Probing the universality of topological defect formation in a quantum annealer: Kibble-Zurek mechanism and beyond

The number of topological defects created in a system driven through a quantum phase transition exhibits a power-law scaling with the driving time. This universal scaling law is the key prediction of the Kibble-Zurek mechanism (KZM), and testing it using a hardware-based quantum simulator is a coveted goal of quantum information science. Here we provide such a test using quantum annealing. Specifically, we report on extensive experimental tests of topological defect formation via the one-dimensional transverse-field Ising model on two different D-Wave quantum annealing devices. We find that the quantum simulator results can indeed be explained by the KZM for open-system quantum dynamics with phase-flip errors, with certain quantitative deviations from the theory likely caused by factors such as random control errors and transient effects. In addition, we probe physics beyond the KZM by identifying signatures of universality in the distribution and cumulants of the number of kinks and their decay, and again find agreement with the quantum simulator results. This implies that the theoretical predictions of the generalized KZM theory, which assumes isolation from the environment, applies beyond its original scope to an open system. We support this result by extensive numerical computations. To check whether an alternative, classical interpretation of these results is possible, we used the spin-vector Monte Carlo model, a candidate classical description of the D-Wave device. We find that the degree of agreement with the experimental data from the D-Wave annealing devices is better for the KZM, a quantum theory, than for the classical spin-vector Monte Carlo model, thus favoring a quantum description of the device. Our work provides an experimental test of quantum critical dynamics in an open quantum system, and paves the way to new directions in quantum simulation experiments.

Figure 1

The annealing schedules on the DW2KQ quantum annealers at (a) NASA Ames Research Center and (b) Burnaby. For the actual quantum annealing processes, $A\left(s\right)/2$ and $B\left(s\right)/2$ are used as in Eq. (1). The energy scale is converted into frequency, i.e., the vertical axis is $E/h$, where $h$ is the Planck constant.

Figure 2

Kink density as a function of annealing time (log-log scale). Error bars indicate the 68% confidence interval. (a) Data from the C16 solver on the DW2KQ at NASA. (b) Data from the D_2000Q_5 solver on the DW2KQ in Burnaby. Data are averaged over $200000$ samples at each value of $t_a$.

Figure 3

The values of the decay exponent $\alpha$ by various methods. D(N): Data from DW2K at NASA. T(O): Open-system theory. D(B): Data from DW2K at Burnaby. C: Classical SVMC (Sec. 5b). T(C): Closed-system theory.

Figure 4

Numerically computed kink density as a function of annealing time (log-log scale), using iTEBD with QUAPI. A linear annealing schedule, $A_{\mathrm{lin}}\left(s\right)/2=1-s$ and $B_{\mathrm{lin}}\left(s\right)/2=s$, is employed here, where the unit of energy is given by $B\left(1\right)/2$ with $B\left(1\right)$ provided in Fig. 1. The Ohmic cutoff frequency is ${\omega }_{\mathrm{c}}=5$ $\left[B\left(1\right)/2\hslash \right]$, the Trotter time slice is $\mathrm{\Delta }t=0.05$ $\left[2\hslash /B\left(1\right)\right]$, the cutoff memory time is ${\tau }_{\mathrm{c}}=10$ $\left[2\hslash /B\left(1\right)\right]$, and the bond dimension is up to 128. (a) Results for various temperatures and a fixed coupling strength $\eta =0.08$. Dashed horizontal lines show the thermal expectation values at temperatures $T=5$, 2, and 1 from the top. The unit of temperature is given by $B\left(1\right)/2k_B$. (b) Results for zero temperature and various coupling strengths. The rightmost eight data points for each coupling strength are fitted with the power law, Eq. (17), and the corresponding exponent $\alpha$ is provided above each data set shown.

Figure 5

Cumulants of $q\mathrm{th}$-order ${\kappa }_q$ of the kink distribution. The chain length is $L=800$. Error bars indicate the 68% confidence interval. Power-law scaling of cumulants from the first-order ${\kappa }_1$ to third-order ${\kappa }_3$ as functions of the annealing time $t_a$ on (a) the DW2KQ at NASA and (b) the DW2KQ in Burnaby. Ratios ${\kappa }_2/{\kappa }_1$ and ${\kappa }_3/{\kappa }_1$ of cumulants on (c) the DW2KQ at NASA and (d) the DW2KQ in Burnaby. The solid lines are optimized fits to constants, ${\kappa }_2/{\kappa }_1\approx 0.61$ and ${\kappa }_3/{\kappa }_1\approx 0.23$ for the NASA case, and ${\kappa }_2/{\kappa }_1\approx 0.63$ and ${\kappa }_3/{\kappa }_1\approx 0.25$ for the Burnaby case.

Figure 6

Histograms of the number of kinks observed on (a) the DW2KQ at NASA and (b) the DW2KQ in Burnaby. The chain length is $L=800$. From right to left, the annealing times $t_a$ are $1,10,100\mu \mathrm{s}$, respectively. Solid, dashed, and dotted lines are the Gaussian distributions of Eq. (5).

Figure 7

The cumulant ratio ${\kappa }_2/{\kappa }_1$ as a function of the annealing time computed by iTEBD with QUAPI. The annealing schedule and other parameters are the same as in Fig. 4. $\eta$ is the strength of the spin-boson coupling. The solid horizontal line denotes ${\kappa }_2/{\kappa }_1=2-\sqrt{2}$, which is the value theoretically predicted for an isolated (closed) system.

Figure 8

Effective temperatures and the optimized Boltzmann distributions on the D-Wave device. The chain length is $L=800$. (a) Effective temperature $1/{\beta }^{\prime }$ as a function of the annealing time $t_a$. Effective temperatures are obtained by equating Eq. (23) to the D-Wave data or minimizing Eq. (22b) with respect to ${\beta }^{\prime }$. (b) The optimized Boltzmann distributions at $t_a=10\mu \mathrm{s}$ and the best fit of the Boltzmann distribution. Here, $P_{\mathrm{N}}$ and $P_{\mathrm{B}}$ are observed distributions in the DW2KQ at NASA and Burnaby, respectively. $Q_{\mathrm{N}}$ and $Q_{\mathrm{B}}$ are the Boltzmann distribution optimized for $P_{\mathrm{N}}$ and $P_{\mathrm{B}}$, respectively.

Figure 9

Numerically computed kink density ${\rho }_{\mathrm{kink}}$ as a function of the dimensionless annealing time $t_a^{\prime }$ from the SVMC model (log-log scale). The error bars indicate the 68% confidence interval. (a) SVMC simulation results using the NASA DW2KQ annealing schedule at $T=12.1$ mK. (b) SVMC simulation results using the Burnaby DW2KQ annealing schedule at $T=13.5$ mK. Dashed straight lines are linear fits to a power-law decay. For $L=50$, the fitting range is limited from $t_a^{\prime }=5$ to $t_a^{\prime }=20$. Contrast with the DW2KQ results shown in Fig. 2.

Figure 10

Numerically computed cumulants of $q\mathrm{th}$-order ${\kappa }_q$ of the kink distribution. The chain length is $L=800$. Error bars indicate the 68% confidence interval. Solid lines are power-law fits of the cumulants from the first-order ${\kappa }_1$ to third-order ${\kappa }_3$ as functions of the dimensionless annealing time $t_a^{\prime }$ with the annealing schedule of (a) the DW2KQ at NASA at $T=12.1\mathrm{mK}$ and (b) the DW2KQ in Burnaby at $T=13.5\mathrm{mK}$. Ratios ${\kappa }_2/{\kappa }_1$ and ${\kappa }_3/{\kappa }_1$ of cumulants for the annealing schedules of (c) the DW2KQ at NASA and (d) the DW2KQ in Burnaby. Solid lines are fits to constants for the power-law decay region of the cumulants. Contrast with the DW2KQ results shown in Fig. 5.

Figure 11

Histograms of the number of kinks computed using the SVMC model with the annealing schedule of (a) the DW2KQ at NASA at $T=12.1\mathrm{mK}$ and (b) the DW2KQ in Burnaby at $T=13.5\mathrm{mK}$. The chain length is $L=800$. From right to left, the annealing times $t_a^{\prime }$ are $1,10,100\mu \mathrm{s}$, respectively. Solid, dashed, and dotted lines are the Gaussian distributions of Eq. (5). Contrast with the DW2KQ results shown in Fig. 6.

Figure 12

Trace-norm distance between the kink density of the D-Wave device or SVMC simulations and the Boltzmann distribution with the same ${\beta }^{\prime }$, chosen such that the distances are minimized at $t_a=100\mu \mathrm{s}$. The chain length is $L=800$. The solid black lines are exponentials, $D_{}\propto e^{-\gamma t_a}$, fitted to the SVMC data. The dotted lines are polynomials, $D_{}\propto t_a^{-\tau }$, fitted to the D-Wave data for $t_a\in [30,100]\mu \mathrm{s}$ using (a) the NASA and (b) Burnaby annealing schedules.

Figure 13

Kink density as a function of the annealing time with different interactions on the DW2KQ at NASA. The error bars indicate the 68% confidence interval. (a) Ferromagnetic interactions ($J=-1)$. (b) Antiferromagnetic interactions ($J=1)$. (c) Random gauge, where we randomly select $L/2$ sites $i$ and flip the sign of interactions between qubits $i-1,i$ and $i,i+1$.

Figure 14

Kink density for $L=800$ as a function of the annealing time from $t_a=1\mu \mathrm{s}$ to $t_a=2000\mu \mathrm{s}$ on (a) the NASA DW2KQ device and (b) the Burnaby DW2KQ device. In both cases, we used antiferromagnetic interactions. The error bars indicate the 68% confidence interval. The dashed lines are power-law fits to the data up to 100 $\mu \mathrm{s}$.

Figure 15

Kink distribution at $t_a=10\mu s$ for different chain lengths $L=50,200,500,800$ (left to right) from the DW2KQ devices at (a) NASA and (b) Burnaby. The dashed lines are the theoretical prediction of Eq. (5).

Figure 16

Same as Fig. 9 but for $T=50$ mK: numerically computed kink density ${\rho }_{\mathrm{kink}}$ as a function of the dimensionless annealing time $t_a^{\prime }$ from the SVMC model (log-log scale). The error bars indicate the 68% confidence interval. (a) SVMC simulation results using the NASA DW2KQ annealing schedule. (b) SVMC simulation results using the Burnaby DW2KQ annealing schedule. Dashed straight lines are linear fits to a power-law decay. For $L=50$, the fitting range is limited from $t_a^{\prime }=5$ to $t_a^{\prime }=20$.

Figure 17

Same as Fig. 10 but for $T=50$ mK: numerically computed cumulants of $q\mathrm{th}$-order ${\kappa }_q$ of the kink distribution. The chain length is $L=800$. Error bars indicate the 68% confidence interval. Solid lines are power-law fits of the cumulants from the first-order ${\kappa }_1$ to third-order ${\kappa }_3$ as functions of the dimensionless annealing time $t_a^{\prime }$ with the annealing schedule of (a) the DW2KQ at NASA and (b) the DW2KQ in Burnaby. Ratios ${\kappa }_2/{\kappa }_1$ and ${\kappa }_3/{\kappa }_1$ of cumulants for the annealing schedules of (c) the DW2KQ at NASA and (d) the DW2KQ in Burnaby. Solid lines are fits to constants for the power-law decay region of the cumulants.

Figure 18

Same as Fig. 11 but for $T=50$ mK: histograms of the number of kinks computed using the SVMC model with the annealing schedule of (a) the DW2KQ at NASA and (b) the DW2KQ in Burnaby. The chain length is $L=800$. From right to left, the annealing times $t_a^{\prime }$ are $1,10,100\mu \mathrm{s}$, respectively. Solid, dashed, and dotted lines are the Gaussian distributions of Eq. (5).