  Quantum annealing in a real device is necessarily susceptible to errors due
to diabatic transitions and thermal noise. Nested quantum annealing correction
is a method to suppress errors by using an all-to-all penalty coupling among a
set of physical qubits representing a logical qubit. We show analytically that
nested quantum annealing correction can suppress errors effectively in
ferromagnetic and antiferromagnetic Ising models with infinite-range
interactions. Our analysis reveals that the nesting structure can significantly
weaken or even remove first-order phase transitions, in which the energy gap
closes exponentially. The nesting structure also suppresses thermal
fluctuations by reducing the effective temperature.
