Landscape Limits of Quantum-Inspired Evolutionary Optimization across 256 continuous functions

Rishi Govind; Ferdin Sagai Don Bosco; Kasturi Venkata Srikanth; Aman Mittal; Abhishek Singh; Aditya Singh; Abhishek Chopra

cs.NESep 25, 2026

Landscape Limits of Quantum-Inspired Evolutionary Optimization across 256 continuous functions

Authors: Rishi Govind, Ferdin Sagai Don Bosco, Kasturi Venkata Srikanth, Aman Mittal, Abhishek Singh, Aditya Singh, Abhishek Chopra

Organizations: BosonQ Psi Corporation, Bangalore, Karnataka, India

Abstract

Quantum-inspired evolutionary optimization (QIEO) represents design variables as a set of qubits and searches a continuous, multi-dimensional landscape through rotation of the qubit's amplitude pair. Every generation rotates those amplitudes toward a single elite, which corresponds to that generation's best. The update is cheap, almost parameter-free, and well-suited for massive parallel implementation, which has encouraged its adoption in engineering, design, and planning applications. However, there are critical issues with this formulation, principally, the treatment of design variables as independent probability components which make it incapable of exploiting local curvature, anisotropy, or variable coupling. Despite this, QIEO is believed to hold promise, and has been used extensively to solve real-world problems, with significant qualitative and computational advantage over its classical counterpart, Genetic Algorithm (GA). A collection of 256 (actually 508; 256 unshifted + 252 shifted, 4 could not be shifted) continuous function are selected from the prior works, in such a way that they represent eleven landscape characteristics, namely continuity, differentiability, separability, scalability, modality, convexity, conditioning, symmetry, maximum dimensionality, dimension dependency, and the coupling pattern of the design variables. These functions are then solved by three QIEO variants, two GA encodings and Hansen's Covariance Matrix Adaptation Evolution Strategy (CMA-ES). The results are evaluated in terms of computational cost, solution precision, and specialization across landscape characteristics. They identify the conditions under which QIEO provides competitive performance, clarify where its independent-variable representation becomes limiting, and establish whether particular QIEO variants offer advantages for specific landscape characteristics.

Figures & tables

Fig. 1: QIEO loop. Blue: initialize amplitudes in superposition, rotate each qubit toward the current elite ( 11 ), and measure a classical population. Orange: evaluate. Green: identify the elite chromosome and update the rotation (interference). The loop is a repeated, coordinate-wise pull onto one elite. Diamonds mark chromosome-level (C) versus gene-level (g) operations.

Fig. 2: GA loop used as the matched-budget baseline. Blue: initialize the population. Orange: evaluate. Green: tournament selection ( $K=3$ ), single-point crossover, and gene-wise mutation ( $15\%$ ). Pink: survivors, with the two elites copied unchanged (“Store Winners”). Diamonds mark chromosome-level (C) versus gene-level (g) operations.

Parameter	Configuration
Implementation	PyGAD
Population size	$1000$
Maximum generations	Same cap as QIEO
Real-coded representation	$D$ bounded coordinate genes
Binary-coded representation	$16D$ binary genes
Binary precision	$16$ bits per decision variable

TABLE I: Genetic Algorithm configuration used as the optimization baseline.

Fig. 3: CMA-ES loop (Hansen default, no restarts). Blue: initialize $(\mathbf{m},\mathbf{C},\sigma,\lambda)$ and sample $\lambda$ offspring from $\mathcal{N}(\mathbf{m},\sigma^{2}\mathbf{C})$ . Orange: evaluate. Green: rank, then update the mean, the global step size $\sigma$ (CSA), and the covariance $\mathbf{C}$ (rank-one and rank- $\mu$ ). The object that evolves is the search distribution, not a set of chromosomes.

Label	Levels used in the analysis
Continuity	Continuous, Discontinuous
Differentiability	Differentiable, Non-differentiable
Separability	Separable, Partially separable, Non-separable
Scalability	Scalable, Non-scalable
Modality	Unimodal, Multimodal
Convexity	Convex, Non-convex

TABLE II: Landscape labels used as predictors of strict success.

Solver	$k/n$	%	$95\%$ CI
QIEO real-coded	$194/256$	$75.8$	$[70.2,80.6]$
QIEO best-of-3 (oracle)	$203/256$	$79.3$	$[73.9,83.8]$
QIEO fixed $\theta$	$174/256$	$68.0$	$[62.0,73.4]$
QIEO adaptive $\theta$	$166/256$	$64.8$	$[58.8,70.4]$
CMA-ES (Hansen default)	$159/256$	$62.1$	$[56.0,67.8]$
GA best-of-2 (oracle)	$119/256$	$46.5$	$[40.5,52.6]$

TABLE III: Unshifted strict success (Excellent+Good) on

n=256

names. Intervals are Wilson

95\%

. Best-of rows are oracles over encodings, not single runs.

Fig. 4: Strict success on 256 unshifted names, with Wilson $95\%$ intervals. Real-coded QIEO leads on raw hit rate; the GA does not approach either QIEO or CMA-ES. Sections V-B and V-C show why this ordering is not the whole story.

Fig. 5: Cost against quality. Filled markers are the median evaluations per function; open diamonds are the mean; bars run from the median to the maximum. The dashed line is the shared cap of $5\times 10^{6}$ evaluations. CMA-ES’s tail stays far below the cap; QIEO’s reaches it. CMA-ES and QIEO occupy the two Pareto-efficient corners; both GA encodings are dominated on cost and quality.

Fig. 6: Accuracy profiles over 256 unshifted names. The dotted and dashed verticals mark the $10^{-6}$ and $10^{-3}$ gates. CMA-ES is nearly bimodal, reaching machine precision or nothing; real-coded QIEO fills the intermediate band that the $10^{-3}$ gate rewards. The QIEO and CMA-ES curves cross near $10^{-7.8}$ .

Fig. 7: Outcome composition for each configuration. CMA-ES converts a smaller number of successes into a higher proportion of Excellent grades; the GA accumulates Good and Fair, rarely converging tightly. The GA is best described as an approximator rather than a solver.

Fig. 8: Radar profiles of strict success over landscape qualities, favourable levels on the left and adverse levels on the right; each axis is one level and $n$ is its subset size. Real-coded QIEO and CMA-ES are shaded. The profiles are nested on the favourable levels but change shape on the adverse ones: CMA-ES closes to within two points on ill-conditioning and exceeds QIEO on discontinuity, while the GA retains QIEO’s shape at a uniformly lower level.

Contrast	% vs %	OR $[95\%]$	$p$	$p_{H}$
Uni- vs multimodal	90.9 / 70.0	4.01 [1.7,9.5]	0.0012	0.011
Convex vs not	94.7 / 72.0	5.70 [1.5,21]	0.0052	0.041
Well- vs ill-cond.	80.4 / 67.3	1.98 [1.1,3.5]	0.028	0.17
Symm. vs asymm.	83.3 / 70.1	2.09 [1.1,3.9]	0.024	0.17
Separable vs not	82.4 / 72.4	1.73 [0.9,3.5]	0.15	0.73
Scalable vs fixed- $D$	78.8 / 73.2	1.34 [0.7,2.4]	0.39	1.00

TABLE IV: Univariate association with real-coded QIEO strict success. OR is the Haldane-corrected odds ratio of the first level against the second.

p_{H}

is Holm-adjusted over the nine tests.

Predictor	QIEO real	CMA-ES	GA best-of-2
Multimodal	$0.24^{**}$	$0.21^{**}$	0.83
Non-separable	0.65	0.86	0.94
Ill-conditioned	0.70	1.63	0.54
Asymmetric	0.72	0.76	0.91
Non-differentiable	0.57	1.27	0.59
$\log_{2}D$ (per doubling)	$0.33^{**}$	$1.01$	$0.18^{**}$

TABLE V: Joint logistic regression of strict success on labels and

\log_{2}D

(

n=256

). Entries are odds ratios; ∗∗ marks

p<0.01

, ∗

p<0.05

Fig. 9: Strict success against coded dimension. QIEO’s advantage is a low-dimensional phenomenon and disappears by $D=6$ ; CMA-ES is nearly flat, and the GA collapses fastest. The highest bin holds only twelve names and should be read as suggestive.

Fig. 10: Modality against coupling; annotations give subset sizes. QIEO stays competent on separable unimodal names and degrades where basins and cross-coordinate terms coincide. CMA-ES follows the same trend, the GA is uniformly weak.

Fig. 11: Conditioning against separability for real-coded QIEO and CMA-ES. Ill-conditioned non-separable names reduce both solvers without inverting their order.

Fig. 12: QIEO encoding gap across characteristics. Real-coded columns are consistently darker; adaptive $\theta$ is not a substitute for a real representation.

Fig. 13: Per-function head-to-head, both axes $\log_{10}$ scaled error, coloured by modality. Points on the lower-left diagonal are solved by both; the horizontal and vertical arms are the exclusive wins that make a two-solver portfolio worthwhile.

Panel of Figure 19 — Fig. 14: Exclusive real-coded QIEO wins (appendix 2-D contour, left of each panel; 3-D surface, right; white marker is campaign-best). Drop-Wave: QIEO real Excellent ( $\hat{f}=-1$ ), GA best Good, CMA-ES Fair. Shubert 3: QIEO real Good ( $\hat{f}=-29.676$ ), GA best Good, CMA-ES Poor ( $\hat{f}=-20.29$ ).

Panel of Figure 20 — Fig. 15: Exclusive CMA-ES wins (same appendix layout as Fig. 14 ). Chen V: CMA-ES Good ( $\hat{f}=-2000$ ); every QIEO encoding and both GAs Poor (real-coded QIEO $\hat{f}=-1000$ ). Thurber: CMA-ES Excellent ( $\hat{f}=5642.71$ ); QIEO real Poor ( $6.33\times 10^{4}$ ), GA best Poor.

Fig. 16: Fate of the $62$ real-coded QIEO failures. CMA-ES is a specialist rescue on ill-conditioned residuals; the GA recovers one name. Most failures are suite-hard rather than QIEO-specific.

Panel of Figure 22 — Fig. 17: Unsolved class 1: nonlinear least-squares residuals (appendix 2-D/3-D). Elattar: all six configurations Poor (QIEO real and CMA-ES both $\hat{f}=1.713$ ). Biggs EXP2: all six Poor ( $\hat{f}\sim 1.40\times 10^{10}$ ).

Panel of Figure 23 — Fig. 18: Unsolved class 2: the table family (appendix 2-D/3-D). Cross-Leg-Table: all six Poor (CMA-ES best, $\hat{f}=-6.71\times 10^{-3}$ ). Crowned-Cross: QIEO real Fair ( $\hat{f}=0.0285$ ), CMA-ES Fair ( $\hat{f}=0.00278$ ), both GAs Poor; none reach Good.

Panel of Figure 24 — Fig. 19: Unsolved class 3: classical hard functions (appendix 2-D/3-D). Damavandi: QIEO real Poor ( $\hat{f}=2$ ), CMA-ES Poor ( $\hat{f}=2$ ), GA binary Fair. Shubert: all six Poor (QIEO real $\hat{f}=-18.10$ against $f^{\star}=-186.73$ ).

Fig. 20: Strict success on the 251 names admitting a tabulated $x^{\star}$ , coded against Gavana-shifted. The ranking and the rates are both essentially unchanged.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Property	Level	$n$	QIEO real	QIEO CI	GA best	CMA-ES
Continuity	Continuous	242	75.6	$[69.8,80.6]$	45.5	59.5
Continuity	Discontinuous	14	71.4	$[45.4,88.3]$	57.1	85.7
Differentiability	Differentiable	207	75.4	$[69.1,80.7]$	46.4	59.9
Differentiability	Non-differentiable	49	75.5	$[61.9,85.4]$	44.9	65.3
Separability	Separable	68	82.4	$[71.6,89.6]$	48.5	63.2
Separability	Partially separable	3	100.0	$[43.8,100]$	0.0	100.0

Appendix

TABLE VI: Strict success (%) by landscape label.

n

is the number of unshifted names in the row.

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Evolutionary Optimization Differential Evolution

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This paper provides a historical analysis of the IEEE CEC Single Objective Optimization competition results (2010-2024). We analyze how benchmark functions shaped winning algorithms, identifying the 2014 introduction of dense rotation matrices as a key performance filter. This design choice introduced parameter non-separability, reduced effectiveness of coordinate-dependent methods (PSO, GA), and established the dominance of Differential Evolution variants capable of preserving the rotational invariance of their difference vectors, specifically L-SHADE. Post-2020 analysis reveals a shift towards high complexity hybrid optimizers that combine different mechanisms (e.g., Eigenvector Crossover, Societal Sharing, Reinforcement Learning) to maximize ranking stability. We conclude by identifying structural similarities between these modern benchmarks and Variational Quantum Algorithm landscapes, suggesting that evolved CEC solvers possess the specific adaptive capabilities required for quantum control.

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