The Quantum Sphere: A Physically Realizable Optimization Benchmark with Provable Linear Convergence in White- and Black-Box Settings

Ofer M. Shir

quant-phSep 30, 2026

The Quantum Sphere: A Physically Realizable Optimization Benchmark with Provable Linear Convergence in White- and Black-Box Settings

Authors: Ofer M. Shir

Organizations: Tel-Hai University, Israel

Abstract

We revisit an established Quantum Control objective, characterize its exact local metric geometry, and reposition it as a rigorous benchmark for Search and Optimization. The trap-free topology of the landscape, established two decades ago, guarantees unhindered optimal pathways, but global topology alone does not govern convergence speed. Although the so-called Quantum Sphere is defined by the fourth power of the control field, we show that its yield gap admits an exact representation as a nonnegative linear combination of $\sin^2$ terms. From this single identity, we derive a two-sided quadratic sandwich with ratio $4/π^2$ on a certified, worst-case quarter-wave region, proving the optimal peak nondegenerate modulo its affine gauge and well-conditioned. For white-box optimization, on a halved (eighth-wave) region, the objective satisfies a Polyak-Łojasiewicz inequality with an explicit constant, granting Gradient Descent a linear rate of $1 - 16/(π^4κ_r)$ , where $κ_r$ is the condition number of the relevant Hessian block. For black-box optimization, we prove that the elitist single-parent Evolution Strategy converges linearly, almost surely and in expected hitting time, from a certified sublevel set, despite the landscape's continuous gauge symmetry and its lattice of periodic optima; the certified level is explicit whenever the landscape is trap-free, and near the optimum the strategy provably sees only the quadratic tangent of the peak. Systematic numerical simulations corroborate the theory. Finally, we show that additive input (actuator) noise reweights each term of the representation by a Debye-Waller factor without disrupting the landscape's structure, and outline open challenges: output (signal) noise, far-field plateau escape, angular-synchronization certificates, and another physically scalable benchmark (SHG with Glass).

Figures & tables

Figure 1: 1D landscape section along the alternating comb

\bm{u}=((-1)^{i})_{i=0}^{N-1}

(normalized by

C/8

, where

C=\bm{u}^{\!\top}H\bm{u}>0

), witnessing the

1{:}2{:}4

radius hierarchy. The exact gap reduces to

\sin^{2}(2c)

(solid blue), invariant to the amplitude profile

\bm{A}

; it lies below the quadratic tangent

Q

(dashed) everywhere, and above the lower bound

\frac{4}{\pi^{2}}Q

(dotted) on the certified region

c\leq\pi/4

only—the touch at

c=\pi/4

is where the sandwich goes tight, and it breaks beyond. Three milestones: (i)

c=\pi/8

(open circle): inflection point marking the boundary of convexity (Thm. 2 ); (ii)

c=\pi/4

(red dot): a critical point with positive gap on

\partial\mathcal{T}_{1}

, where the sandwich constant

4/\pi^{2}

is attained; it caps the quarter-wave barrier,

c_{\star}\leq C/8

(Lemma 1 ); and (iii)

c=\pi/2

(open square): a zero of

G

outside the gauge—a wrapped affine phase of slope

\pi

, i.e., a point of the wrap lattice.

	Gradient Descent (white-box)	$(1{+}1)$ -ES (black-box)
Landscape region	convexity and PL on $\mathcal{T}_{1/2}$ (eighth-wave) [Thm. 2 ]	sandwich, star shape and no critical points on $\mathcal{T}_{1}$ (quarter-wave) [Cor. 4 , Lem. 1 ]
Certified start	$\left\lVert P\bm{\delta}^{0}\right\rVert_{2}\leq\tfrac{1}{8\sqrt{\kappa_{r}}}$ , hence $G(\bm{\delta}^{0})\leq\tfrac{\lambda_{r}}{128}$ [Thm. 3 ]	any $\bm{\delta}^{0}$ with $G(\bm{\delta}^{0})<b_{\star}=\min\{G_{\mathrm{trap}},\tfrac{\lambda_{r}}{128}p_{\mathrm{tgt}}^{2/r}\}$ [Thm. 4 ]
— sharp?	radii yes: the comb pins the convexity radius; PL fails at $\partial\mathcal{T}_{1}$ [Props. G.1 , G.2 ]	no; the barrier obeys $c_{\star}\leq\min\{\tfrac{C}{8},\min_{j}G(\pi\bm{e}_{j})\}$ [Lem. 1 ]
Rate (per step)	$\displaystyle 1-\frac{16}{\pi^{4}\kappa_{r}}$ (contraction factor) [Thm. 3 ]	linear, a.s. and in expected hitting time; constant not explicit [Thm. 4 ]
Terminal regime	strong cvx. modulo gauge on $\mathcal{T}_{\theta}$ , $\theta<\tfrac{1}{2}$ : modulus $\cos(\theta\pi)\lambda_{r}$ [Thm. 2 ]	the ES on the quadratic tangent $Q$ [Prop. 1 ] ; measured $0.88$ – $0.96$ of the Sphere reference $2\psi^{\star}_{1/5}/r$ [Tab. 5 ]
Gauge coordinates	frozen — constants of the motion [Thm. 3 ]	neutral martingale; freezes as $\sigma_{t}\to 0$ [Lem. F.1 ]

Table 1: The Quantum Sphere’s breakdown of results: white-box (gradient) versus black-box (ES) guarantees.

Test	Identity checked	Max. deviation
T1	shell sum $=$ DFT yield (rel.)	$3.3\times 10^{-17}$
T2	exact gauges: const. / int.-slope (rel. gap)	$8.5\times 10^{-16}$
	ripple ampl. vs. wraparound closed form	$10^{-13}$ (rel.)
T3	$H=\sum_{k}2w_{k}\,\bm{a}_{k}^{\vphantom{\top}}\bm{a}_{k}^{\!\top}$ vs. 2nd diff. (rel.)	$8.5\times 10^{-8}$
	zero modes (gauge, circular eval.)	$1$ at every $N$
	$\kappa_{\mathrm{DW}}$ (nonzero modes)	$804$ – $828$

Table 2: Shell validation over

N\in\{32,64,80,160\}

	Property checked	Result
S1	direct yield $=$ shell $\sin^{2}$ form (rel.)	$1.2\times 10^{-15}$
S2	gauge kernel exact (incl. $\pi$ -slope comb)	$\leq 1.0\times 10^{-9}$
S3	analytic vs. FD Hessian; rank $=N-2$	$2.4\times 10^{-6}$ ; exact
S4	sandwich on $\mathcal{T}_{1}$ , $16{,}000$ draws	$0$ violations
	observed $G/Q$ (allowed $[0.405,1]$ )	$[0.725,\,0.932]$
S5	lower const. attained by $\tfrac{\pi}{4}(-1)^{i}$ comb	$1.1\times 10^{-16}$

Table 3: Sandwich-theorem validation: worst case over

N\in\{32,64,80,160\}

and three amplitude profiles.

	Property checked	Result
W1	Hessian PSD on $\mathcal{T}_{1/2}$ ( $3{,}000$ pts/size)	confirmed
	comb curvature $=(\bm{u}^{\!\top}\!H\bm{u})\cos 4c$ (rel.)	$2.3\times 10^{-8}$
W2	PL on $\mathcal{T}_{1/2}$ , $12{,}000$ draws total	$0$ violations
	min ratio / bound $32\lambda_{r}/\pi^{4}$	$\geq 6.4$
	strong convexity margin at $\theta=\tfrac{1}{4}$	positive
W3	per-step ratio $\leq 1-16/(\pi^{4}\kappa_{r})$ (to $10^{-24}G_{0}$ )	$0$ violations

Table 4: Validating Thms. 2 – 3 . W1/W2/W4:

N\in\{32,64,80\}

; W3:

N\in\{12,\dots,80\}

(

\kappa_{r}=1.36

–

1.48

$N$	$r$	$\kappa_{r}$	$b/Y(0)$	$c_{\star}/Y(0)$	rate on $G$	$G/Q$	$v\,r/2\psi^{\star}_{1/5}$
$32$	$30$	$1.45$	$6.8\times 10^{-4}$	$[0.012,\,0.17]$	$11.06\;[9.72,\,12.03]$	$1.00$	$0.88$
$64$	$62$	$1.48$	$3.5\times 10^{-4}$	$[0.006,\,0.089]$	$5.45\;[5.22,\,5.97]$	$1.01$	$0.90$
$80$	$78$	$1.48$	$2.9\times 10^{-4}$	$[0.005,\,0.072]$	$4.45\;[4.01,\,4.70]$	$1.00$	$0.92$
$160$	$158$	$1.49$	$1.4\times 10^{-4}$	$[0.002,\,0.037]$	$2.28\;[2.13,\,2.48]$	$1.02$	$0.96$

Table 5: Validating Theorem 4 and Proposition 1 on the top-hat instance. Certified level

b=\frac{\lambda_{r}}{128}p_{\mathrm{tgt}}^{2/r}

and barrier bracket

\lambda_{r}/8\leq c_{\star}\leq\min_{j}G(\pi\bm{e}_{j})

(lower end conditional on the absence of traps), relative to

Y(0)

; per-iteration log-rate of

G_{t}

on the window

[10^{-9}Y(0),\,b]

, mean [min, max] over

10

seeds (

\times 10^{-3}

); ratio to the rate of the same ES on the quadratic tangent

Q

; and fraction of the Sphere reference

2\psi^{\star}_{1/5}/r

$G/Y(0)$	$[10^{-2},10^{-1})$	$[10^{-3},10^{-2})$	$[10^{-4},10^{-3})$	$[10^{-9},10^{-4})$
$N=32$	$1.1\times 10^{-3}$	$9.5\times 10^{-5}$	$2.3\times 10^{-5}$	$4.7\times 10^{-6}$ ( $1/2.1\times 10^{5}$ )
$N=160$	$1.4\times 10^{-3}$	$1.4\times 10^{-4}$	$3.3\times 10^{-5}$	$0$ ( $0/1.0\times 10^{6}$ )

Table 6: Proposition 1 in action: frequency with which

G

and

Q

disagree on accepting the same candidate, along ES trajectories (

10

seeds,

20

probe candidates per iteration), binned by the gap level.

Optimization Convergence Analysis

Optimization convergence analysis studies whether and how quickly iterative algorithms approach solutions, and which assumptions ensure reliable progress.

See all 74 papers

Oct 23, 2026

Acquisition for Spherical-Linear Bayesian Optimization on the Cube

Recent work demonstrated that executing Bayesian Optimization (BO) using simply a linear surrogate with a clever transformation is enough to approximate the performance of the standard Gaussian-Process (GP) based version on high-dimensional test functions. When the decision region is a sphere, this strategy admits rapid exact acquisition. However, it is the unit cube we encounter most often in the BO literature, and in this setting a standard approach, namely a multi-start quasi-Newton method, is employed for solving the acquisition subproblem. In this article, we will discuss a series of results, both positive and negative, regarding the complexity of global resolution of various acquisition subproblems over the unit cube. On the basis of this discussion, we will propose a new acquisition function which is globally optimizable using significantly less execution time than the incumbent approach while maintaining or improving performance. The resulting method is competitive with GP-based BO in terms of accuracy per evaluation, and is superior in terms of accuracy per walltime on black-boxes of low to moderate computational intensity.

Nathan Wycoff

Department of Mathematics and Statistics, University of Massachusetts Amherst

Bayesian Optimization Surrogate-Assisted Optimization Black-Box Optimization

Ask PDF

May 27, 2026

Information-Geometric Optimization on Spheres

We consider the black-box optimization problem on a sphere. Two information-geometric optimization flows (IGO flows) are designed with rigorous calculation of natural search gradients based on hyperbolic (information) geometry of Poincar' e and Bergman balls. We demonstrate that ensembles of generalized Kuramoto oscillators on spheres compute natural search gradients and realize IGO algorithms on both manifolds. The relationship between natural gradient policies in Bergman balls and quantum decision making is pointed out.

Vladimir Ja\' cimović

Faculty of Natural Sciences and Mathematics University of Montenegro Cetinjski put bb., 81000 Podgorica Montenegro

Information Geometry Kuramoto Model Riemannian Optimization Black-Box Optimization

Ask PDF

Mar 12, 2026

Quantum mechanical framework for quantization-based optimization: from Gradient flow to Schroedinger equation

This work presents a quantum mechanical framework for analyzing quantization-based optimization algorithms. The sampling process of the quantization-based search is modeled as a gradient-flow dissipative system, leading to a Hamilton-Jacobi-Bellman (HJB) representation. Through a suitable transformation of the objective function, this formulation yields the Schroedinger equation, which reveals that quantum tunneling enables escape from local minima and guarantees access to the global optimum. By establishing the connection to the Fokker-Planck equation, the framework provides a thermodynamic interpretation of global convergence. Such an analysis between the thermodynamic and the quantum dynamic methodology unifies combinatorial and continuous optimization, and extends naturally to machine learning tasks such as image classification. Numerical experiments demonstrate that quantization-based optimization consistently outperforms conventional algorithms across both combinatorial problems and nonconvex continuous functions.

Jinwuk Seok, Changsik Cho

On-Device AIM Research Section AIC Research Lab, ETRI Daejeon, Rep. Korea

HJB Equations Global Optimization Nonconvex Optimization

Ask PDF

Abstract

Optimization Convergence Analysis

Explore similar work

Acquisition for Spherical-Linear Bayesian Optimization on the Cube

Information-Geometric Optimization on Spheres

Quantum mechanical framework for quantization-based optimization: from Gradient flow to Schroedinger equation