(Sub)Exponential Quantum Speedup for Optimization

We demonstrate provable (sub)exponential quantum speedups in both discrete and continuous optimization, achieved through simple and natural quantum optimization algorithms, namely the quantum adiabatic algorithm for discrete optimization and quantum Hamiltonian descent for continuous optimization. Our result builds on the Gily\'en--Hastings--Vazirani (sub)exponential oracle separation for adiabatic quantum computing. With a sequence of perturbative reductions, we compile their construction into two standalone objective functions, whose oracles can be directly leveraged by the plain adiabatic evolution and Schr\"odinger operator evolution for discrete and continuous optimization, respectively.