Quantum thermal search: computing ground states via quantum mixing thermal operations

The BBBV theorem is known to largely prohibit exponential speedup of quantum search over classical search, though not entirely, with potential loopholes such as adiabatic quantum computing. Recently, Chen-Huang-Preskill-Zhou (CHPZ) quantum (thermal) gradient descent proposal suggested another potential way to go around the BBBV theorem. We simplify the heavily complicated CHPZ analysis by focusing on the final equilibrium in the quantum thermal operation framework that has already been rigorously formulated in quantum thermodynamics, resulting in quantum thermal search. In particular, repeated applications of an identical quantum mixing thermal operation result in exponential convergence (in the number of repeated applications) of the system state to the equilibrium Gibbs state for the given system Hamiltonian at initial bath temperature. This allows for an efficient computation of the system ground state. Quantum mixing thermal operations evade the BBBV theorem by transferring initial system state information to the bath. Despite computational advantage of CHPZ and quantum thermal search for computing the ground state, it is also noted that some ground state computations corresponding to NP decision problems may require bath states that are exponentially close to the bath ground state, which translates to polynomially-scaling inverse bath temperature, potentially limiting usefulness of quantum thermal search. Potential implications for black hole physics, in light of pure to mixed and back to pure state evolution, are briefly noted.