and versatile classical-to-quantum data encoding for real-world applications.

Modified amplitude encoding refers to a set of techniques that alter or generalize standard amplitude encoding to enhance efficiency, scalability, expressivity, hardware compatibility, or reconstruction capabilities in both quantum information processing and classical-to-quantum data mapping. These methods address critical bottlenecks of exact amplitude encoding, including exponential gate-depth, limited fidelity under noise, inflexibility for learnable mappings, and the need to encode additional data properties (such as magnitude or structured polynomial features).

1. Standard Amplitude Encoding and Its Limitations

In standard amplitude encoding, a normalized classical vector $x\in\mathbb{R}^N$ (%%%%1%%%% for $n$ qubits) is mapped to a quantum state

$$\ket{x} = \frac{1}{\|x\|_2} \sum_{i=0}^{N-1} x_i \ket{i}\,.$$

Quantum state preparation (QSP) routines realize this encoding via multi-level controlled rotations and ancillary registers with gate depth $\mathcal{O}(N)$, leading to two principal limitations:

• Exponential Circuit Depth: The gate count scales as $\mathcal{O}(2^n)$, restricting practical implementations to small $N$ on near-term hardware.
• Fixed Amplitude Preparation: The mapping is non-learnable and may not align with downstream variational circuit evolution or dataset-specific structural priors (Wang et al., 14 Aug 2025).

These constraints motivate a spectrum of modified amplitude encoding schemes targeting resource reductions, expressivity, or robustness (Morgan et al., 22 Aug 2025, Pagni et al., 21 Mar 2025, Gonzalez-Conde et al., 2023).

2. Resource-Efficient Quantum State Preparation

Several modified amplitude encoding algorithms have been developed to reduce state-preparation complexity while preserving high fidelity:

2.1. Circuit Depth Reduction via Parallelization and Amplitude Amplification

Pagni et al. introduce a shallow amplitude encoding protocol leveraging parallel block encoding and Grover-style amplitude amplification (Pagni et al., 21 Mar 2025):

• Block-Parallel Encoding: The $N$-vector is partitioned into $M$ parallel blocks, each handled by a corresponding ancillary sub-register. This yields a qubit cost of $\mathcal{O}(M\log N)$ and block-depth of $\mathcal{O}((N/M)\log(M+1))$.
• Success Probability and Amplitude Amplification: The raw success is $O(\rho)$, with data-density parameter $\rho = (1/N)\sum_{i=0}^{N-1}(x_i/\|x\|_\infty)^2$. Amplitude amplification reduces the number of trials from $O(1/\rho)$ to $O(1/\sqrt{\rho})$, providing a quadratic speedup for non-uniform data.
• Worst-Case and Average Runtime:
  • Fully parallel mode ($M=N$): $$\tau = O((1/\sqrt{\rho})\log(N+1)) \leq O(\sqrt{N}\log N)$$.
  • For random data, $\mathbb{E}[\tau] = O((\log N)^{1.5})$.

2.2. Polynomial Function Encoding

García-Pérez et al. demonstrate specialized methods for polynomial function amplitude encoding (Gonzalez-Conde et al., 2023):

• Matrix Product State (MPS) Representation: Exploits efficient tensor network representations, offering circuit-depth $O(n\,d^2)$ for polynomial degree $d$, and exactness if bond dimension $\chi = d+1$.
• Hadamard–Walsh and QSVT Encoding: Encodes linear or polynomial functions using a truncated Hadamard–Walsh expansion, block encoding, and Quantum Singular Value Transformation (QSVT), with rigorous error bounds and explicit trade-offs between ancillary qubits and fidelity.
• Complexity Table:

Method	Ancillas	Depth	Error Scaling
MPS (exact, $\chi$)	0	$O(n\chi^2)$	exact
DHWT+QSVT (exact, $n$)	$n+5$	$O(n\,d)$	exact $P(j)$
DHWT+QSVT (approx, $k_0$)	$k_0+5$	$O(k_0\,d/\mathcal{F})$	$O(2^{-k_0})$

3. Expressive, Data-Adapted Encodings: Neural Amplitude Encoding

Neural amplitude encoding replaces fixed, deterministic amplitude mapping with task-adaptive, learnable schemes (Wang et al., 14 Aug 2025):

• Energy-Based Amplitude Mapping: For input coordinate $\boldsymbol\Theta$ (plus optional latent code $\mathbf z$), an MLP $f_\phi$ outputs a vector of energies $E_i$, which is transformed into a categorical (Gibbs) distribution:

$$P_i = \frac{\exp(-\beta E_i)}{Z},\quad Z = \sum_{j=1}^N \exp(-\beta E_j).$$

• Amplitude State Preparation: The quantum input state is

$$\ket{\psi_{\rm in}} = \sum_{i=1}^N \sqrt{P_i} \ket{i},$$

and is always real-valued (phases $\phi_i=0$).

• Fully Parameterized PQC: Follows with $Y$-rotation layers and entangling gates, constellation designed to avoid barren plateaus and preserve gradient variance.
• Empirical Advantages: Improved trainability, convergence on high-frequency details, and joint handling of heterogeneous data collections, with demonstrable performance gains (e.g., MSE and PSNR metrics in Tables 1–2 of (Wang et al., 14 Aug 2025)).

4. Fast and Robust Amplitude Encoding for NISQ Devices

Approximate amplitude encoding strategies further focus on near-term quantum devices, where hardware noise and circuit depth are limiting factors (Morgan et al., 22 Aug 2025):

• EnQode: Implements approximate amplitude encoding by $k$-means clustering of training data, learning a low-depth, machine-specific parametrized ansatz for each centroid. For a new input, the nearest centroid’s circuit is used, with optional fine-tuning via gradient correction. For practical feature sets $(N=4,8,16,32)$, EnQode maintains fidelities $\gtrsim 0.74$ at hardware-compatible circuit depths.
• Magnitude Augmented Encodings: Augment the amplitude-encoded input with a normalized magnitude feature, e.g., via MinMax or MaxMin scaling, thereby restoring otherwise discarded norm information. This yields robust generalization improvements (up to 36% test MSE reduction on time-series forecasting benchmarks).

Encoding Variant	Description	Empirical MSE
Baseline (QRNN)	Canonical amplitude encoding	0.0088
+ MinMax amplitude	Adds MinMax magnitude feature	0.0067
+ MaxMin amplitude	Adds MaxMin magnitude feature	0.0056

• Alternating-Register Circuits: By pipelining feature map preparation across two register sets, overall quantum circuit depth is reduced from $\mathcal{O}(TN)$ to $\mathcal{O}(T\log N)$ for sequence models, with just one extra $\log N$ qubits.

5. Modified Amplitude Encoding in Classical Signal Processing

Amplitude encoding concepts generalize to classical signal acquisition. The delta-ramp encoder (or “modified amplitude encoding” in the classical context) replaces uniform time sampling with adaptive, time-encoded amplitude sampling (Martínez-Nuevo et al., 2018):

• Ramp-Segmentation: A sawtooth waveform of slope $\alpha$ is added to the input $x(t)$, converting it into $y(t)=x(t)+s(t)$, a strictly monotonic function for suitable $\alpha > \|x'(t)\|_\infty$.
• Time-Amplitude Duality: Level crossing times $\{t_k\}$ correspond to uniform amplitude samples of a (generally) monotonic transform $g(t)=\alpha t + x(t)$, with exact forward/inverse duality formulas.
• Iterative Reconstruction: The original signal $x(t)$ can be recovered from $\{t_k\}$ using a rapidly converging iterative amplitude-sampling algorithm, with error bounds controlled via the decay of the associated residual function’s spectrum.
• Comparison With Standard ADCs: By trading amplitude quantization for timing precision, this encoding enables adaptive sampling density where the input changes rapidly, outperforming conventional frame-based (uniform time) reconstruction as sampling approaches the Landau limit.

6. Applications and Practical Impact

Modified amplitude encoding underpins performance and resource improvements in quantum machine learning, quantum signal processing, and data-conditioned quantum circuit design:

• Quantum Machine Learning and QRNNs: Enables fast, near-term-implementable encoding for large feature spaces and sequential data (Morgan et al., 22 Aug 2025).
• Quantum Visual Fields (QVF): Offers learnable, data-aligned amplitude structures for geometric and signal field learning (Wang et al., 14 Aug 2025).
• Polynomial Function Solvers: Facilitates efficient initialization in quantum PDE solvers and quantum simulation tasks requiring structured initial states (Gonzalez-Conde et al., 2023).
• End-to-End Quantum Advantage: Preserves overall quantum algorithm speedup, as in quantum Fourier transform algorithms where the state preparation now becomes a sublinear or nearly polylogarithmic bottleneck for typical inputs (Pagni et al., 21 Mar 2025).
• Signal Acquisition: Classical delta-ramp encoding enhances fidelity and reconstruction rates for analog-to-digital conversion, motivating further generalizations in mixed classical/quantum architectures (Martínez-Nuevo et al., 2018).

7. Summary of Methodological Trade-offs

A compact comparison of paradigms for modified amplitude encoding:

Method/Class	Circuit/Algorithmic Depth	Fidelity/Error	Key Features	References
Standard AE (QSP)	$\mathcal{O}(N)$	Exact	Exponential depth	(Morgan et al., 22 Aug 2025)
Shallow+Amplified QAE	$\mathcal{O}((1/\sqrt{\rho})(N/M)\log(M+1))$	$\gtrsim 1 - \epsilon$	Parallel blocks, amplitude amp	(Pagni et al., 21 Mar 2025)
Neural AE (NAE)	Task-dependent	Learned	MLP-driven, learnable	(Wang et al., 14 Aug 2025)
Polynomial MPS+QSVT	$\mathcal{O}(n d)$	$O(2^{-k_0})$	Truncated expansion	(Gonzalez-Conde et al., 2023)
EnQode	$\mathcal{O}(\log N)$	$f\gtrsim 0.9$ (typ.)	Cluster-based, NISQ-ready	(Morgan et al., 22 Aug 2025)
Delta-ramp (classical)	Algorithm-dependent (iterative)	Rapid convergence	Adaptive sampling rate	(Martínez-Nuevo et al., 2018)

The choice among these approaches depends on hardware constraints, desired fidelity, downstream algorithm compatibility, and computational resources for classical preprocessing (e.g., clustering, SVD, or pre-training).

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Modified amplitude encoding thus encompasses a set of quantum and classical methods that generalize the amplitude mapping paradigm for improved efficiency, flexibility, and performance, with concrete instantiations addressing current bottlenecks in quantum machine learning, signal representation, and quantum state engineering.