control-theoretic-quantum-advantage

name

control-theoretic-quantum-advantage

description

Control-theoretic framework for understanding Quantum Advantage (QA). Recasts quantum computation as operator controllability problem on SU(N), identifying QA with polynomial-in-n upper bound on minimal-time function. Use when: analyzing quantum advantage from control theory perspective, studying operator controllability of quantum systems, deriving time bounds for quantum algorithms (QFT, QAOA), or designing quantum control protocols for superconducting or neutral-atom processors.

Control Theoretic Quantum Advantage

Overview

Methodology from arXiv:2606.13481 (Dario Pighin, June 2026). Provides a systematic control-theoretic route to characterize when and how Quantum Advantage arises, using the bilinear controlled Schrödinger equation as the common thread.

Core Framework

Mathematical Foundation

1. Target Recasting: Recast target quantum computation as operator controllability problem on SU(N)
2. QA Definition: Quantum Advantage = polynomial-in-n upper bound on minimal-time function
3. Common Thread: Bilinear controlled Schrödinger equation

Paradigmatic Applications

QFT on Superconducting Processors (e.g., IBM)

• Prove operator controllability via Lie-algebraic argument
• Derive O(n²) upper bound on minimal time via gate-concatenation lemma + standard QFT circuit decomposition

MIS on Neutral-Atom Processors (e.g., Pasqal)

• Analyze Rydberg-blockade Hamiltonian as bilinear control system
• Reformulate QAOA as continuous-time optimal control problem
• Show problem solvable via controllability result
• Define control-based QA for MIS

When to Use

• Analyzing quantum advantage from a control theory lens
• Designing quantum control protocols for specific hardware platforms
• Deriving time complexity bounds for quantum algorithms
• Comparing different quantum hardware platforms (superconducting vs neutral-atom)

Key Insights

• QA is fundamentally a controllability question, not just a computational one
• Lie algebra provides the bridge between control theory and quantum complexity
• Different hardware platforms require different control-theoretic formulations
• Open problems chart directions at intersection of Control Theory and Quantum Computing

Pitfalls

• Controllability proofs require careful Lie-algebraic analysis
• Minimal-time bounds depend on specific hardware constraints
• QAOA reformulation as continuous-time control requires proper discretization analysis