Can programmable radio propagation serve as computational depth rather than only as a communication channel or one-shot analog transform? We introduce the Radio Alternating Operator Ansatz (RAOA), a recurrent computing architecture that alternates an energy-derived problem update with a mixing update over a persistent latent state. Recomputing the problem field after each mix makes repeated passes compositional even when the same learned controls are reused across depth. We evaluate this idea through exact discrete optimization, constrained programmable-propagation simulation, and pretrained-model adaptation. On discrete objectives, repeated execution can improve solution quality without increasing the learned-control count, and the same formulation handles higher-order interactions directly. A passive phase-only free-space model further shows that the required operators can be approximated by programmable propagation while retaining useful downstream behavior despite realization error. When inserted as a zero-initialized residual adapter, RAOA adapts pretrained language models with WikiText performance close to a matched shallow MLP across three model families, while reasoning-task transfer remains model-dependent. Together, these results connect alternating-operator computation, programmable radio propagation, and neural adaptation within one recurrent framework. The RF realization evidence is simulation-based rather than a hardware demonstration.
Figures & tables
Figure 1: Conceptual RAOA realization and Transformer insertion. (a) One stage alternates problem and mixer configurations of the same programmable path; receiver state is fed back, so L stages correspond to 2L modeled RF passes. (b) A pretrained Transformer projects hidden state into the RAOA state, returns it to model width, and adds it through a zero-initialized residual scale, making the insertion exactly behavior-preserving at initialization. The physical loop is conceptual; all foundation-model experiments execute the differentiable operator in software.
Figure 2: Operator ladder motivating RAOA. A fixed programmed path is linear in its transmitted state; signal-conditioned programming can create nonlinear input-output maps; feedback adds memory; RAOA repeatedly alternates problem and mixer roles over that recurrent state. The quantitative annotations are controlled primitive demonstrations, not application benchmarks or hardware measurements. “Severe” denotes the preregistered synthetic impairment profile.
Figure 3: Cross-depth reuse isolates recurrent execution depth. Twelve independent instances per family and size; error bars are standard deviations across instances after averaging five optimizer seeds within each instance. The same (γ,β) controls are optimized only at two passes and then reused unchanged at deeper execution depths. Mean exact-optimum hit probability increases monotonically in all eight family-size groups.
Problem / representation
Executed vars.
Ancillas
Success
3-SAT, native HUBO
8
0
0.989±0.003
3-SAT, quadratized QUBO
17
9
0.161±0.023
Hypergraph Max-Cut, native HUBO
8
0
0.830±0.008
Hypergraph Max-Cut, quadratized QUBO
16
8
0.119±0.011
Table 1: Native higher-order execution versus exact quadratization at eight modeled passes. Values are mean ± seed standard deviation over five seeds.
Model
Metric
Frozen
LoRA
RAOA ( L=1 )
MLP
Qwen3-4B
WikiText NLL
3.3007
2.6426±.0016
2.6851±.0040
2.6596±.0023
GSM8K EM (768)
87.95
81.65±1.96
87.95±.49
88.05±.25
Gemma-3-4B-it
WikiText NLL
5.9833
2.9046±.0051
2.9009±.0055
2.8763±.0061
GSM8K EM (768)
81.12
79.83±.98
81.43±.27
81.25±.29
Phi-4-mini
WikiText NLL
3.1010
2.6232±.0006
2.6503±.0058
2.6210±.0023
GSM8K EM (384)
79.23
83.04±.23
79.45±.39
83.73±.53
Table 2: Foundation-model results. WikiText NLL (lower is better) and GSM8K EM in percent (higher is better). Qwen and Gemma use the frozen-schedule 768-token sensitivity audit; Phi retains the canonical 384-token evaluation. Trainable entries are mean ± seed standard deviation; bold marks the best mean, not significance.
Figure 4: Structural and realizability controls. Left: full-test Gemma GSM8K accuracy versus RAOA trainable parameters; every-second-layer placement nearly matches dense placement with half the adapters. Right: Max-Cut success after compiling problem/mixer operators into 1, 2, or 4 passive 128-element phase-only panels (coherent parallel single-bounce paths) and reoptimizing only the same two scalar controls. More panel resources reduce the gap to the ideal effective operators, although substantial variability across geometry seeds remains at large depth.
Panels
Phases
Max-Cut P
Partition P
Mixer M
1
128
0.687±.016
0.711±.034
0.758±.048
2
256
0.519±.021
0.535±.059
0.573±.055
4
512
0.357±.092
0.348±.142
0.406±.092
Table 3: Phase-only compilation. Normalized Frobenius error for problem ( P ) and mixer ( M ) operators (mean ± standard deviation over three geometries); 128 phases per panel and a shared mixer target.
Appendix figures & tables17 assets
Supplementary material from the paper’s appendix.
Appendix
Problem
Order
Variables
Optimum degeneracy
Uniform optimum rate
Max-Cut-8
2
8
2
0.78%
Partition-8
2
8
10
3.91%
3-SAT-8
3
8
36
14.06%
Hypergraph Max-Cut-8
4
8
28
10.94%
Appendix
Table 4: Exact small discrete-optimization instances used in the appendix study.
Problem
2 RF passes
4 RF passes
8 RF passes
16 RF passes
Max-Cut-8
0.071±0.017
0.204±0.014
0.433±0.034
0.690±0.347
Partition-8
0.064±0.007
0.025±0.012
0.041±0.006
0.008±0.005
3-SAT-8
0.617±0.020
0.890±0.026
0.980±0.013
1.000±0.000
Hypergraph Max-Cut-8
0.352±0.021
0.622±0.014
0.856±0.011
0.575±0.467
Appendix
Table 5: Exact-optimum hit probability for tied-control RAOA (mean ± seed standard deviation, five seeds). The learned-control count is two in every cell; RF-pass counts are modeled.
Problem
Representation
Vars.
Anc.
4 passes
8 passes
3-SAT-8
Native HUBO
8
0
0.896±0.013
0.989±0.003
3-SAT-8
Quadratized QUBO
17
9
0.052±0.011
0.161±0.023
Hypergraph Max-Cut-8
Native HUBO
8
0
0.603±0.012
0.830±0.008
Hypergraph Max-Cut-8
Quadratized QUBO
16
8
0.056±0.006
0.119±0.011
Appendix
Table 6: Native higher-order RAOA versus exact quadratization. Success is exact-optimum hit probability, mean ± seed standard deviation over five seeds.
Figure 5: AI-generated conceptual system illustration. This appendix-only raster image provides qualitative intuition for programmable propagation, signal-conditioned control, feedback, repeated alternating passes, and Transformer insertion. It is not experimental evidence. The image is pre-cropped to remove the original project-title band and generated caption; all quantitative plots in this paper are produced programmatically from frozen numerical results.
Profile
Angle bits
σα (rad)
σz
Ideal
–
0
0
Mild
8
0.005
0.002
Moderate
6
0.020
0.010
Severe
4
0.060
0.040
Appendix
Table 7: Modeled RAOA impairment profiles. These are synthetic perturbation levels, not device measurements.
Model
Family
Steps
Learning rate
Qwen3-4B
LoRA
300
10−4
RAOA
300
10−3
MLP
300
10−3
Gemma-3-4B-it
LoRA
300
10−4
RAOA
300
10−3
MLP
300
3×10−4
Appendix
Table 8: Validation-selected WikiText schedules used for full-test confirmation.
Model
Frozen
LoRA
RAOA
Shallow MLP
Qwen3-4B
3.3007
2.6426±0.0016
2.6851±0.0040
2.6596±0.0023
Gemma-3-4B-it
5.9833
2.9046±0.0051
2.9009±0.0055
2.8763±0.0061
Phi-4-mini
3.1010
2.6232±0.0006
2.6503±0.0058
2.6210±0.0023
Appendix
Table 9: Full-test WikiText NLL (mean ± seed standard deviation). Lower is better.
Model
Family
Steps
Learning rate
Qwen3-4B
LoRA
100
10−5
RAOA
25
10−5
MLP
50
10−5
Gemma-3-4B-it
LoRA
10
3×10−6
RAOA
5
3×10−5
MLP
10
3×10−5
Appendix
Table 10: Holdout-selected GSM8K schedules used for full-test confirmation.
Model
Frozen
LoRA
RAOA
Shallow MLP
Qwen3-4B
81.35
76.55±1.70
81.88±0.12
81.65±0.16
Gemma-3-4B-it
73.77
74.75±1.29
73.97±0.19
73.21±0.26
Phi-4-mini
79.23
83.04±0.23
79.45±0.39
83.73±0.53
Appendix
Table 11: Canonical 384-token full-test GSM8K exact-match accuracy (%, mean ± seed standard deviation for trainable methods).
Model
Family
EM
Seed std.
Cap hit
Qwen3-4B
Frozen
87.95
–
0.38
LoRA
81.65
1.96
3.23
RAOA
87.95
0.49
0.53
MLP
88.05
0.25
0.63
Gemma-3-4B-it
Frozen
81.12
–
4.17
LoRA
79.83
0.98
3.36
Appendix
Table 12: 768-token GSM8K sensitivity audit for Qwen and Gemma. Accuracy is full-test EM (%); cap is max-token-limit-hit rate (%).
Model
Pair
Delta
95% CI
Qwen3-4B
RAOA − Frozen
0.00
[−0.78,0.78]
RAOA − MLP
-0.10
[−0.66,0.43]
LoRA − Frozen
-6.29
[−8.11,−4.50]
Gemma-3-4B-it
RAOA − Frozen
+0.30
[−0.53,1.16]
RAOA − MLP
+0.18
[−0.45,0.81]
LoRA − Frozen
-1.29
[−2.68,0.13]
Appendix
Table 13: Paired example-bootstrap differences at 768 tokens (A − B, percentage points; 95% CI).
Placement
Blocks
Params (M)
GSM8K EM (%)
All
34
11.370
74.30±0.38
Every 2
17
5.685
74.00±0.51
Every 4
9
3.010
73.69±0.54
Last 2
2
0.669
73.87±0.19
First+last
2
0.669
73.21±0.50
First 2
2
0.669
73.01±0.27
Appendix
Table 14: Full-test Gemma placement results.
Family
n
2 passes
4 passes
8 passes
16 passes
3-SAT
8
0.543±0.148
0.862±0.114
0.971±0.038
0.997±0.008
3-SAT
12
0.367±0.104
0.745±0.117
0.923±0.067
0.980±0.026
Hypergraph Max-Cut
8
0.586±0.062
0.817±0.050
0.945±0.026
0.995±0.003
Hypergraph Max-Cut
12
0.441±0.044
0.725±0.048
0.914±0.030
0.984±0.014
Max-Cut
8
0.0785±0.0273
0.175±0.069
0.291±0.130
0.390±0.226
Max-Cut
12
0.0158±0.0086
0.0520±0.0324
0.111±0.069
0.175±0.131
Appendix
Table 15: Random-instance exact-optimum success when controls are trained only at two passes and reused unchanged at deeper execution depths. Values are mean ± standard deviation across 12 independent instances after averaging optimizer seeds within instance.
Problem
Full
β=0
Frozen field
Identity
Rand. orth.
3-SAT
0.985
0.978
0.483
0.980
0.987
Hypergraph Max-Cut
0.850
0.853
0.040
0.879
0.856
Max-Cut
0.430
0.319
0.060
0.320
0.317
Partition
0.0356
0.105
0.0062
0.111
0.076
Appendix
Table 16: Mechanism ablation at eight modeled RF passes. Values are mean exact-optimum success over five seeds. “Frozen field” computes the local field once on the initial state and reuses it at every problem pass.
Problem
Operator
1 panel
2 panels
4 panels
Max-Cut
Problem
0.687±0.016
0.519±0.021
0.357±0.092
Max-Cut
Mixer
0.758±0.048
0.573±0.055
0.406±0.092
Partition
Problem
0.711±0.034
0.535±0.059
0.348±0.142
Partition
Mixer
0.758±0.048
0.573±0.055
0.406±0.092
Appendix
Table 17: Normalized Frobenius error of phase-only compiled operators (mean ± standard deviation across three geometry seeds).
Figure 6: Synthetic RAOA perturbation sensitivity on Qwen3-4B WikiText. Ideal-trained RAOA is nearly invariant through the preregistered severe angle/state profile; domain-randomized training is consistently slightly worse. These are modeled perturbations, not measured RF impairments.
Evaluation profile
Ideal-trained
Domain-randomized
Ideal
2.68188±0.00151
2.68469±0.00385
Mild
2.68162±0.00153
2.68447±0.00399
Moderate
2.68159±0.00156
2.68433±0.00377
Severe
2.68239±0.00137
2.68523±0.00377
Appendix
Table 18: Modeled-impairment WikiText NLL (mean ± seed standard deviation).
Recurrent-depth language models improve inference-time reasoning by repeatedly applying a shared network to refine internal representations, but the additional recurrent computation also increases inference cost. This overhead is particularly pronounced during long-context prefill, where the full input sequence is repeatedly processed across recurrent depth. Weight-and-activation quantization offers a natural way to accelerate this computation through lower-precision arithmetic, yet repeatedly applying quantized computation introduces numerical perturbations that accumulate along the recurrent trajectory. We systematically analyze how such errors propagate across recurrent depth and show that quantization errors introduced earlier are attenuated by more subsequent recurrent steps and therefore have a smaller effect on the final representation. Matched-budget experiments strongly support this prediction, showing that placing quantized computation earlier in recurrent depth consistently yields the best fidelity. Based on this insight, we introduce RAMP (Recurrent Adaptive Mixed Precision), which performs early recurrent refinement using low-precision weight-and-activation computation and restores high precision for terminal refinement. Across context lengths up to 4K, RAMP preserves near-full-precision prefill quality while achieving up to 1.16x end-to-end first-prefill speedup on an NVIDIA H100, corresponding to a 13.9% latency reduction.
Ke Wan, Chen Chen
Department of Computer Science, University of Virginia
Recursive models create computational depth through parameter reuse, offering a parameter-efficient alternative to increasing model size. However, most recursive models repeatedly apply one learned transformation or a prescribed sequence of transformations, restricting computation to a fixed layer order. We introduce the Random Recursive Model (RRM), which maintains a pool of L learned layers and performs T recursive steps by sampling one layer independently with replacement for each example and step. This enables flexible layer reuse while retaining the parameter efficiency of recurrence. We evaluate RRM on challenging reasoning tasks, where it matches or exceeds the baselines, often with 50-75 % fewer parameters. RRM can vary its depth at inference, including beyond that seen during training, without retraining or adding parameters, improving tasks that benefit from deeper iterative computation. RRM also supports Monte Carlo inference and probabilistic test-time scaling, both of which improve performance without retraining. These insights may open new directions in neural network architecture design.
Jama Hussein Mohamud, Mirco Ravanelli
Mila – Quebec AI Institute · Université de Montréal · Concordia University
Radio maps are essential for wireless decision-making tasks such as access-point placement, coverage planning, and localization, but their fine spatial details are governed by complex propagation effects and are costly to simulate accurately. Machine learning offers a path to high-fidelity radio-map prediction without running expensive high-fidelity simulations for every scene. However, generating high-quality training labels at scale is also difficult: the affordable labels come from finite-ray simulations, which are richer than low-fidelity inputs but carry residual Monte Carlo noise. We address this challenge with Physics-Unrolled Hybrid Neural Operator (PU-HNO), a three-stage cascade that predicts high-fidelity indoor radio maps from low-fidelity ray-tracing outputs and scene priors by progressively capturing reflection, diffraction, and scattering effects, rather than treating radio maps as generic images. We prove that, under conditionally unbiased label noise, the model can learn stable propagation structure and outperform its own training labels. Experiments across diverse floorplans show that PU-HNO outperforms image-to-image baselines, wireless learning models, and monolithic neural operators across both image-quality and wireless deployment metrics.
Rafid Umayer Murshed, Saif Ur Rahman, Mingyue Tang +1
Department of Computer Science University of Illinois Urbana-Champaign Urbana, IL 61801, USA