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.

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