Adaptive Rotation for iSOMA: Geometry, Benchmarking, and Noise Robustness in Variational Quantum Objectives

Vojtěch Novák; Ivan Zelinka

cs.NESep 29, 2026

Adaptive Rotation for iSOMA: Geometry, Benchmarking, and Noise Robustness in Variational Quantum Objectives

Authors: Vojtěch Novák, Ivan Zelinka

Organizations: Department of Computer Science, Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava, Ostrava, Czech Republic. · IT4Innovations National Supercomputing Center, VSB-Technical University of Ostrava, Ostrava, 708 00, Czech Republic. · Department of Informatics and Statistics, Marine Research Institute, Klaipeda University, Klaipeda, Lithuania.

Abstract

We study whether the coordinate dependence of the improved Self-Organizing Migrating Algorithm (iSOMA) can be reduced while retaining its inexpensive leader-directed migration mechanism. We introduce iSOMA-AR, which learns a basis from successful migration displacements and selectively applies the standard perturbation mask in that basis. On the complete noiseless BBOB suite, iSOMA- AR significantly outperformed baseline iSOMA across matched conditions, with the largest gains on geometrically difficult landscapes. A targeted ablation shows that the learned orientation is beneficial on a rotated ill-conditioned landscape and that moderate changes of the gate threshold and rotation cap preserve the qualitative result. On CEC 2011 Real World Optimization Problems, iSOMA-AR outperformed iL-SHADE on most problems, although its advantage over baseline iSOMA was not statistically significant. A canonical-jSO rerun is reported as a post-hoc sensitivity check alongside the original jSO-derived comparator. On frustrated-spin variational quantum objectives, adaptive rotation improved most transverse-field conditions, while gains on the diagonal and anisotropic models were absent or selective. Under strong effective sampling noise, the SOMA variants were the most robust population-based methods in the comparison, but iSOMA-AR was not significantly better than baseline iSOMA. Repairing all-zero PRT masks greatly reduced repeated-point evaluations without changing endpoint quality significantly, making this implementation detail unlikely to explain the noise result. Overall, adaptive rotation is most useful on coordinate-sensitive deterministic problems, while the observed noise robustness appears to arise mainly from the underlying SOMA migration mechanism.

Figures & tables

Figure 1: Illustrative proposal geometry on a rotated Rosenbrock valley. With the same partially active perturbation mask, standard iSOMA applies masking in the original coordinate system, whereas iSOMA-AR applies it in the learned basis $Q_{s}$ . The rotated proposals follow the valley geometry more directly and make faster progress toward the leader. Discrete points denote evaluated path locations and diamonds denote accepted replacements.

	BBOB	CEC 2011 real-world
Problems	$24$ noiseless functions	$22$ real-world problems
Dimensions	$D\in\{5,10,20\}$	problem-specific ( $1$ – $240$ )
Instances	$1$ – $5$	fixed problem definitions
FE budgets	$500D,\ 2000D$	$50{,}000$
Stochastic runs	$10$ / condition	$10$ / problem
Primary optimizers	iSOMA, iSOMA-AR, jSO-derived, iL-SHADE, CMA-ES

Table 1: Benchmark protocols used in the final study.

Algorithm	Mean rank	Top-1	Target-hit frac.	Success $10^{-8}$	Med. FE/D to $10^{-8}$	s/1000 FE
jSO-derived	1.86	49.9%	65.4%	48.9%	1073	0.0279
CMA-ES	2.49	51.0%	55.1%	44.4%	417	0.0676
iL-SHADE	3.13	16.9%	35.8%	15.0%	552	0.0334
iSOMA-AR	3.46	10.6%	36.4%	15.5%	1212	0.0204
iSOMA	4.05	8.3%	29.5%	12.2%	1154	0.0182

Table 2: Overall BBOB results across 720 matched function–dimension–instance–budget conditions. Target-hit fraction is the fraction of all run–target pairs attained among the 51 fixed targets

\Delta f\in[10^{2},10^{-8}]

at the

2000D

budget. Lower rank and runtime are better; higher shares are better.

Group	Most frequent condition winner	AR/base W/T/L	AR/base ratio	AR/iL-SHADE W/T/L	AR/iL-SHADE ratio
G1 separable	jSO-derived 115/150	84/39/27	0.93	71/33/46	1.00
G2 low/moderate conditioning	CMA-ES 90/120	99/0/21	0.55	45/0/75	1.32
G3 high-conditioning unimodal	CMA-ES 131/150	137/0/13	0.45	45/0/105	2.53
G4 multimodal, adequate structure	jSO-derived 80/150	100/0/50	0.90	21/0/129	2.44
G5 multimodal, weak structure	jSO-derived 67/150	87/3/60	0.99	69/3/78	1.01

Table 3: BBOB official-group pairwise final-error outcomes. W/T/L is the number of matched conditions won/tied/lost by the first method. The ratio is the median error ratio; values below one favor iSOMA-AR.

Algorithm	Mean rank	Top-1	Target-hit frac.	Success $10^{-4}$	Success $10^{-6}$	s/1000 FE
jSO-derived	1.50	77.3%	63.1%	59.5%	29.1%	0.0202
CMA-ES	2.77	18.2%	48.6%	45.5%	7.7%	0.0834
iSOMA-AR	3.27	13.6%	42.0%	33.2%	11.8%	0.0193
iSOMA	3.41	9.1%	43.2%	34.1%	12.3%	0.0110
iL-SHADE	4.05	9.1%	18.3%	11.4%	6.8%	0.0224

Table 4: Overall CEC 2011 results across 22 problems at 50,000 evaluations. The normalized target-hit fraction is diagnostic: it is the fraction of run–target pairs attained for six relative gap reductions

10^{-1}

–

10^{-6}

reconstructed from saved checkpoints.

Group	Most frequent problem winner	AR/base W/T/L	AR/base ratio	AR/iL-SHADE W/T/L	AR/iL-SHADE ratio
A1 control / parameter estimation	jSO-derived 3/3	0/2/1	1	1/2/0	1
A2 molecular / material potentials	jSO-derived 3/3	0/0/3	1.3	1/0/2	1.9
A3 signal / antenna design	CMA-ES 1/2, iL-SHADE 1/2	2/0/0	0.97	0/0/2	4.9
A4 power systems	jSO-derived 10/12	8/1/3	0.87	10/0/2	0.0002
A5 spacecraft trajectory	iSOMA-AR 1/2, jSO-derived 1/2	1/0/1	0.88	2/0/0	0.038

Table 5: CEC 2011 diagnostic application-group outcomes. The groups are interpretive and outside the official CEC 2011 taxonomy. W/T/L compares median final regret over the problems in each group; values below one favor iSOMA-AR.

Study	Comparison	W/T/L	Median ratio	$p_{\rm Holm}$
BBOB	iSOMA-AR / iSOMA	507/42/171	0.814	1.34e-50
BBOB	iSOMA-AR / iL-SHADE	251/36/433	1.35	8.44e-12
BBOB	iSOMA-AR / CMA-ES	198/46/476	2.78	2.34e-45
BBOB	iSOMA-AR / jSO-derived	58/54/608	22	1.91e-101
BBOB	jSO-derived / CMA-ES	310/115/295	1	0.99
CEC2011	iSOMA-AR / iSOMA	11/3/8	0.99	0.338

Table 6: Selected paired comparisons. Ratios are median error/regret ratios of the first method over the second; Holm-adjusted Wilcoxon

p

-values are descriptive because benchmark conditions are not fully independent experimental units.

Model	Condition	iSOMA	iSOMA-AR	AR/base	Best core method
Q1	$N=12$ , 10k	$1.50\text{\times}{10}^{-05}$	$1.46\text{\times}{10}^{-04}$	9.708	jSO-derived
Q1	$N=12$ , 30k	0	$1.17\text{\times}{10}^{-13}$	–	iSOMA
Q1	$N=14$ , 10k	$6.93\text{\times}{10}^{-08}$	$5.10\text{\times}{10}^{-07}$	7.360	jSO-derived
Q1	$N=14$ , 30k	0	0	–	iSOMA
Q1	$N=16$ , 10k	0.458	0.458	1.000	iSOMA
Q1	$N=16$ , 30k	0.458	0.458	1.000	jSO-derived

Table 7: Exact-objective VQE extension using the original 25-run Q1–Q3 benchmark protocol. Values are median final errors. For Q1 and Q2 the error is relative to the exact physical ground energy; for Q3 it is relative to the unchanged saved best-known variational reference. AR/base is the ratio of iSOMA-AR to iSOMA median error; values below one favor adaptive rotation.

Noise	Model	Budget	iSOMA	iSOMA-AR	jSO-derived	iL-SHADE	CMA-ES	SPSA
Low	Q1	10k	0.066	0.087	0.115	0.177	1.712	2.022
Low	Q1	30k	0.062	0.080	0.083	0.158	1.755	2.017
Low	Q2	10k	0.317	0.308	0.374	0.446	0.342	0.587
Low	Q2	30k	0.319	0.316	0.260	0.272	0.322	0.203
Low	Q3	10k	1.928	1.927	3.689	1.555	1.382	0.356
Low	Q3	30k	1.907	1.910	1.960	1.244	1.216	0.166

Table 8: Sampling-noise robustness at

N=12

over 25 runs. Entries are the median exact errors of the parameter vector selected using the noisy objective. Bold marks the smallest median in each condition. Low and high noise correspond to 8192 and 128 effective shots, respectively.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Role / implementation	Settings used	Notes
iSOMA	Baseline SOMA used in all three benchmark families.	Pop $50$ ; $N_{\mathrm{jump}}=10$ ; Step $=0.3$ ; $m=10$ , $n=5$ , $k=15$ ; masks regenerated every proposal; $\mathrm{PRT}=0.1+0.9\,N_{\mathrm{FE}}/B$ ; path $t=3.0,2.7,\ldots,0.3$ ; first non-worsening proposal accepted; box clipping; 10% pop reinitialization after $>50\,\mathrm{PopSize}$ failed attempts without global best improvement.	FE budget is effective stopping rule. BBOB/CEC driver uses nonbinding migration cap of $10^{5}$ ; VQE wrapper uses $10^{4}$ .
iSOMA-AR	Proposed variant; identical baseline selection, path, repair, and acceptance with occasional rotated masking.	All iSOMA settings above plus $C_{s}(0)=I$ , success update scale $0.12$ (cap $0.35$ ), eigenspace update every 5 accepted improving steps, ridge $10^{-4}I$ , min. successful-step count $\max(6,\lceil 1.5D\rceil)$ , off-axis threshold $0.18$ , max rotation probability $0.45$ .	No extra FEs spent on basis learning. Covariance memory and rotation gate reset at baseline stagnation restart.
jSO-derived	Fixed custom DE used throughout BBOB/CEC and VQE comparison.	$N_{0}=\max(30,\lfloor 25\sqrt{D}\log_{10}(\max(D,2))\rfloor)$ , linear reduction to $N_{\min}=4$ ; $H=5$ ; $M_{F}=0.5$ , $M_{CR}=0.8$ ; Cauchy $F$ (scale $0.1$ ), Gaussian $CR$ ( $\sigma=0.1$ ); $p$ decreases linearly $0.25\to 0.125$ ; external archive with pop capacity; binomial crossover; clipping repair; improvement-weighted memory.	Trial mutation: $x_{i}+F(x_{p\mathrm{best}}-x_{i})+F(x_{r1}-x_{r2})$ . Reported as jSO-derived ; specific differences are listed below.
jSO-canonical	Post-hoc BBOB/CEC implementation-sensitivity comparator following the published jSO/iL-SHADE mechanisms.	$N_{0}=\max(4,\mathrm{round}(25\ln(\max(D,2))\sqrt{D}))$ , linear reduction to $N_{\min}=4$ ; $H=5$ ; $M_{F}=0.3$ , $M_{CR}=0.8$ with the special final $(0.9,0.9)$ memory entry; $p:0.25\to 0.125$ ; early $CR$ lower bounds and $F$ cap; weighted current-to- $p$ best mutation; external archive; midpoint-to-parent bound repair.	Uses the paper-prose $p$ schedule. Run on the complete BBOB and CEC 2011 grids as a sensitivity comparator; VQE uses jSO-derived.
iL-SHADE	Standardized adaptive-DE reference through PyADE.	pyade.ilshade.get_default_params(D) with pop size $4D$ , explicit box bounds, supplied replicate seed, and max_evals $=B$ . All other settings are package defaults.	No internal fallback results are included in reported experiments.
CMA-ES	Covariance-adaptation reference.	BBOB/CEC: pycma , uniform $x_{0}$ , $\sigma_{0}=0.15\times\text{mean box width}$ , maxfevals $=B$ , box bounds, supplied seed. VQE: fixed wrapper with $\lambda=4+\lfloor 3\ln D\rfloor$ , $\sigma_{0}=1$ , tolx = tolfun = $10^{-10}$ , generation cap $\lfloor B/\lambda\rfloor$ .	Objective-level FE guard enforces VQE budget if internal termination occurs later.

Appendix

Table 9: Optimizer implementations and numerical settings.

B

denotes the FE budget and

D

the decision-space dimension.

Algorithm	BBOB s/1000 FE	vs base	CEC s/1000 FE	vs base
iSOMA	0.0182	1.00 $\times$	0.0110	1.00 $\times$
iSOMA-AR	0.0204	1.12 $\times$	0.0193	1.75 $\times$
jSO-derived	0.0279	1.54 $\times$	0.0202	1.83 $\times$
jSO-canonical †	0.0304	1.67 $\times$	0.0253	2.30 $\times$
iL-SHADE	0.0334	1.84 $\times$	0.0224	2.03 $\times$
CMA-ES	0.0676	3.72 $\times$	0.0834	7.55 $\times$

Appendix

Table 10: Median computational overhead normalized per 1000 objective evaluations. Wall-clock values are implementation- and hardware-dependent and are included only as relative overhead indicators.

Problem	Control / default AR	W/T/L	Median ratio	Wilcoxon $p$	AR rotation frac.
Ellipsoid (axis-aligned)	random basis / AR default	10/0/15	1.41	0.692	7.4%
Ellipsoid (axis-aligned)	iSOMA / AR default	7/0/18	12.74	0.0451	7.4%
Ellipsoid (rotated)	random basis / AR default	3/0/22	3.16	$1.20\text{\times}{10}^{-04}$	22.5%
Ellipsoid (rotated)	iSOMA / AR default	6/0/19	4.33	$1.20\text{\times}{10}^{-04}$	22.5%
Rastrigin (axis-aligned)	random basis / AR default	11/0/14	1.00	0.874	12.4%
Rastrigin (axis-aligned)	iSOMA / AR default	14/0/11	0.90	0.491	12.4%

Appendix

Table 11: Controlled adaptive-rotation mechanism ablation at

D=10

and 20,000 FEs (25 paired runs). W/T/L and the median error ratio compare the listed control with default learned AR; ratios above one favor learned AR.

Problem	Variant	W/T/L	Median ratio	Wilcoxon $p$
Ellipsoid (axis-aligned)	AR $p_{\max}=0.25$	11/0/14	2.85	0.23
Ellipsoid (axis-aligned)	AR $p_{\max}=0.65$	13/0/12	0.79	0.916
Ellipsoid (axis-aligned)	AR $\tau=0.10$	15/0/10	0.45	0.474
Ellipsoid (axis-aligned)	AR $\tau=0.30$	9/0/16	8.51	0.21
Ellipsoid (rotated)	AR $p_{\max}=0.25$	6/0/19	1.98	0.0125
Ellipsoid (rotated)	AR $p_{\max}=0.65$	12/0/13	1.07	0.23

Appendix

Table 12: One-factor sensitivity of the adaptive-rotation gate. Each variant is paired against the default setting

\tau=0.18

p_{\max}=0.45

Function	base W/T/L	base/AR	$p$	random W/T/L	random/AR	$p$
$f_{2}$	6/0/14	6.88	0.0973	14/0/6	0.05	0.0637
$f_{3}$	10/0/10	1.00	0.784	10/0/10	1.00	0.898
$f_{8}$	4/0/16	3.53	0.00639	8/0/12	1.79	0.245
$f_{9}$	4/0/16	1.41	0.00365	3/0/17	1.67	$7.08\text{\times}{10}^{-04}$
$f_{10}$	4/0/16	1.94	0.00315	4/0/16	3.01	0.00315
$f_{15}$	6/0/14	1.37	0.114	11/0/9	0.84	0.596

Appendix

Table 13: Selected BBOB corroboration of the AR mechanism at

D=10

, instances 1–2, 2000 FE/D (20 paired runs per function after pooling instances). Ratios above one favor default learned AR.

Model	Algorithm	FE	W/T/L	Error ratio	Wilcoxon $p$	Same-point allowed	Same-point repaired
Q1	iSOMA	10k	14/0/11	0.90	0.833	2.34%	0.50%
Q1	iSOMA	30k	15/0/10	0.88	0.596	2.71%	0.92%
Q2	iSOMA	10k	10/0/15	1.18	0.396	0.49%	0.46%
Q2	iSOMA	30k	9/0/16	1.15	0.252	0.91%	0.40%
Q1	iSOMA-AR	10k	14/0/11	0.82	0.672	2.26%	0.37%
Q1	iSOMA-AR	30k	14/0/11	0.96	0.833	2.56%	0.53%

Appendix

Table 14: All-zero PRT-mask control under high effective-shot Gaussian noise at

N=12

(25 paired runs). The nonzero variant repairs an all-zero Bernoulli mask by activating one uniformly random coordinate. Performance ratios below one favor the repair.

Study	Algorithm	Mean rank	Top-1	All targets	Tight target	Tight runtime
BBOB	jSO-derived	2.00	46.8%	65.4%	48.8%	$1.08\times 10^{3}$
BBOB	jSO-canonical	3.33	18.6%	57.8%	36.8%	$1.52\times 10^{3}$
Direct derived/canonical: 557/104/59 W/T/L; median error ratio 0.48; Wilcoxon $p=$ 6.32\text{\times}{10}^{-79}$$ .
CEC 2011	jSO-derived	1.77	63.6%	62.9%	29.1%	0.504
CEC 2011	jSO-canonical	2.73	27.3%	56.7%	23.6%	0.222
Direct derived/canonical: 15/2/5 W/T/L; median regret ratio 0.47; Wilcoxon $p=0.0853$ .

Appendix

Table 15: Post-hoc canonical-jSO sensitivity analysis. Ranks and Top-1 shares use the six-method sensitivity panels. The tight target is

\Delta f=10^{-8}

for BBOB and the

10^{-6}

relative-gap target for CEC 2011; tight runtime is FE/D for BBOB and a fraction of the 50,000-FE budget for CEC. Direct rows compare jSO-derived with jSO-canonical, and ratios below one favor jSO-derived.

Group	Frequent winner	AR/base W/T/L	AR/base ratio	AR/iLS W/T/L	AR/iLS ratio
C1 ill-conditioned / anisotropic	CMA-ES 137/210	186/2/22	0.49	60/2/148	2.72
C2 valley / ridge path-following	CMA-ES 122/150	109/0/41	0.80	35/0/115	1.76
C3 structured multimodality	jSO-derived 125/210	138/0/72	0.91	80/0/130	1.42
C4 basin selection / weak structure	jSO-derived 60/120	72/3/45	0.99	65/3/52	0.92
C5 plateau / nonsmooth / rugged	jSO-derived 48/120	83/0/37	0.86	24/0/96	1.67
C6 asymmetry / deception	CMA-ES 42/90	63/0/27	0.92	39/0/51	1.37

Appendix

Table 16: Diagnostic BBOB groups used to separate landscape mechanisms. These groups overlap by design.

$f$	Function	Landscape feature	Frequent winner	AR/base	AR/iLS	jSO-derived/CMA
1	Sphere	smooth, isotropic, unimodal	CMA-ES 30/30	19/9/2	1/10/19	0/20/10
2	Ellipsoidal (sep.)	separable; unimodal; $\kappa\!\approx\!10^{6}$	CMA-ES 25/30	27/0/3	4/0/26	5/15/10
3	Rastrigin (sep.)	separable; regular multimodality; $\sim\!10^{D}$ optima	jSO-derived 23/30	22/0/8	29/0/1	28/0/2
4	Bueche–Rastrigin	asymmetric/deceptive Rastrigin; $\sim\!10^{D}$ optima	jSO-derived 22/30	16/0/14	30/0/0	29/0/1
5	Linear Slope	linear; optimum at domain boundary	4-way tie (30/30)	0/30/0	7/23/0	0/30/0
6	Attractive Sector	strongly asymmetric unimodal	CMA-ES 30/30	29/0/1	9/0/21	0/10/20

Appendix

Table 17: Function-level BBOB outcomes across 30 matched dimension–instance–budget conditions per function. AR/base and AR/iLS are W/T/L for iSOMA-AR against baseline iSOMA and iL-SHADE.

$F$	Problem	Application family	$D$	Frequent winner	AR/base	AR/iLS	jSO-derived/CMA
1	FM sound parameter estimation	A1 Control & param.	6	jSO-derived 1/1	0/0/1	1/0/0	1/0/0
2	Lennard–Jones potential	A2 Materials	30	jSO-derived 1/1	0/0/1	0/0/1	1/0/0
3 †	Bifunctional catalyst opt. control	A1 Control & param.	1	All tied (1/1)	0/1/0	0/1/0	0/1/0
4 †	Stirred-tank reactor opt. control	A1 Control & param.	1	jSO-derived 1/1	0/1/0	0/1/0	1/0/0
5	Tersoff potential Si(B)	A2 Materials	30	jSO-derived 1/1	0/0/1	0/0/1	1/0/0
6	Tersoff potential Si(C)	A2 Materials	30	jSO-derived 1/1	0/0/1	1/0/0	1/0/0

Appendix

Table 18: CEC 2011 problem-level outcomes at 50,000 evaluations. A dagger ( † ) marks problems for which the Minion C++ rewrite has documented discrepancies relative to the Octave/MATLAB reference.

Model	Condition	iSOMA median	AR median	$p_{\mathrm{Holm}}$	Effect
Q1	$N=12$ , 10k	$1.50\text{\times}{10}^{-05}$	$1.46\text{\times}{10}^{-04}$	0.886	-0.136
Q1	$N=12$ , 30k	0	$1.17\text{\times}{10}^{-13}$	0.005	-0.485
Q1	$N=14$ , 10k	$6.93\text{\times}{10}^{-08}$	$5.10\text{\times}{10}^{-07}$	0.159	-0.341
Q1	$N=14$ , 30k	0	0	1.000	-0.110
Q1	$N=16$ , 10k	0.458	0.458	1.000	0.014
Q1	$N=16$ , 30k	0.458	0.458	0.030	-0.469

Appendix

Table 19: Direct exact-objective comparison of iSOMA-AR with baseline iSOMA.

p_{\mathrm{Holm}}

is the Holm-adjusted two-sided Mann–Whitney

U

result within each benchmark condition. The effect column is oriented so that positive values favor iSOMA-AR because lower error is better.

	(a) Condition-wise mean rank		(b) Median selection penalty
Optimizer	Low noise	High noise	Low noise	High noise
iSOMA	3.00	2.17	0.034	0.313
iSOMA-AR	3.00	1.83	0.046	0.381
jSO-derived	4.00	3.67	0.069	0.620
iL-SHADE	3.67	3.50	0.073	0.583
CMA-ES	3.83	4.50	0.044	0.410

Appendix

Table 20: Performance metrics in the

N=12

noisy VQE extension across low- and high-noise regimes. Lower is better for both metrics.

High noise
Model	Budget	iSOMA median	AR median	$p_{\mathrm{Holm}}$	Effect
Q1	10k	1.037	0.958	1.000	0.181
Q1	30k	0.719	0.649	1.000	0.034
Q2	10k	0.593	0.707	0.898	-0.043
Q2	30k	0.433	0.522	0.664	-0.162
Q3	10k	2.318	2.425	0.923	0.018

Appendix

Table 21: Direct noisy-objective comparison of iSOMA-AR with baseline iSOMA. Endpoints are scored by their exact energy after noisy selection. No direct AR–baseline comparison remains significant after Holm correction.

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