  Standard quantum mechanics has been formulated with complex-valued
Schrodinger equations, wave functions, operators, and Hilbert spaces. However,
previous work has shown possible to simulate quantum systems using only real
numbers by adding extra qubits and exploiting an enlarged Hilbert space. A
fundamental question arises: are the complex numbers really necessary for the
quantum mechanical description of nature? To answer this question, a non-local
game has been developed to reveal a contradiction between a multiqubit quantum
experiment and a player using only real numbers. Here, based on deterministic
and high-fidelity entanglement swapping with superconducting qubits, we
experimentally implement the Bell-like game and observe a quantum score of
8.09(1), which beats the real number bound of 7.66 by 43 standard deviations.
Our results disprove the real-number description of nature and establish the
indispensable role of complex numbers in quantum mechanics.
