Probing many-body localization on a noisy quantum computer

A disordered quantum system of interacting particles exhibits localized behavior when the disorder is large compared to the interaction strength. Studying this phenomenon on a quantum computer with no, or limited, error correction is challenging because even weak coupling to a thermal environment destroys most signatures of localization. Fortunately, spectral functions of local operators are known to contain features that can survive the presence of noise. In these spectra, discrete peaks and a soft gap at low frequencies compared to the thermal phase indicate localization. Here, we present the computation of spectral functions on a trapped-ion quantum computer for a one-dimensional Heisenberg model with disorder. Further, we design an error-mitigation technique which is effective at removing the noise from the measurement allowing clear signatures of localization to emerge as the disorder increases. Thus, we show that spectral functions can serve as a robust and scalable diagnostic of many-body localization on current and future generations of quantum computers.

Figure 1

The circuit used to simulate time evolution under the Heisenberg model Hamiltonian. (a) After the qubits are prepared into the desired initial state, $m=6$ Trotter steps are used to evolve the system to time $t$. All qubits are then measured in the $z$ basis. (b) Each Trotter step consists of several one- and two-qubit gates, as described by Eq. (6). The single-body interactions are implemented as rotations about the $X$ or $Z$ axis ($R_x$ and $R_z$ gates). (c) The two-body interactions $\widehat{U}\left(J\delta \right)$ are implemented as three XX(Ising) gates sandwiched between single-qubit rotations. This segment of the circuit is equivalent to a sequential application of XX, YY, ZZ gates, which describes evolution under the Heisenberg interaction exactly.

Figure 2

The spectrum of ${\widehat{\sigma }}_i^z$ at different values of $J$ (with $w=1$) for a three-site system for two sample disorder realizations (top and bottom). Each panel shows both simulation (curves) and experimental (symbols) results. The different colors are for the different sites. The lack of distinguishing characteristics between the spectra at different values of $J$ for individual samples shows the necessity of averaging over several disorder realizations. With 2400 repetitions, the statistical error bars are smaller than the plot markers.

Figure 3

(a)–(c) The spectral function of ${\widehat{\sigma }}_i^z$ averaged over position and 24 disorder realizations for different values of $J$ (with $w=1$) for a three-site system. Lines show simulation results and circles show experimental results. (d)–(f) The spectral function normalized by its value at $\omega =0$. At $J=0.7$, the spectral response increases as $\omega$ decreases while at $J=0.1$, the spectrum is damped at low frequencies. The bands on the simulation curves and error bars on the experimental data signify the uncertainty after averaging over different disorder realizations. The points at $\omega =0$ in the top row are discontinuous with the rest of the curves since they arise from the diagonal elements of the observable in the eigenstate basis which have qualitatively different behavior than the off-diagonal ones.

Figure 4

The distribution histogram of linewidths $\mathrm{\Gamma }$ calculated as described in the text at different values of $J$ from data taken on three qubits for 24 realizations. The bins are [0-1], [1-2]$,...,$ etc. The number of peaks used to generate the distribution is $\sim 200$ for each value of $J$. We derive the errorbars shown in the plot by assuming each bin approximately follows a binomial distribution. The inset shows the Pearson's first coefficient of skewness, Sk1, and the average linewidth $\overline{\mathrm{\Gamma }}$.

Figure 5

(a) The simulated spectrum for $n=7$ spins averaged over 100 disorder configurations for fixed $J=1$, while $w$ is varied. (b) The same plot with the frequency scaled by $w$.

Figure 6

A schematic example of the splitting of spectral lines according to the theoretical model in [20] for a localized system in the limit $J\ll w$.

Figure 7

An example of how the linewidths are extracted. The blue stars correspond to experimental data. The blue line consists of a polynomial fit. The red diamonds mark the peaks and the red horizontal lines mark the widths of the peaks.