RAOA: Alternating-Operator Neural Computation with Programmable Radio Propagation

Toshiaki Koike-Akino

cs.LGOct 2, 2026

RAOA: Alternating-Operator Neural Computation with Programmable Radio Propagation

Authors: Toshiaki Koike-Akino

Organizations: Mitsubishi Electric Research Laboratories (MERL), 201 Broadway, Cambridge, MA 02139, USA

Abstract

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

Panel of Figure 1 — 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.

Panel of Figure 2 — 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

(\gamma,\beta)

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	$0.989\pm 0.003$
3-SAT, quadratized QUBO	17	9	$0.161\pm 0.023$
Hypergraph Max-Cut, native HUBO	8	0	$0.830\pm 0.008$
Hypergraph Max-Cut, quadratized QUBO	16	8	$0.119\pm 0.011$

Table 1: Native higher-order execution versus exact quadratization at eight modeled passes. Values are mean

\pm

seed standard deviation over five seeds.

Model	Metric	Frozen	LoRA	RAOA ( $L{=}1$ )	MLP
Qwen3-4B	WikiText NLL	3.3007	$\mathbf{2.6426}\!\pm\!.0016$	$2.6851\!\pm\!.0040$	$2.6596\!\pm\!.0023$
	GSM8K EM (768)	87.95	$81.65\!\pm\!1.96$	$87.95\!\pm\!.49$	$\mathbf{88.05}\!\pm\!.25$
Gemma-3-4B-it	WikiText NLL	5.9833	$2.9046\!\pm\!.0051$	$2.9009\!\pm\!.0055$	$\mathbf{2.8763}\!\pm\!.0061$
	GSM8K EM (768)	81.12	$79.83\!\pm\!.98$	$\mathbf{81.43}\!\pm\!.27$	$81.25\!\pm\!.29$
Phi-4-mini	WikiText NLL	3.1010	$2.6232\!\pm\!.0006$	$2.6503\!\pm\!.0058$	$\mathbf{2.6210}\!\pm\!.0023$
	GSM8K EM (384)	79.23	$83.04\!\pm\!.23$	$79.45\!\pm\!.39$	$\mathbf{83.73}\!\pm\!.53$

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

\pm

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	$0.687\!\pm\!.016$	$0.711\!\pm\!.034$	$0.758\!\pm\!.048$
2	256	$0.519\!\pm\!.021$	$0.535\!\pm\!.059$	$0.573\!\pm\!.055$
4	512	$0.357\!\pm\!.092$	$0.348\!\pm\!.142$	$0.406\!\pm\!.092$

Table 3: Phase-only compilation. Normalized Frobenius error for problem (

P

) and mixer (

M

) operators (mean

\pm

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	$0.071\pm 0.017$	$0.204\pm 0.014$	$0.433\pm 0.034$	$0.690\pm 0.347$
Partition-8	$0.064\pm 0.007$	$0.025\pm 0.012$	$0.041\pm 0.006$	$0.008\pm 0.005$
3-SAT-8	$0.617\pm 0.020$	$0.890\pm 0.026$	$0.980\pm 0.013$	$1.000\pm 0.000$
Hypergraph Max-Cut-8	$0.352\pm 0.021$	$0.622\pm 0.014$	$0.856\pm 0.011$	$0.575\pm 0.467$

Appendix

Table 5: Exact-optimum hit probability for tied-control RAOA (mean

\pm

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	$0.896\pm 0.013$	$0.989\pm 0.003$
3-SAT-8	Quadratized QUBO	17	9	$0.052\pm 0.011$	$0.161\pm 0.023$
Hypergraph Max-Cut-8	Native HUBO	8	0	$0.603\pm 0.012$	$0.830\pm 0.008$
Hypergraph Max-Cut-8	Quadratized QUBO	16	8	$0.056\pm 0.006$	$0.119\pm 0.011$

Appendix

Table 6: Native higher-order RAOA versus exact quadratization. Success is exact-optimum hit probability, mean

\pm

seed standard deviation over five seeds.

Profile	Angle bits	$\sigma_{\alpha}$ (rad)	$\sigma_{z}$
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	$3\times 10^{-4}$

Appendix

Table 8: Validation-selected WikiText schedules used for full-test confirmation.

Model	Frozen	LoRA	RAOA	Shallow MLP
Qwen3-4B	3.3007	$\mathbf{2.6426}\pm 0.0016$	$2.6851\pm 0.0040$	$2.6596\pm 0.0023$
Gemma-3-4B-it	5.9833	$2.9046\pm 0.0051$	$2.9009\pm 0.0055$	$\mathbf{2.8763}\pm 0.0061$
Phi-4-mini	3.1010	$2.6232\pm 0.0006$	$2.6503\pm 0.0058$	$\mathbf{2.6210}\pm 0.0023$

Appendix

Table 9: Full-test WikiText NLL (mean

\pm

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	$3\times 10^{-6}$
	RAOA	5	$3\times 10^{-5}$
	MLP	10	$3\times 10^{-5}$

Appendix

Table 10: Holdout-selected GSM8K schedules used for full-test confirmation.

Model	Frozen	LoRA	RAOA	Shallow MLP
Qwen3-4B	81.35	$76.55\pm 1.70$	$\mathbf{81.88}\pm 0.12$	$81.65\pm 0.16$
Gemma-3-4B-it	73.77	$\mathbf{74.75}\pm 1.29$	$73.97\pm 0.19$	$73.21\pm 0.26$
Phi-4-mini	79.23	$83.04\pm 0.23$	$79.45\pm 0.39$	$\mathbf{83.73}\pm 0.53$

Appendix

Table 11: Canonical 384-token full-test GSM8K exact-match accuracy (%, mean

\pm

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	$74.30\pm 0.38$
Every 2	17	5.685	$74.00\pm 0.51$
Every 4	9	3.010	$73.69\pm 0.54$
Last 2	2	0.669	$73.87\pm 0.19$
First+last	2	0.669	$73.21\pm 0.50$
First 2	2	0.669	$73.01\pm 0.27$

Appendix

Table 14: Full-test Gemma placement results.

Family	$n$	2 passes	4 passes	8 passes	16 passes
3-SAT	8	$0.543\pm 0.148$	$0.862\pm 0.114$	$0.971\pm 0.038$	$0.997\pm 0.008$
3-SAT	12	$0.367\pm 0.104$	$0.745\pm 0.117$	$0.923\pm 0.067$	$0.980\pm 0.026$
Hypergraph Max-Cut	8	$0.586\pm 0.062$	$0.817\pm 0.050$	$0.945\pm 0.026$	$0.995\pm 0.003$
Hypergraph Max-Cut	12	$0.441\pm 0.044$	$0.725\pm 0.048$	$0.914\pm 0.030$	$0.984\pm 0.014$
Max-Cut	8	$0.0785\pm 0.0273$	$0.175\pm 0.069$	$0.291\pm 0.130$	$0.390\pm 0.226$
Max-Cut	12	$0.0158\pm 0.0086$	$0.0520\pm 0.0324$	$0.111\pm 0.069$	$0.175\pm 0.131$

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

\pm

standard deviation across 12 independent instances after averaging optimizer seeds within instance.

Problem	Full	$\beta{=}0$	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	$0.687\pm 0.016$	$0.519\pm 0.021$	$0.357\pm 0.092$
Max-Cut	Mixer	$0.758\pm 0.048$	$0.573\pm 0.055$	$0.406\pm 0.092$
Partition	Problem	$0.711\pm 0.034$	$0.535\pm 0.059$	$0.348\pm 0.142$
Partition	Mixer	$0.758\pm 0.048$	$0.573\pm 0.055$	$0.406\pm 0.092$

Appendix

Table 17: Normalized Frobenius error of phase-only compiled operators (mean

\pm

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	$2.68188\pm 0.00151$	$2.68469\pm 0.00385$
Mild	$2.68162\pm 0.00153$	$2.68447\pm 0.00399$
Moderate	$2.68159\pm 0.00156$	$2.68433\pm 0.00377$
Severe	$2.68239\pm 0.00137$	$2.68523\pm 0.00377$

Appendix

Table 18: Modeled-impairment WikiText NLL (mean

\pm

seed standard deviation).

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RAMP: Recurrent Adaptive Mixed Precision for Efficient Prefill in Recurrent-Depth Language Models

Ke Wan, Chen Chen

Department of Computer Science, University of Virginia

Recurrent Language Models LLM Inference Acceleration Mixed-Precision Quantization LLM Quantization Long-Context LLM Inference

Open paper AskPDF arXiv

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.

Abstract

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