Griffiths-McCoy singularity on the diluted Chimera graph: Monte Carlo simulations and experiments on quantum hardware

The Griffiths-McCoy singularity is a phenomenon characteristic of low-dimensional disordered quantum spin systems, in which the magnetic susceptibility shows singular behavior as a function of the external field even within the paramagnetic phase. We study whether this phenomenon is observed in the transverse-field Ising model with disordered ferromagnetic interactions on the quasi-two-dimensional diluted Chimera graph both by quantum Monte Carlo simulations and by extensive experiments on the D-Wave quantum annealer used as a quantum simulator. From quantum Monte Carlo simulations, evidence is found for the existence of the Griffiths-McCoy singularity in the paramagnetic phase. The experimental approach on the quantum hardware produces results that are less clear cut due to the intrinsic noise and errors in the analog quantum device but can nonetheless be interpreted to be consistent with the existence of the Griffiths-McCoy singularity as in the Monte Carlo case. This is the first experimental approach based on an analog quantum simulator to study the subtle phenomenon of Griffiths-McCoy singularities in a disordered quantum spin system, through which we have clarified the capabilities and limitations of the D-Wave quantum annealer as a quantum simulator.

Figure 1

Connectivity in the diluted Chimera graph. The original Chimera graph has much more interactions. For example, the site on the top left corner has interactions with all four sites within the unit cell of $4+4$ sites but only two of them remain here.

Figure 2

(a) The Binder ratio $g$ in the scale $-ln(1-g)$ for different system sizes $L=6,8,10,12$ with the inverse temperature $\beta =20$. Curves cross at $\mathrm{\Gamma }\simeq 1.7$. (b) Finite-size scaling analysis of the Binder ratio with ${\mathrm{\Gamma }}_{\mathrm{c}}=1.75\left(4\right)$ and $\nu =1.4\left(2\right)$.

Figure 3

(a) Critical point ${\mathrm{\Gamma }}_{\mathrm{c}}$ and (b) the critical exponent $\nu$ as functions of the temperature $T$. Black dashed lines represent the linear fitting of the data. Extrapolation shows that ${\mathrm{\Gamma }}_{\mathrm{c}}\simeq 1.78$ and $\nu \simeq 1.5$ in the zero-temperature limit.

Figure 4

Log-log plot of the histogram of the local susceptibility $P\left({\chi }_{\mathrm{loc}}\right)$ for different sizes at inverse temperature $\beta =50$ and transverse field $\mathrm{\Gamma }=1.895$. The black solid line shows a fit to the data whose slope is $-13.83\pm 0.15$, which indicates that $d/z^{\prime }\simeq 12.83\pm 0.15$.

Figure 5

Log-log plot of the histogram of the local susceptibility $P\left({\chi }_{\mathrm{loc}}\right)$ for different sizes at inverse temperature $\beta =50$ and transverse field $\mathrm{\Gamma }=1.79$. The slope clearly depends on the system size compared to Fig. 4. The black solid line shows a fit to the data with the largest system size $L=12$ whose slope is $-5.40\pm 0.33$, which indicates that $d/z^{\prime }$ is at least smaller than about $4.40\pm 0.33$.

Figure 6

Log-log plot of the histogram of the local susceptibility $P\left({\chi }_{\mathrm{loc}}\right)$ for different sizes at inverse temperature $\beta =50$ and transverse field $\mathrm{\Gamma }=1.4$ (ferromagnetic phase). $P\left({\chi }_{\mathrm{loc}}\right)$ grows monotonically as the susceptibility ${\chi }_{\mathrm{loc}}$ increases especially for the largest system size $L=12$.

Figure 7

Log-log plot of the histogram of the nonlinear local susceptibility $P\left({\chi }_{\mathrm{nlloc}}\right)$ for different sizes at inverse temperature $\beta =50$ and transverse field $\mathrm{\Gamma }=1.895$. The black solid line shows the fitting to the data whose slope is $-4.02\pm 0.06$, which indicates that $d/3z^{\prime }$ is around $3.02\pm 0.06$.

Figure 8

Log-log plot of the histogram of the nonlinear local susceptibility $P\left({\chi }_{\mathrm{nlloc}}\right)$ for different sizes at inverse temperature $\beta =50$ and transverse field $\mathrm{\Gamma }=1.79$. The black solid line shows the fitting to the data with the largest system size $L=12$ whose slope is $-1.29\pm 0.08$, which indicates that $d/3z^{\prime }$ is smaller than $0.29\pm 0.08$.

Figure 9

The exponent $d/z^{\prime }$ as a function of $\mathrm{\Gamma }$ measured at the inverse temperature $\beta =50$. The data in circles and crosses are taken from ${\chi }_{\mathrm{loc}}$ and ${\chi }_{\mathrm{nlloc}}$, respectively. The vertical dashed line shows the critical point in the zero temperature limit ${\mathrm{\Gamma }}_{\mathrm{c}}=1.78$. The critical point corresponding to this finite-temperature ($\beta =50$) data would be smaller than ${\mathrm{\Gamma }}_{\mathrm{c}}=1.78$ for the zero-temperature value. The horizontal dotted line is for $d/z^{\prime }=3$, where the nonlinear susceptibility starts to diverge. Notice that the present data for $d/z^{\prime }$ from finite-size simulations give upper bounds.

Figure 10

Histogram of the global susceptibility at $\mathrm{\Gamma }=1.925$ (far from the critical point). The black solid line shows the fitting to the data with slope $-8.42\pm 1.8$, indicating that $d/z^{\prime }$ is about $7.42\pm 1.8$.

Figure 11

Histogram of the global susceptibility at $\mathrm{\Gamma }=1.79$ (close to the critical point). The black solid line is the fitting to the data with slope $-2.5\pm 0.22$, or $d/z^{\prime }$ being around $1.5\pm 0.22$.

Figure 12

Histogram of the global nonlinear susceptibility at $\mathrm{\Gamma }=1.91$. The black solid line is the fitting to the data with slope $-2.5\pm 0.25$, meaning that $d/3z^{\prime }$ is around $1.5\pm 0.25$.

Figure 13

Similar plot to Fig. 9 but for the exponent extracted from the global linear susceptibility.

Figure 14

$P\left({\chi }_{\mathrm{nl}}\right)$, which is similar to Fig. 7 but for the global nonlinear susceptibility at $\mathrm{\Gamma }=1.865$. The black solid line shows a fit to the data whose slope is $-1.616\pm 0.25$, which indicates that $d/3z^{\prime }$ is around $0.616\pm 0.25$ or $d/z^{\prime }\simeq 1.85\pm 0.75$, which seems to be quite small compared to the data from global linear susceptibility in Fig. 13.

Figure 15

The annealing schedules $A\left(s\right)$ and $B\left(s\right)$ on the D-Wave 2000Q.

Figure 16

(Left column) Histograms (heat map) of magnetization with system sizes (a) $L=8$, (c) $L=12$, and (e) $L=16$. The horizontal axis $s_{*}$ denotes the pause point during the anneal-pause-quench protocol, corresponding to a finite $\mathrm{\Gamma }$ of the transverse field. The vertical axis is for the magnetization. Up to a certain point $s_{*}\simeq 0.39$, the magnetization tends to be distributed around zero, which implies that the system is in the paramagnetic phase. In the region with $s_{*}$ larger than 0.39, the magnetization tends to have values of saturation $\pm 1$, and the system is in the ferromagnetic phase. Note that the color code is in logarithmic scale. (Right column) Cross sections of the heat map on the left column with (b) $L=8$, (d) $L=12$, and (f) $L=16$, respectively, at select values of $s_{*}$, 0.365 (in the presumed paramagnetic phase) and 0.405 (in the presumed ferromagnetic phase).

Figure 17

(a) The Binder ratio for three linear sizes from the D-Wave experiment. (b) Averaged magnetization for three linear sizes obtained from the D-Wave experiment.

Figure 18

(a) D-Wave data for the global linear susceptibility for system sizes $L=8,12$, and 16. (b) Position of peaks as a function of the inverse system size $1/L$. The red dashed line shows a linear fit.

Figure 19

(a) D-Wave data for the global nonlinear susceptibility. The aspect ratio of this graph is the same as in Fig. 18 for direct comparison. (b) Peak position of the nonlinear susceptibility as a function of the inverse system size $1/L$.

Figure 20

Log-log plot of the D-Wave data for $P\left(\chi \right)$ of the global linear susceptibility $\chi$ at the pause point $s_{*}=0.365$ (far from the critical point). The black solid line is a fit to a line of slope $-14.7\pm 1.74$, implying that $d/z^{\prime }\simeq 13.7\pm 1.7$.

Figure 21

Log-log plot of the D-Wave data for the histogram $P\left(\chi \right)$ of the global linear susceptibility $\chi$ with the pause point $s_{*}=0.369$. The black solid line shows a fit to data whose slope is $-12.3\pm 2.0$, which indicates that the exponent $d/z^{\prime }$ is around $11.3\pm 2.0$.

Figure 22

Log-log plot of the D-Wave data for the histogram $P\left(\chi \right)$ of the global linear susceptibility $\chi$ with the pause point $s_{*}=0.375$. The black solid line shows a fit to data whose slope is $-9.7\pm 2.1$, which indicates that the exponent $d/z^{\prime }$ is around $8.7\pm 2.1$.

Figure 23

Log-log plot of the D-Wave data for the histogram $P\left(\chi \right)$ of the global linear susceptibility $\chi$ with the pause point $s_{*}=0.394$ (above the critical point). No power-law decay behavior is observed on this figure and $P\left(\chi \right)$ roughly grows monotonically as the global linear susceptibility increases.

Figure 24

Log-log plot of the D-Wave data for the global nonlinear susceptibility ${\chi }_{\mathrm{nl}}$ at the pause point $s_{*}=0.365$ (far from the critical point). The black solid line is a fit to data with slope $-8.0\pm 1.1$, i.e., $d/3z^{\prime }$ is around $7.0\pm 1.1$.

Figure 25

Log-log plot of the D-Wave data for $P\left({\chi }_{\mathrm{nl}}\right)$ of the global nonlinear susceptibility obtained at the pause point $s_{*}=0.369$ (far from the critical point). The black solid line shows a fit to data whose slope is $-5.8\pm 0.9$, which indicates that the exponent $d/3z^{\prime }$ is around $4.8\pm 0.9$.

Figure 26

The exponent $d/z^{\prime }$ as a function of the pause point $s_{*}$ estimated from the linear and nonlinear susceptibilities. The vertical dashed line represents the critical point $s_c\simeq 0.386$ and the horizontal dotted line is for $d/z^{\prime }=3$, where the nonlinear susceptibility starts to diverge.

Figure 27

The relation between the pause point $s_{*}$ and the parameters (a) $\beta$ and (b) $\mathrm{\Gamma }$. The black dashed line shows the point $s_{*}=s_c=0.386$ at which $\beta =2.49$ and $\mathrm{\Gamma }=1.37$.

Figure 28

(a) Global susceptibility $\chi$ with different system sizes $L=6,8,10,12$ with the inverse temperature $\beta =20$. (b) Finite-size scaling analysis of the global susceptibility with $\gamma =0.96\left(6\right)$.

Figure 29

(a) Magnetization $\left|m\right|$ with different system sizes $L=6,8,10,12$ with the inverse temperature $\beta =20$. (b) Finite-size scaling analysis of the global susceptibility with the critical exponent $\beta =0.95\left(6\right)$ for magnetization, which is not to be confused with the inverse temperature.

Figure 30

Critical exponents of the global susceptibility $\gamma$ in (a) and the magnetization $\left|m\right|$ in (b) as functions of the temperature $T$. Black dashed lines represent linear fitting of the data.

Figure 31

Binder ratio $g$ with different sizes as a function of $\beta /L^z$ at the critical point. The dynamical critical exponent $z$ is set to (a) $z=0.5$, (b) $z=1.0$, (c) $z=1.25$, and (d) $z=1.5$. The plots use logarithmic scales on both horizontal and vertical axes.

Figure 32

Mean-square error (MSE) as a function of the dynamical exponent $z$. This figure shows that MSE is minimum at around $z\simeq 1$.

Figure 33

Average spin configuration for each site over 100 runs of annealing. Purple dots (denoted by “$+$”) show the result without applying error mitigation and green dots (denoted by “$\times$”) shows the one with error mitigation.

Figure 34

(a) Histogram (heat map) of magnetization without calibration as a function of the pause point. The notation is the same as in Fig. 16. (b) Histogram of the magnetization without calibration with the pause point $s_{*}$ is fixed to 0.21 (paramagnetic phase) and 0.41 (ferromagnetic phase).

Figure 35

(a) Histogram (heat map) of magnetization after calibration. (b) Histogram of the magnetization after calibration with the pause point $s_{*}$ fixed to 0.21 (paramagnetic phase) and 0.41 (ferromagnetic phase). These figures are the same as Fig. 16.