Fig. 2: The energy dispersions of 1D lattice models through the demonstration on the IBM quantum devices and tensor-network simulation.

a The excitation spectra G(ω) of an 11-site Ising model. b The excitation spectra of a 13-site Heisenberg model. The results are obtained from measurements on IBMQ devices. The red lines in a and b represent the energy dispersions of infinite long spin chains obtained by analytic calculations. The intensities at k = 0 have been removed. c The energy dispersion for the 1D Heisenberg chain of length N = 51 with h_z = 0 and J = 1. The total time evolution is set as T_tot = τ = 5. d Experimental simulation error for the measurement results of \({\sum }_{i}\left\langle {\hat{\sigma }}_{i}^{y}(t)\right\rangle /N\) under real-time evolution conducted on IBMQ devices, which is compared to results using noiseless Trotter formulae for both the Ising Hamiltonian (red line) and the Heisenberg Hamiltonian (blue line). e Experimental simulation error for the measurement results G(ω) from IBMQ experiments in the frequency domain averaged over different momenta. The results in d and e are compared to those using noiseless Trotter formulae. The maximal time length is T_tot = 5. f Scalability analysis for the gapless Heisenberg chain. The numerical simulation results for different values of τ are compared with the approximately ideal case in which τ → ∞. The figure shows the maximum error of the intensity G(ω) in the momentum space for different time scales τ and system sizes. The error is calculated as \({\max }_{k}\overline{| {G}_{k,\tau }(\omega )-{G}_{k,\tau \to \infty }(\omega )| }\) where the averaging has been taken over ω.