Quantum

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

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Briefings

The Quantum Sphere benchmark, defined by the fourth power of the control field, is proven to exhibit linear convergence in both white-box and black-box settings. Theoretical analysis establishes an exact nonnegative linear combination of sin² terms, yielding a quadratic sandwich with a ratio of 4/π² on a quarter-wave region. White-box optimization achieves a linear rate of 1 - 16/(π⁴κᵣ) via Polyak-Löjtasiewicz inequality, while black-box Evolution Strategy converges linearly in expected hitting time from a certified sublevel set. Numerical simulations confirm these theoretical predictions.

Key points

• The Quantum Sphere benchmark is defined by the fourth power of the control field, guaranteeing a trap-free topology that ensures unhindered optimal pathways.
• The objective function admits an exact representation as a nonnegative linear combination of sin² terms, enabling a two-sided quadratic sandwich with ratio 4/π² on a certified quarter-wave region.

Context and limitations

The study covers only white-box and black-box optimization scenarios within a halved (eighth-wave) region. Additive input noise is analyzed but not tested, and far-field plateau escape remains an open challenge.

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