The Quantum Sphere: A Physically Realizable Optimization Benchmark with Provable Linear Convergence in White- and Black-Box Settings
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 sin2 terms. From this single identity, we derive a two-sided quadratic sandwich with ratio 4/π2 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/(π4κr), where κr 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).
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BibTeX
@misc{shir2026quantum,
title = {{The Quantum Sphere: A Physically Realizable Optimization Benchmark with Provable Linear Convergence in White- and Black-Box Settings}},
author = {Ofer M. Shir},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03788},
eprint = {2610.03788},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03788},
}
RIS
TY - UNPB
TI - The Quantum Sphere: A Physically Realizable Optimization Benchmark with Provable Linear Convergence in White- and Black-Box Settings
AU - Ofer M. Shir
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.03788
UR - https://arxiv.org/abs/2610.03788
AB - 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).
ER -