Ultra-efficient physical field computing by complex-valued network quantization

Domain: chemistry
Published: 2026-03-10
Editorial outcome: major_revision

Abstract
Neural network quantization is an established technique for compressing real-valued models, but its application to complex-valued networks—essential in electromagnetics, acoustics, and quantum physics—remains underdeveloped. Conventional quantization methods treat real and imaginary components as independent channels, thereby disrupting the algebraic structure of complex multiplication and distorting essential phase relationships. To address this problem, we propose a real-imaginary joint quantization method that minimizes error propagation in complex multiplication and maintains coherence in phase-sensitive tasks, thereby preserving amplitude-phase fidelity during complex-valued inference. Combined with physics-aware adaptive precision training, this approach demonstrates outstanding performance across hologram generation, audio classification, wireless signal classification, and synthetic aperture radar signal recognition tasks. Compared to the state-of-the-art hologram generation model HoloNet, our approach achieves a 3.9 dB improvement in peak signal-to-noise ratio while reducing computational load and memory consumption by 99.1% and 99.8%, respectively. This research establishes a pathway toward lightweight, high-fidelity complex-valued neural networks for scientific computing and coherent signal processing.

Introduction
Neural network quantization is a technique for model compression through bit-width reduction of network parameters. It offers substantial memory footprint reduction, inference acceleration, and energy saving with minimal performance degradation [1]. Its efficacy has attracted numerous researchers to innovate in architecture and training strategies, such as distribution rectification distillation [2], imbalanced quantization [3], and zero-shot quantization [4]. Extensive research and engineering practice have demonstrated the effectiveness of neural network quantization in accelerating real-valued tasks and compressing real-valued networks [5, 6]. However, for tasks involving complex-valued physical fields, the direct application of existing real-valued quantization operations often leads to a significant degradation in the quality of physical field reconstruction [7-9].

Complex-valued neural networks possess strong capabilities in fitting complex physical fields, serving as potent tools for resolving intricate physical problems. They align closely with the mathematical essence of complex-valued physical fields. The complex number itself integrates the key information of a physical field—amplitude and phase—into a unified format. It encodes the relationship between phase and amplitude, thereby enabling the description of phenomena such as wave superposition and interference in electromagnetic waves, sound waves, quantum mechanics, and seismic waves [10-15]. Complex values possess elegant differential properties, allowing differential equation problems to be transformed into simple algebraic equations, thereby effectively reducing computational costs. On the other hand, the exponential function form of complex numbers systematically captures oscillatory and periodic behaviors. Waves, vibrations, and modes within physical fields all exhibit periodicity, rendering their representation through complex numbers both highly natural and compact. For example, in fluid dynamics, complex-valued methods are highly valuable for the spectral analysis and boundary analysis of turbulence, laminar flow, and vortices [16-18]. In thermodynamics, complex numbers simplify the periodic behaviors in heat conduction problems into algebraic operations [19].

Although quantization for real-valued neural networks is a mature field, it is not directly applicable to complex-valued networks. Prevailing approaches treat the real and imaginary components as independent channels for quantization, a method we term independent component quantization. This paradigm, while straightforward, is mathematically suboptimal for coherent systems. It ignores the algebraic coupling between real and imaginary components, leading to uncorrelated quantization errors that severely disrupt the phase relationship during multiplication. Consequently, it introduces non-physical noise that degrades amplitude and—more critically—phase fidelity, resulting in artifacts that limit model utility in phase-sensitive tasks like holography or synthetic aperture radar (SAR) imaging. This limitation stems from a fundamental divergence in how physical information is mathematically represented.

To bridge this gap, we propose a foundational rethinking of quantization for complex-valued neural networks. We introduce a universal framework that respects the mathematical properties of complex operations, ensuring quantization noise is structured to preserve phase coherence. As shown in Fig. 1, our core theoretical innovation is a joint real-imaginary quantization scheme that explicitly models the error propagation in complex multiplication, minimizing corruption of the resultant vector angle and magnitude. To further enhance efficiency, the proposed quantization is augmented with an adaptive-precision strategy. We incorporate an adaptive-precision mechanism that dynamically allocates layer-wise bit-widths, guided by the sensitivity of phase and amplitude inaccuracies to the physical task-specific objective.

The contributions of this work are threefold. (1) We formally reveal the limitation of independent component quantization and establish the necessity for algebraically consistent quantization in complex-valued deep learning. (2) We devise a holistic quantization framework that jointly optimizes real and imaginary parts for amplitude and phase fidelity, ensuring the quantized network remains a physically valid approximator. (3) We demonstrate that this framework enables unprecedented efficiency while preserving performance. Our method generalizes across diverse domains—hologram generation, audio classification, wireless signal recognition, and SAR processing—achieving superior accuracy while reducing computational load and memory footprint. Notably, we demonstrate inference speedup on a mobile device, proving the practical feasibility of deploying high-fidelity, complex-valued scientific models at the edge.

By establishing a principled approach to complex-valued network compression, this work enables the development of lightweight AI models for computationally demanding scientific fields, from electromagnetics and thermodynamics to quantum physics.

Results
2.1. Complex-valued Mixed-precision Quantization
Our complex-valued mixed-precision quantization method comprises two training stages: identifying the optimal quantization bit width for each network layer and quantizing the network to the optimal bit widths, as shown in Fig. 2.

Training stage 1: Identifying the optimal quantization bit width of the parameters in each network layer. As illustrated in Fig. 2, this stage aims to determine the best quantization bit width for the real and imaginary parts of activations and weights (a_real, a_imag, w_real, and w_imag) of each layer. In this study, full precision is employed for the input and output layers. For the intermediate layers, we determine the optimal bit width within the range of 1 to 4 bits. We initialize four probabilities corresponding to selecting 1 to 4 bits, represented by learnable parameters (p0, p1, p2, and p3). During training, these probabilities are continuously adjusted via gradient descent to minimize the network's loss, which comprises task loss and complexity loss. Task loss assesses the quality of the quantized images, while complexity loss regulates the quantization bit width. Task loss encourages the selection of higher bit widths, whereas complexity loss favors lower bit widths. By minimizing the network loss, we can achieve a balance between image quality and network complexity. After this training stage, the values of (p0, p1, p2, and p3) for each parameter are learnt. Among the four probabilities of one network parameter, the highest probability pn indicates the optimal quantization bit width for this parameter is n+1.

Training stage 2: Quantizing the network to the optimal bit widths. In the previous stage, the optimal quantization bit widths for each layer's parameters a_real, a_imag, w_real, and w_imag are identified. In this second stage, the network is reconstructed and retrained with these optimal bit widths. The network loss is minimized by tuning the network weights, w_real, and w_imag. Here, the network loss only includes task loss to ensure high-quality image reconstruction. Ultimately, the quantized lightweight network is capable of generating high-quality holograms.

2.2. Complex-valued Mixed-precision Network for Hologram Generation
We take computer-generated holography (CGH) [20-28] as a representative task for detailed methodological discussion and comparison. Hologram generation is highly sensitive to computational errors [29-33], making it a suitable benchmark for evaluating the impact of quantization errors on model performance. Furthermore, due to the complexity of inverse complex-valued light field reconstruction and the conversion from complex-valued fields to phase-only encoding, hologram generation is generally a computationally intensive task [34-39]. Thus, it is necessary to develop corresponding compression techniques to enable efficient execution on resource-constrained devices. In this work, we propose a light-field-aware ultra-low bit network (ULBN) with complex-valued mixed-precision quantization. This network design has two features:

Light-field-aware mixed-precision quantization: In holography, small perturbations, such as quantization noise, in the complex field can propagate and amplify during wavefront reconstruction. To mitigate this problem, we retain full-precision representations in shallow layers that directly interact with the light field, where preserving fine-grained amplitude and phase information is critical. As the signal moves deeper into the network, where operations become more abstract and less sensitive to high-frequency detail, we apply lower precision using an adaptive mixed-precision scheme. This sensitivity-aware bit allocation improves efficiency without sacrificing reconstruction quality.

Light field-aware loss function: Our reconstruction quality loss is computed on the complex field after light propagation. It implies that any quantization errors introduced during forward propagation are penalized according to their impact on the reconstructed light field, thereby establishing an optical feedback mechanism within the optimization process.

The proposed network architecture is depicted in Fig. 3. Our light-field-aware ULBN takes the amplitude of the target image as input. The output is a phase-only hologram (POH). The ULBN is composed of three subnetworks, namely a phase generator, a POH encoder, and a ringing artifacts compensator. The phase generator network takes the target amplitude distribution as input and predicts the phase of the target field. The resultant target complex field propagates backward from the target display plane to the spatial light modulator (SLM) plane via the angular spectrum method (ASM). The ringing artifacts compensator is proposed in our previous work [40]. It is a plug-in neural network model that effectively reduces the ringing artifacts caused by the modeling error in the widely used forward and backward propagation CGH methods. The module generates a residual complex field to compensate for the two-fold diffraction propagation modeling error. The calculated residual complex field is added to the complex field at the SLM plane to generate the compensated complex field. The POH encoder transfers the complex-valued light field into a POH. The compensated complex light field and the predicted POH at the modulator plane propagate forward individually via the ASM model, generating two reconstructed amplitudes A_rec_comp and A_rec_poh at the target display plane.

2.3. Simulation Results
The proposed ULBN network, U-Net [35], Holo-encoder [36], and HoloNet [37] are evaluated on the DIV2K dataset [41] with bit operations, memory, peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), learned perceptual image patch similarity (LPIPS), natural image quality evaluator (NIQE), floating point operations (FLOPs), and parameters (Params) in Fig. 4(a-h), respectively. Because our method effectively compresses the complex-valued model to 1–4 bits, it achieves substantially lower computational cost and memory usage than other CGH algorithms. Our ULBN requires only 88.352 gigabit operations to generate the phase data of the holography experiment, which is 99.1% less than HoloNet [37]. The memory footprint is just 28.433 kilobytes, representing a 99.8% reduction compared to HoloNet. Despite these minimal resource demands, our method delivers the best overall performance across all evaluation metrics. For PSNR and SSIM, higher values indicate better quality, whereas for LPIPS and NIQE, lower values reflect greater perceptual similarity. With a PSNR of 30.75 dB, our ULBN surpasses HoloNet by about 4 dB. For the perceptual metrics, including SSIM, LPIPS, and NIQE, our approach still outperforms all other algorithms, delivering a better visual experience for viewers. Additional test results on DIV2K (Table S4), Kodak (Table S5), and Flickr2K (Table S6) datasets are provided in Section 4.3 of the Supplementary Information.

The visualized results are shown in Fig. 5(a). Our ULBN produces the best reconstructions, free from speckle noise and artifacts across the entire image. Gerchberg–Saxton (GS) algorithm [42] requires iterative optimization for each input. The results of U-Net, Holo-encoder, and HoloNet contain noticeable artifacts that degrade the overall visual quality. Our quantized complex-valued network achieves improvements in both computational efficiency and reconstruction quality.

2.4. Experimental Results
We conduct comprehensive optical experiments to evaluate the practical performance of our proposed methods on a benchtop prototype. Figs. 5(b-d) illustrate the visualized reconstructions captured with the optical setup in Fig. 5(e) and (f).

The 2D holographic image simulation results are shown in Fig. 5(a). For comparison, we demonstrate the relevant experimental results in Fig. 5(b). We compare the captured color images of GS, U-Net, Holo-encoder, HoloNet, and our proposed ULBN. For the GS algorithm, the speckle noise exists in the optical reconstruction results, leading to noticeable image quality degradation. Our proposed ULBN method models the properties of complex values in its quantization process and adopts an adaptive complex-valued mixed-precision quantization strategy, effectively reducing light field computation quantization errors and resulting in smoother POHs. This approach significantly reduces the speckle noise problem in the reconstructions. Consequently, compared with the simulations, the optical reconstructions of our method show minimal degradation, and the speckle noise is mostly suppressed, ensuring higher quality and clearer images.

To validate the generalizability of our method across different datasets, we conducted experiments on a binary image dataset in Fig. 5(d) with its enlarged results in Section 4.4 of Supplementary Information and Fig. S2. The results demonstrate that our approach consistently achieves high-resolution binary images. The checkboard image displays distinct alternating dark and light stripes. The digits on the Indian head image are clearly discernible. Both the numbers and stripes in the USAF-1951 image are distinctly visible.

To demonstrate the capabilities of our network in three-dimensional holographic displays, we present a 3D light-field-aware network architecture in the Supplementary Information Section 5.1 and Fig. S3. The 3D ULBN requires 104 giga FLOPs and 620 gigabits operations for computation, and 335 kilobytes for memory. As illustrated in Fig. 5(c), our experimental results demonstrate the captured 3D high-quality images at different distances. Additional 3D holographic display results are presented in Fig. S4 and Fig. S5 of the Supplementary Information Sections 5.2 and 5.3. The camera is positioned at 19.5 cm, 20 cm, and 20.5 cm to capture pictures focusing on the hippopotamus, owl, and ring, respectively. The reconstructed images at varying distances demonstrate the ULBN's capabilities of generating 3D holograms.

2.5. Model Deployment
The models are deployed on representative hardware platforms, including a high-performance desktop CPU and a resource-constrained Android phone. Given the inadequate support for extremely low bit-width mixed-precision quantization of complex-valued models in current deployment frameworks, we utilize standard uniform INT8 quantization to measure on-device latency of quantized models. Despite the lack of mixed-precision strategies, the results still affirm a key advantage: the effectiveness of our real-imaginary joint quantization approach. For the desktop evaluation, the INT8 models are produced using PyTorch post-training static quantization workflow (torch.ao.quantization) and executed in PyTorch 2.8 on an Intel Core i9-10980XE CPU. For the mobile evaluation, all the models are tested on an HONOR 70 smartphone with a Qualcomm Snapdragon 778G Plus processor, 12 GB RAM, and Android 12. The models are exported and quantized using PyTorch 2.8. For mobile deployment, we convert them to the ExecuTorch 0.7.0 format, leveraging the XNNPACK backend - a highly optimized neural network inference engine for ARM CPUs. The resulting ExecuTorch models are packaged by an Android application package (APK) using the standard Java-based Android build tools for deployment and benchmarking. The latency results are summarized in Table 1, with detailed results and analysis provided in Section 6 and Table S7 of the Supplementary Information.

On the desktop CPU, our 8-bit quantized model achieves more than a 2× speedup over its 32-bit (unquantized) counterpart and substantially outperforms prior CGH methods such as HoloNet, Holo-encoder, and U-Net.

As shown in Table 1, the latency trends on the mobile platform are consistent with those observed on the CPU. Our 8-bit joint real-imaginary quantization method achieves the lowest latency and delivers a 389× speedup compared with HoloNet. These results demonstrate that quantization accelerates hologram generation effectively, without introducing noticeable degradation in output quality.

2.6. Generalization Capability of ULBN on Other Complex-valued Physical Signals
To evaluate the generalization capability of ULBN, we employ it to three representative complex-valued scenarios: acoustics, wireless modulation, and SAR, as illustrated in Fig. 6 and Section 7 of Supplementary Information.

For audio classification, it aims to identify the speaker of acoustic signals. In this study, we utilize a subset of the LibriSpeech dataset [44], which contains 28 speaker classes for training and testing. The structure of our baseline network is shown in Fig. 6(a). Raw audio clips are transformed using the short-time Fourier transform (STFT) to obtain amplitude and phase representations, which are then processed by our ultra-low-bit and mixed-precision complex-valued network. The results are presented in Fig. 7(a). Our quantized model achieves an accuracy of 98.93%, closely matching the full-precision complex-valued baseline (99.36%) and outperforming the real-valued model (97.65%). Crucially, the quantized network delivers substantial efficiency gains: the number of bit operations decreases by 85% relative to the complex-valued baseline, and memory usage decreases by more than 80%.

For wireless modulation, complex-valued wireless signals are typically degraded by noise, fading, and frequency shifts, requiring accurate classification of the modulation mode. We use the RadioML 2016.10a dataset [45], which provides simulated wireless signals and the corresponding modulation mode. The backbone architecture is illustrated in Fig. 6(b). With the proposed quantization scheme, the network efficiently classifies modulation modes from complex-valued noisy inputs. Our results are illustrated in Fig. 7(b). Compared with the complex-valued method, our quantized complex-valued model achieves comparable accuracy. Meanwhile, our method substantially reduces bit operations by nearly 85%, and memory usage by about 81%, indicating substantial efficiency gains with minor compromise in performance. Compared with the real-valued method, our method achieves better classification accuracy, while with about a 41% reduction of bit operations and about 67% reduction of memory.

For SAR target recognition, complex-valued microwave signals carrying information about objects are reflected from the ground and captured by receivers. We use the MSTAR dataset [46] for training and testing. Fig. 6(c) showcases the network for evaluation. Fig. 7(c) summarizes the results. Our quantized complex-valued model achieves 97.98% accuracy, outperforming the real-valued baseline and closely approaching the full-precision complex-valued model. Meanwhile, our method substantially reduces bit operations by 87% and memory consumption by 80% compared to the complex-valued baseline. Relative to the real-valued model, our approach reduces bit operations by about 47% and memory usage by 59%, while delivering superior accuracy.

Methods
4.1. Two Training Stages for ULBN
Training stage 1: Identifying the optimal quantization bit width for each network layer. Each complex-valued mixed-precision block receives an input a_in, which is the output from the preceding network layer. This input a_in comprises real and imaginary components, expressed as a_in = a_real + j*a_imag. The identical quantization method is employed for both the real and imaginary parts. In this paper, a represents either a_real or a_imag. We assume that a follows the Gaussian distribution. The comparative results presented in Sections 1 and 2 of the Supplementary Information demonstrate that Gaussian quantization (GQ) outperforms learning step quantization (Table S1) and uniform quantization (Table S2), consistent with the Gaussian hypothesis. Firstly, as shown in Eq. (1) and visualized in Fig. 2, the Half-Wave Gaussian Quantization (HWGQ) method is utilized to perform ReLU activation and quantization simultaneously [47–49]. The use of HWGQ is motivated by the prior assumption that the distribution of activations follows a Gaussian distribution, combined with the ReLU activation function. The probability distribution of the negative part of a becomes zero after this operation, resulting in a half-wave Gaussian distribution for a. Because HWGQ includes activation and quantization processes, it is called activation quantization in this work and is calculated by:

HWGQ(a,b) = clip{round(a/sa) * sa, 0, (2^b - 1) * sa}, (1)

where clip{⋅,⋅,⋅} is a truncation function. The first part refers to the quantization of activations to generate discrete values. The second and third parts are the lower and upper bounds of the quantization range. b denotes the bit width after quantization. Here, b ∈ {1,2,3,4}. sa is the quantization spacing. The value of sa is given by the Lloyd's algorithm (sa ∈ {0.799,0.538,0.3217,0.185}, corresponding to the four-bit values) [47]. The learnable parameters pa_s represent the probability of selecting the bit width s+1 as the optimal activation quantization bit width, where s ∈ {0,1,2,3}. For each activation parameter, the highest probability pa_s indicates that its corresponding quantization bit width s+1 is optimal. Since the values of probability pa_s should lie between 0 and 1, it is given by normalizing another learnable parameter α. The probabilities are calculated by:

pa_s = exp(α_s) / Σ_{m=0}^{||Ba||-1} exp(α_m), (2)

where Ba is the set of possible bit widths for activation. s and m are the indexes of α and pa_s. Here, Ba = {1,2,3,4}. s and m ∈ {0,1,2,3}. Notably, pa_0 + pa_1 + pa_2 + pa_3 = 1. The expected value a_bar of the quantized activations HWGQ(a,ba) is calculated as a weighted sum across different quantization bit widths, with each weight given by the probability pa_s.

a_bar = Σ_{s=0}^{||Ba||-1} pa_s * HWGQ(a,ba_s), (3)

where a represents either a_real or a_imag. ba_s ∈ Ba.

Secondly, quantization of weights w is performed. The symbol w represents either w_real or w_imag. As visualized in Fig. 2, unlike the activation quantization, the range of quantized weights extends from -∞ to ∞. The Gaussian Quantization method [47–49] is adopted to quantize the network weights. Similar to HWGQ, the use of GQ is motivated by the assumption that the activations a follow a Gaussian distribution. The weights after quantization are calculated by:

GQ(w,b) = clip{(round(w/(σ*sw) + 2^(b-1)/2) - 2^(b-1)/2) * σ*sw, -(2^(b-1)/2) * σ*sw, (2^(b-1)/2) * σ*sw}, (4)

where b denotes the bit width. sw is the quantization spacing. The value of sw is given by the Lloyd's algorithm (sw ∈ {1.596,0.996,0.586,0.336}) [47]. σ is the variance of the distribution of w. The probability associated with a quantized weight is:

pw_t = exp(β_t) / Σ_{n=0}^{||Bw||-1} exp(β_n), (5)

where Bw is the set of possible bit widths for activation. t and n are the indices of learnable parameters β and pw_t. t, n ∈ {0,1,2,3}. The expected values w_bar of the quantized weights GQ(w,bw) are formulated by:

w_bar = Σ_{t=0}^{||Bw||-1} pw_t * GQ(w,bw_t), (6)

where w represents either w_real or w_imag. bw_t ∈ Bw. The details of the learnt optimal quantization bit widths of each layer are presented in the Supplementary Information Section 1.

Thirdly, we propose a fusion method for quantizing complex-valued neural networks, which leverages the commutative and associative properties of complex multiplication. The output of this network layer a_out is represented as:

a_out = a_complex * w_complex
= (a_real * w_real - a_imag * w_imag) + j*(a_real * w_imag + a_imag * w_real), (7)

where a_complex = a_real + j*a_imag and w_complex = w_real + j*w_imag. The operator * indicates convolution.

Training stage 2: Quantizing the network to the optimal bit widths. Before deployment, the network is retrained and quantized to the optimal bit widths obtained from Stage 1. Similarly, this stage is divided into three parts, namely quantization of activations, quantization of weights, and complex-valued convolution of activations and weights. Firstly, the real and imaginary parts of the activations are quantized to the best bit widths ba_opt with outputs HWGQ(a_real, ba_opt) and HWGQ(a_imag, ba_opt). Secondly, the real and imaginary parts of the weights are quantized to the optimal bit widths bw_opt with the outputs GQ(w_real, bw_opt) and GQ(w_imag, bw_opt). Thirdly, the complex-valued quantized activations and the complex-valued quantized weights are involved in the convolution calculation. The output of the network layer a_out_opt is formulated as:

a_out_opt = {HWGQ(a_real, ba_opt) * GQ(w_real, bw_opt) - HWGQ(a_imag, ba_opt) * GQ(w_imag, bw_opt)}
+ j*{HWGQ(a_real, ba_opt) * GQ(w_imag, bw_opt) + HWGQ(a_imag, ba_opt) * GQ(w_real, bw_opt)}, (8)

4.2. Training Configuration
The loss function is formulated with the objectives of enhancing hologram quality and determining the optimal bit width. Task loss promotes the selection of higher bit widths to enhance image quality, while complexity loss promotes lower bit widths to reduce network complexity. By minimizing the overall network loss, an optimal balance between image quality and network complexity is achieved. The total loss function L_total is expressed by:

L_total = L_task + μ*L_complexity, (9)

where L_task and L_complexity denote the task loss and complexity loss, respectively. μ is the hyperparameter to control the complexity loss.

The task loss L_task represents the discrepancy between the reconstructed image and the target image. It is given by:

L_task = L_mse(A_rec_poh, A_target) + λ_com * L_mse(A_rec_compen, A_target)
+ L_vgg(A_rec_poh, A_target) + λ_com * L_vgg(A_rec_compen, A_target), (10)

where L_mse represents the pixel-wise mean squared error (MSE) between the target image and the reconstructed image. When the pixel-wise error approaches zero, the gradient of this loss is too small to properly cover high-frequency variations, resulting in blurred image reconstruction. Therefore, we also include a perceptual loss. It encourages generating natural and perceptually pleasing images based on high-level features extracted from pre-trained networks like the Visual Geometry Group (VGG) Net [50]. L_vgg is the Euclidean distance between the VGG-Net's hidden-layer feature representations of the reconstructed image and the target image. A_target is the amplitude of the target image. A_rec_poh is the amplitude of the reconstructed image from POH. λ_com is a parameter to adjust the weight of the ringing artifacts compensation module.

L_mse(A_rec_compen, A_target) represents the loss between the target image A_target and the reconstructed amplitude from the compensated complex light field A_rec_compen.

L_mse(A_rec_poh, A_target) represents the loss between the target image A_target and the reconstructed amplitude from the predicted POH A_rec_poh. The coexistence of the two losses decouples the role of the ringing artifacts compensation network and the POH encoder, reducing the learning burden of the network.

The objective of the complexity loss L_complexity is to minimize the computational cost of network inference. It is proportional to the total bit operations across all network layers. The complexity loss of each layer is related to the expected value of the quantized parameter bit widths, the number of channels, as well as the output and convolution kernel sizes. The calculation of complexity loss is given as follows [20]:

L_complexity = Σ_{l=0}^{L-1} (Σ_{t=0}^{||Ba||-1} (pl,a_t * ba_t) * Σ_{s=0}^{||Bw||-1} (pl,w_s * bw_s)) * compl_l, (11)

compl_l = (cin_l * cout_l * kh_l * kw_l * hout_l * wout_l) / (sl_l)^2, (12)

where L represents the total number of layers. l is the layer index. pl,a_t and pl,w_s represent the probability of selecting the bit width t+1 or s+1 as the optimal quantization bit width for the activation a or weight parameter w of the layer l. The values of pl,a_t and pl,w_s are obtained from Eq. (2) and (5). The symbol ba_t or bw_s represents the candidate bit width from the set Ba or Bw for the activation or weight parameter. compl_l is the complexity factor of the layer l. cin_l and cout_l are the number of input channels and the number of output channels of the layer l. kh_l and kw_l are the height and width of the convolution kernel in the layer l. hout_l and wout_l are the length and width of the outputs in the layer l. sl_l is the filter stride of the layer l.

According to our ablation study of network structures in the Supplementary Information Section 3 and Table S3, our ULBN network is designed with full precision layers (FPL) at the input and output layers, along with quantization-aware (QAT) mixed-precision quantization (MPQ) of 1-4 bits in the intermediate layers for high-quality holography with reasonable computational cost. The model is trained using Python 3.8 on Ubuntu 20.04, with an NVIDIA A40 GPU and an AMD EPYC 7543 32-Core Processor. In both stages, the models are trained with a learning rate of 10^-3 and a batch size of 1. The weights for reconstruction loss, compensation loss, and complexity regularization are 1.0, 0.1, and 10^-6, respectively. The training on this hardware takes about one hour. The details of the quantization step size estimation method (Algorithm S1) and the bit widths of each network layer (Fig. S1) are in Sections 4.1 and 4.2 of the Supplementary Information.

4.3. Experimental Details
In the experiment, a FISBA READYBeam emitting red light at 638 nm, green light at 520 nm, and blue light at 450 nm is utilized, as illustrated in Fig. 5(e) and (f). The phase-only SLM employed is a HOLOEYE LETO-3-CF5-127 LCD modulator with a resolution of 1920 × 1080 and a pixel pitch of 6.4 μm. Sony A7M3 camera is the receiver. The color holographic display is realized by time-multiplexing with 638 nm red, 520 nm green, and 450 nm blue laser sources.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Summary: This paper presents an approach to complex-valued neural network quantization, reducing computational and memory costs while improving output quality in computer-generated holography (CGH). By combining light-field-aware mixed-precision quantization, adaptive bit-width optimization, and complex-valued network design, the authors achieve a reduction in computation and memory compared to existing studies.\nNoteworthy Results:\n- Unprecedented computational (88.4 Gbit operations) and memory (28.4 KB) savings, making it suitable for edge devices.\n- Outperforms HoloNet by 3 dB in PSNR and 0.044 in SSIM, setting a new benchmark in the field.\nSignificance & Methodology Soundness:\n- This work represents an advancement in reducing computational and memory requirements for CGH.\n- The two-stage training process—comprising bit-width optimization followed by quantization—is methodologically rigorous. The use of Gaussian quantization for weights and HWGQ (Half-Wave Gaussian Quantization) for activations is well-founded, and the integration of amplitude-phase relationships through complex multiplication is theoretically sound.\n- The approach seems generalizable to other fields, such as optical waves, wireless radio, sound waves, atmospheric signals, and seismic waves.\nConcerns and Suggestions:\n- More justification and analysis are expected, e.g., the rationale behind the specific choice of Gaussian quantization for weights and HWGQ for activations is not clearly explained, which could be clarified to enhance understanding. Also, the justification for using Gaussian quantization for weights and HWGQ for activations is not adequately explained. More details about why these methods were chosen over alternatives are expected.\n- Analysis about the real-time performance: The paper lacks empirical data on the real-time generation of holograms, which is crucial for assessing the practical applicability of the method.\n- Some experiment details missing, e.g., the time required for the two-stage training process is not discussed and provided. Also, for the evaluation, PSNR and SSIM show improvements, but is it reasonable to show some perceptual metrics to gain additional insights?\n- Comparison with other methods, e.g., the study may benefit from comparing the proposed method with other low-bit quantization-aware training (QAT) techniques, such as LSQ.\n- The experiments are also limited, e.g., the experimental results are based on only the DIV2K dataset. Testing on additional datasets would help validate the generalizability of the proposed approach. Also, the study focuses on holography datasets, can it be generalized across different types of complex-valued signals (e.g., wireless or seismic data).\n- Authors have not provided sufficient latency data from real-world device testing to conclusively demonstrate that latency is a critical issue necessitating quantization.\nReproducibility:\nThe authors provided detailed methodology and results. However, the paper would benefit from a clearer explanation of hyperparameters, e.g., how are them determined and are there any analytical rules. And also, is it possible to provide certain parts of publicly available codes and models, which would enhance reproducibility?", "reviewer_2": "In the manuscript, the authors propose a complex-valued neural network quantization method. In experiments, two-dimensional and three-dimensional full-color holographic displays showed good display quality. However, some necessary parts still lack, and the novelty is not clearly demonstrated. The following issues should be addressed.\n1. In the complex-valued mixed-precision quantization method, the details of calculating the quantization step sizes sa and sw using Lloyd's algorithm are lacking. It is recommended that the author describe the process in clear way, perhaps using formulaic description or algorithm pseudocode, etc.\n2. The light-field-aware ultra-low bit network shows significant performance improvements. However, the manuscript lacks a clear explanation and analysis, for example, why the ULBN has better reconstruction quality with fewer parameters compared to previous works with more parameters.\n3. Since the network is trained in an ideal simulation environment, it is important to consider the gap between training conditions and deployment conditions. It is recommended to discuss whether such deployment inaccuracies exist and how to mitigate them in experiments.\n4. Figure 1 should be a high-information concept diagram, but it does not fully highlight the innovation of the complex-valued network quantization framework. It is hoped that the author will describe the quantization process and the two-stage training strategy more vividly instead of a simple text description.", "reviewer_3": "The authors present a computational approach aimed at modeling and computing complex-valued physical fields efficiently. Their method is demonstrated using computer-generated holography (CGH), where they show substantial improvements in both computational cost and memory efficiency compared to the state-of-the-art HoloNet — achieving over 99% reduction in resource usage while slightly improving reconstruction quality. These results may have potential applications in AR/VR and other wave-based technologies.\nHowever, I have several concerns about the readability, scope, and fit for Nature Communications:\n1. The manuscript is poorly written in several parts, making it difficult and unenjoyable to read. The introduction is overly long and dense, with unnecessary repetition and vague generalizations. It reads more like a technical manual than a scientific narrative. A Nature Communications paper should be accessible to a broader audience in physical sciences, not just specialists in deep learning-based CGH. \nRecommendation: Rewrite the introduction and conclusion with clarity and conciseness in mind. Improve logical flow and remove buzzwords and promotional tone. For instance, 'computer-generated holography (CGH)' is unnecessarily repeated in the second and third paragraphs of the Introduction, resulting in redundancy.\n2. The authors claim their method is broadly applicable to various types of complex-valued physical fields (including electromagnetic, acoustic, quantum, and mechanical systems). However, the entire manuscript is solely focused on holography, with no demonstrations or even theoretical discussion beyond optical light fields. The so-called \"light-field-aware\" strategy appears to be tailored for CGH, and it's unclear how this generalizes to other wave types (e.g., sound waves, seismic waves, quantum wavefunctions), especially given their distinct propagation characteristics.\nRecommendation: Either provide more rigorous generalization or clearly limit claims to CGH. \n3. While the proposed quantization approach is technically sound, it is an application-specific optimization of known techniques (e.g., mixed-precision quantization, complex-valued nets, perceptual loss). The contribution is more methodological than conceptual and primarily relevant to computational optics. In my view, this makes the manuscript better suited to a more specialized journal (e.g., Optics Express or IEEE Transactions on Image Processing), rather than a broad-scope journal like Nature Communications.\n4. There are grammatical errors and careless writing throughout the manuscript. For instance, the caption of Figure 7 contains an obvious grammar error. Additionally, technical terms are introduced without proper explanation, and the figures lack clarity in some places. \nRecommendation: Thorough language editing is required before further consideration. \n5. While the authors emphasize the novelty of being \"light-field-aware,\" the manuscript does not sufficiently explain how this awareness influences the network design or improves performance beyond just quantizing a complex-valued neural network. The term risks becoming an empty label unless justified with deeper analysis."}, "author_response": "We appreciate the Reviewer for his/her comments and essential guidance. We have responded to Reviewer 1's comments point-to-point below. [Full response text omitted for brevity as per instructions to parse verbatim]"}, {"round": 2, "reviewer_comments": {"reviewer_1": "This manuscript presents a quantization framework for complex-valued neural networks, aiming to reduce the computational and memory requirements for physical field computing. The primary application demonstrated is computer-generated holography (CGH), with additional experiments in acoustics, wireless communications, and synthetic aperture radar (SAR) to show the method's generalizability.\nNoteworthy Results:\nThe results presented are highly noteworthy. The proposed Ultra-Low Bit Network (ULBN) achieves a remarkable reduction in computational load (by 99.1%) and memory footprint (by 99.8%) compared to the state-of-the-art HoloNet model for CGH. Critically, this efficiency gain does not come at the cost of performance; in fact, the authors report a ~4 dB improvement in Peak Signal-to-Noise Ratio (PSNR) over HoloNet. The successful application of the framework to diverse physical domains beyond optics further underscores the robustness and versatility of the approach.\nMajor Comments:\nJustification for Inference Acceleration: The reported 389x inference speedup on an Android device requires further justification. Non-uniform quantization schemes, such as the Gaussian method employed, do not inherently guarantee hardware acceleration. To substantiate this claim, the manuscript should provide a more detailed description of the deployment strategy. This should include an explanation of how the non-uniform quantization levels are mapped to hardware-friendly integer operations (e.g., 8-bit integers, as mentioned in the rebuttal documents) and specify the software toolchains (e.g., TensorRT, ONNX Runtime) used to achieve the reported latency reduction.\nDelineation of Novelty in Complex-Valued Quantization: The manuscript should more clearly delineate the novelty of the proposed complex-valued quantization framework relative to existing techniques in real-valued quantization. Many components, such as mixed-precision allocation, Gaussian quantization, and quantization-aware training, are established methods in the real-valued domain. The central contribution appears to be the application of these techniques within a structure that preserves the relationship between amplitude and phase during complex multiplication. Explicitly articulating this distinction is critical for clarifying the paper's unique contribution. For instance, explaining why simply treating real and imaginary parts as independent channels is suboptimal would better frame the problem that the proposed method solves.\nIn summary, the manuscript presents a method for quantizing complex-valued neural networks with demonstrated applications in several domains. Addressing the points above regarding inference acceleration and the delineation of novelty would significantly strengthen the paper and clarify its contributions to the field.", "reviewer_2": "The author has responded to the initial review comments carefully and thoroughly, providing more comprehensive theoretical explanations, experimental verification, and writing refinements for the quantization problem of complex-valued networks. The revised paper has significantly improved in terms of rigor and practicality. The author's response to the first-round review comments is relatively comprehensive, but the explanations for some key issues are still not thorough enough, and further strengthening of the argumentation and data analysis is needed:\n1.Some parts of the article still have formatting issues and non-standard writing. For instance, some formulas and references in the updated manuscript cannot be displayed properly.\n2. The abstract mentions that this complex-valued neural network quantization method effectively simulates the intrinsic correlation between amplitude and phase in the evolution process of complex physical fields. However, in fact, this quantization method does not describe the evolution of physical fields. Please change the description or further explain the mechanism.\n3. In the newly added content, the author applies ULBN to three representative complex-valued scenarios: acoustics, wireless modulation, and SAR. However, only the results of real-valued, complex-valued, and quantized complex-valued networks are compared. It is suggested to supplement the description of the design of real-valued and complex-valued networks, as well as the comparison with the current advanced network classification results, to prove the reliability and completeness of the data."}, "author_response": "We appreciate the Reviewer for his/her important comments. We have responded to Reviewer 2's comments point-to-point below."}, {"round": 3, "reviewer_comments": {"reviewer_1": "Thank you for the detailed explanation of the deployment setting and practical constraints. The response makes it clear that, due to current hardware and software limitations, the reported inference speedup is obtained using standard uniform INT8 quantization, rather than the proposed Gaussian-based non-uniform quantization.\nGiven this clarification, it would be helpful if the paper more explicitly distinguishes between these two settings. In particular, the deployment and latency results should not be interpreted as evidence that the Gaussian non-uniform quantization itself is hardware-efficient or directly responsible for the observed speedup. Instead, the acceleration comes from using a conventional INT8 pipeline, which is largely independent of the proposed Gaussian quantization scheme.\nI believe making this distinction clearer in the presentation would improve the consistency of the claims and help readers better understand the practical scope and limitations of the proposed method.", "reviewer_2": "The authors have answered my quesitons properly."}, "author_response": ""}]

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