Digital controllability of transverse field Ising chains

Abstract: Quantum Annealing (QA) encounters limitations when the energy gap along the annealing path becomes exponentially small, leading to impractically long runtimes. In contrast, the success of hybrid digital methods like the Quantum Approximate Optimization Algorithm (QAOA), which operate via discrete unitary operations, relies on the optimization of the variational parameters appearing in the state. We analyze a class of transverse-field Ising models which includes problems with exponentially small spectral gaps, but whose dynamics is described in terms of fermionic Gaussian states after Jordan-Wigner mapping. We show that, for digital alternating QAOA-like states, the number of unitaries required to reach the exact ground state scales quadratically with system size and is independent of the annealing gap. This number can be exactly computed from the algebraic properties of the Ansatz, revealing a fundamental distinction between digital methods and their analog counterpart.