  Motivated by recent realizations of two-dimensional (2D)
superconducting-qubit lattices, we propose a protocol to simulate Hofstadter
butterfly with synthetic gauge fields in superconducting circuits. Based on the
existing 2D superconducting-qubit lattices, we construct a generalized
Hofstadter model on zigzag lattices, which has a fractal energy spectrum
similar to the original Hofstadter butterfly. By periodically modulating the
resonant frequencies of qubits, we engineer a synthetic gauge field to mimic
the generalized Hofstadter Hamiltonian. A spectroscopic method is used to
demonstrate the Hofstadter butterfly from the time evolutions of experimental
observables. We numerically simulate the dynamics of the system with realistic
parameters, and the results show a butterfly spectrum clearly. Our proposal
provides a promising way to realize the Hofstadter butterfly on the latest 2D
superconducting-qubit lattices and will stimulate the quantum simulation of
novel properties induced by magnetic fields in superconducting circuits.
