URformer: Deep-Unfolding Transformer for RAQR

• URformer is a model-driven deep-unfolding neural network that addresses biased phase retrieval in Rydberg atomic quantum receivers.
• It replaces fixed EM-GS filtering with learnable modules such as a trainable gate and a channel Transformer to enhance stability and performance.
• Empirical results show URformer achieves lower NMSE at low SNR with reduced pilot overhead, outperforming standard iterative and black-box methods.

Searching arXiv for URformer and the cited paper to ground the article. URformer is a model-driven deep-unfolding neural network for channel estimation in Rydberg atomic quantum receivers (RAQRs). It is designed for the RAQR measurement physics, in which channel estimation becomes a biased phase retrieval problem rather than a standard linear estimation problem. Concretely, URformer is a Transformer-based unrolling architecture derived from a stabilized variant of the expectation-maximization Gerchberg-Saxton (EM-GS) algorithm; each unrolled layer contains a learnable filter, a trainable gate, and a channel Transformer residual corrector, with the explicit goal of combining the physics interpretability of EM-GS, the adaptivity of neural networks, and the long-range modeling capability of Transformers (Xiao et al., 16 Sep 2025).

1. Problem setting in Rydberg atomic quantum receivers

RAQRs are not conventional RF receivers. Their readout is based on atomic spectroscopy, and the measured output is proportional to the magnitude of the total incident RF field, which includes both the unknown user signal field and a known reference or LO field. For pilot slot $p$, the observation model is

$$\mathbf{z}_p = |\mathbf{H}\mathbf{s}_p + \mathbf{b} + \mathbf{w}_p|,$$

where $\mathbf{H}\in \mathbb{C}^{M\times K}$ is the MIMO channel, $\mathbf{s}_p\in \mathbb{C}^{K\times 1}$ is the pilot vector, $\mathbf{b}\in \mathbb{C}^{M\times 1}$ is the known reference or LO field, and $\mathbf{w}_p$ is noise. The associated estimation problem is

$$\hat{\mathbf{H}}=\arg\min_{\mathbf{H}\in\mathbb{C}^{M\times K}} \frac{1}{MP}\sum_{p=1}^P \left\|\mathbf{z}_p-|\mathbf{H}\mathbf{s}_p+\mathbf{b}|\right\|_2^2.$$

This is a structured MIMO biased phase retrieval problem rather than a standard linear channel-estimation problem (Xiao et al., 16 Sep 2025).

The term biased phase retrieval is used because only magnitudes are observed, while the unknown signal is measured jointly with a known offset or reference field $\mathbf{b}$. That bias alters the inverse-problem structure and invalidates the classical orthogonality-based pilot decoupling intuition. The nonseparability is explicit in

$$\left|\sum_k \mathbf{h}_k s_{k,p}\right| \neq \sum_k |\mathbf{h}_k s_{k,p}|,$$

so user channels cannot be separated independently after the magnitude operation. Joint estimation of the full channel matrix $\mathbf{H}$ is therefore required.

A plausible implication is that RAQR channel estimation is governed less by conventional linear estimation theory than by the interaction between nonlinear magnitude measurements, known bias terms, and MIMO coupling across users and antennas.

2. Derivation from stabilized EM-GS

URformer is derived by unrolling a stabilized version of EM-GS. In the baseline EM-GS iteration, the current channel estimate first reconstructs the complex field,

$$\mathbf{z}_p = |\mathbf{H}\mathbf{s}_p + \mathbf{b} + \mathbf{w}_p|,$$0

then forms the local SNR-like proxy

$$\mathbf{z}_p = |\mathbf{H}\mathbf{s}_p + \mathbf{b} + \mathbf{w}_p|,$$1

and applies a fixed Bessel-ratio filter,

$$\mathbf{z}_p = |\mathbf{H}\mathbf{s}_p + \mathbf{b} + \mathbf{w}_p|,$$2

The channel is then updated by least squares as

$$\mathbf{z}_p = |\mathbf{H}\mathbf{s}_p + \mathbf{b} + \mathbf{w}_p|,$$3

URformer preserves this backbone while replacing rigid components with trainable modules (Xiao et al., 16 Sep 2025).

The motivation for this replacement is twofold. First, GS and EM-GS use fixed update rules and a fixed noise/statistics-based filter, so they cannot adapt well to practical non-idealities. Second, the objective is highly non-convex, and these iterative methods may become trapped in poor local minima and become highly sensitive to initialization, especially at low SNR. URformer retains the physically grounded iterative scaffold but allows learning to modify exactly those parts of EM-GS that are most brittle.

This suggests that URformer belongs to the class of model-driven unrolled estimators in which the iterative prior is not discarded; instead, the algorithmic prior becomes the architecture.

3. Layer architecture and internal modules

URformer is a $$\mathbf{z}_p = |\mathbf{H}\mathbf{s}_p + \mathbf{b} + \mathbf{w}_p|,$$4-layer unrolled network in which each layer corresponds to one modified EM-GS iteration. Each layer takes $$\mathbf{z}_p = |\mathbf{H}\mathbf{s}_p + \mathbf{b} + \mathbf{w}_p|,$$5 and outputs $$\mathbf{z}_p = |\mathbf{H}\mathbf{s}_p + \mathbf{b} + \mathbf{w}_p|,$$6. The first component is the Gated Filtering Module. Given the reconstructed field

$$\mathbf{z}_p = |\mathbf{H}\mathbf{s}_p + \mathbf{b} + \mathbf{w}_p|,$$7

URformer extracts the phase

$$\mathbf{z}_p = |\mathbf{H}\mathbf{s}_p + \mathbf{b} + \mathbf{w}_p|,$$8

forms the direct GS-style reconstruction

$$\mathbf{z}_p = |\mathbf{H}\mathbf{s}_p + \mathbf{b} + \mathbf{w}_p|,$$9

and computes

$\mathbf{H}\in \mathbb{C}^{M\times K}$0

Instead of using the fixed EM-GS filter $\mathbf{H}\in \mathbb{C}^{M\times K}$1, URformer uses a small MLP named FilterNet,

$\mathbf{H}\in \mathbb{C}^{M\times K}$2

followed by

$\mathbf{H}\in \mathbb{C}^{M\times K}$3

A trainable gate then interpolates between the conservative direct update and the learned filtered update: $\mathbf{H}\in \mathbb{C}^{M\times K}$4 The gate is introduced specifically to stabilize the unrolling (Xiao et al., 16 Sep 2025).

The second component is the Linear Estimation Module,

$\mathbf{H}\in \mathbb{C}^{M\times K}$5

This is structurally analogous to the least-squares step in EM-GS, but it operates on a reconstruction that has already been modified by the learnable filter and the trainable gate.

The third component is the residual correction module, implemented as a channel Transformer: $\mathbf{H}\in \mathbb{C}^{M\times K}$6 Its role is to correct residual estimation errors by capturing non-local dependencies across the channel matrix. The architecture therefore combines a physically interpretable reconstruction stage with a learned nonlocal residual corrector rather than replacing the iterative solver wholesale.

4. Transformer parameterization and representational role

The channel Transformer operates on a tokenized representation of the complex channel matrix. The complex-valued input is split into real and imaginary parts,

$\mathbf{H}\in \mathbb{C}^{M\times K}$7

and reshaped into $\mathbf{H}\in \mathbb{C}^{M\times K}$8 tokens,

$\mathbf{H}\in \mathbb{C}^{M\times K}$9

These tokens are projected to model dimension and augmented with positional embeddings,

$\mathbf{s}_p\in \mathbb{C}^{K\times 1}$0

where $\mathbf{s}_p\in \mathbb{C}^{K\times 1}$1 and $\mathbf{s}_p\in \mathbb{C}^{K\times 1}$2 (Xiao et al., 16 Sep 2025).

The encoder then applies standard Transformer blocks: $\mathbf{s}_p\in \mathbb{C}^{K\times 1}$3

$\mathbf{s}_p\in \mathbb{C}^{K\times 1}$4

After $\mathbf{s}_p\in \mathbb{C}^{K\times 1}$5 encoder blocks, the output is projected back,

$\mathbf{s}_p\in \mathbb{C}^{K\times 1}$6

reshaped to

$\mathbf{s}_p\in \mathbb{C}^{K\times 1}$7

and recombined into a complex residual matrix,

$\mathbf{s}_p\in \mathbb{C}^{K\times 1}$8

The intended representational effect is explicit: the Transformer models interactions across all user-channel tokens simultaneously. This is relevant because RAQR measurements entangle users through the magnitude operation, and the residual errors remaining after the linear step are not assumed to be localized to a single antenna or a single user. The design therefore treats the residual as a structured matrix-valued object rather than as independent per-entry noise.

5. Training protocol and reported empirical behavior

URformer is trained end-to-end with normalized MSE,

$\mathbf{s}_p\in \mathbb{C}^{K\times 1}$9

Its inputs are the pilot measurements $\mathbf{b}\in \mathbb{C}^{M\times 1}$0, the known pilot matrix $\mathbf{b}\in \mathbb{C}^{M\times 1}$1, and the known reference or LO channel $\mathbf{b}\in \mathbb{C}^{M\times 1}$2, and its output is the estimated channel matrix $\mathbf{b}\in \mathbb{C}^{M\times 1}$3. The reported training data are simulated RAQR channel realizations generated under a clustered Saleh–Valenzuela channel model (Xiao et al., 16 Sep 2025).

The reported simulation configuration uses $\mathbf{b}\in \mathbb{C}^{M\times 1}$4 users, $\mathbf{b}\in \mathbb{C}^{M\times 1}$5 atomic antennas, $\mathbf{b}\in \mathbb{C}^{M\times 1}$6 pilots, $\mathbf{b}\in \mathbb{C}^{M\times 1}$7 clusters, $\mathbf{b}\in \mathbb{C}^{M\times 1}$8 subrays, $\mathbf{b}\in \mathbb{C}^{M\times 1}$9 URformer layers, $\mathbf{w}_p$0 Transformer encoder blocks, $\mathbf{w}_p$1, and 20,000 training samples. Baselines are GS, EM-GS, CNN, and standard Transformer; for fairness, the classic iterative methods are run with $\mathbf{w}_p$2 iterations.

The reported findings are consistent across several axes. In NMSE-versus-SNR evaluation, URformer significantly outperforms GS and EM-GS, especially at low SNR, and around $\mathbf{w}_p$3 dB the classic methods degrade strongly while URformer remains much more accurate. In comparison with black-box neural baselines, both CNN and vanilla Transformer exhibit a high error floor across SNR values, which is interpreted as evidence that pure black-box learning does not solve the biased phase retrieval problem effectively without physical structure in the architecture. In NMSE-versus-pilot-count evaluation, URformer improves much faster as the number of pilots increases and can achieve about $\mathbf{w}_p$4 dB NMSE with only $\mathbf{w}_p$5 pilots. The claimed significance is lower pilot overhead together with improved spectral efficiency.

6. Conceptual significance, limitations, and nomenclature

URformer is best understood as a Transformer-enabled deep-unfolded EM-GS network rather than as a generic Transformer estimator. A recurrent misconception would be to regard it as a purely data-driven architecture; the defining property is instead the combination of physics-informed unrolling, learned noise-aware filtering, stabilized gating, and Transformer-based correction of global residual structure. In that sense, its contribution is architectural hybridization: the useful inductive bias of EM-GS is preserved, while the rigid parts of the classical method are replaced by trainable components (Xiao et al., 16 Sep 2025).

Several limitations or open challenges are either stated or directly implied. The architecture is evaluated on representative RAQR simulations rather than on a broader deployment matrix. The performance claims are tied to the reported setup, including the clustered Saleh–Valenzuela channel model and the specified $\mathbf{w}_p$6, $\mathbf{w}_p$7, and $\mathbf{w}_p$8. The broader approximation-versus-fidelity issue in efficient learned inference for nonlinear inverse problems is not eliminated; a plausible implication is that generalization beyond the evaluated tasks and operating conditions remains an important follow-up question.

A separate terminological issue is that “URformer” should not be conflated with “UniFormer: Unified and Efficient Transformer for Reasoning Across General and Custom Computing,” which is a different 2025 work on cross-platform efficient Transformer design for GPUs and FPGAs. That work concerns dual-branch attention built around GEMM and hardware-aware deployment, whereas URformer concerns RAQR channel estimation under biased phase retrieval (Ran et al., 11 Nov 2025). The shared suffix reflects Transformer-based design, but the problem domain, algorithmic derivation, and target hardware assumptions are distinct.