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Motivated by recent realizations of two-dimensional (2D)superconducting-qubit lattices, we propose a protocol to simulate Hofstadterbutterfly with synthetic gauge fields in superconducting circuits. Based on theexisting 2D superconducting-qubit lattices, we construct a generalizedHofstadter model on zigzag lattices, which has a fractal energy spectrumsimilar to the original Hofstadter butterfly. By periodically modulating theresonant frequencies of qubits, we engineer a synthetic gauge field to mimicthe generalized Hofstadter Hamiltonian. A spectroscopic method is used todemonstrate the Hofstadter butterfly from the time evolutions of experimentalobservables. We numerically simulate the dynamics of the system with realisticparameters, and the results show a butterfly spectrum clearly. Our proposalprovides a promising way to realize the Hofstadter butterfly on the latest 2Dsuperconducting-qubit lattices and will stimulate the quantum simulation ofnovel properties induced by magnetic fields in superconducting circuits.

Quantum simulation of Hofstadter butterfly with synthetic gauge fields on two-dimensional superconducting-qubit lattices