Figure 3: The four-qubit quantum phase transition experimental results in comparison with numerical simulation.

(a) Four-qubit joint occupation probabilities (in logarithmic scale) as functions of the sweep time t for Δ_r/2π=−30 MHz, Ω/2π=4 MHz and τ=600 ns (error bars, on the order of 1%, are not shown for clarity). λ_c/2π varies from 60 MHz at t=0 ns to 12 MHz at t=600 ns. The 16 probabilities are grouped by their corresponding excitation quanta as marked by different colours: green for no excitation (P₀₀₀₀), purple for one quantum excitation (P₀₀₀₁, P₀₀₁₀, P₀₁₀₀, P₁₀₀₀), blue for two quantum excitation (P₀₀₁₁, P₀₁₀₁, ⋯, P₁₁₀₀), red for three (P₀₁₁₁, P₁₁₀₁, P₁₀₁₁, P₁₁₁₀) and black for four (P₁₁₁₁). The critical point is approached when the two-quantum-excitation curves start to gain finite probability values. Curves in the same group behave similarly, validating the identical spin assumption in the QPT theory. (b) 〈J_z〉/(N/2) dynamics calculated from data in a (points with error bars). Line is a numerical simulation. Error bars are s.d. of repetitive measurements, during each measurement we add a random bias sequence to each qubit to simulate the frequency uncertainties of ±1 MHz (the uncertainty level of our calibration of the qubit frequency). Experimentally measured error bars agree with numerical calculations considering all known uncertainties in our experiments, with the majority of the errors coming from the frequency uncertainties in biasing the qubits, which accumulate over the sweep time, and the readout uncertainties of the occupation probability. (c,d) 〈J_z〉/(N/2) as functions of λ/λ_c showing the existence of QPT (points with error bars). Lines are numerical simulations including decoherence. Error bars are obtained similarly to those in b. The choice of a negative Δ_r is to experimentally minimize the state leakage, which should not affect the dynamics and the QPT physics as calculated using a positive Δ_r in Fig. 2. Experimental signal of 〈J_z〉 is slightly larger than theory prediction because of the slight state leakage, which is miscounted as 〈J_z〉’s signal in the measurements (Supplementary Figs 3 and 4).