Reexamination of the evidence for entanglement in a quantum annealer

A recent experiment [T. Lanting et al., Phys. Rev. X 4, 021041 (2014)] claimed to provide evidence of up to eight-qubit entanglement in a D-Wave quantum annealing device. However, entanglement was measured using qubit tunneling spectroscopy, a technique that provides indirect access to the state of the system at intermediate times during the anneal by performing measurements at the end of the anneal with a probe qubit. In addition, an underlying assumption was that the quantum transverse-field Ising Hamiltonian, whose ground states are already highly entangled, is an appropriate model of the device and not some other (possibly classical) model. This begs the question of whether alternative classical or semiclassical models would be equally effective at predicting the observed spectrum and thermal state populations. To check this, we consider a recently proposed classical rotor model with classical Monte Carlo updates, which has been successfully employed in describing features of earlier experiments involving the device. We also consider simulated quantum annealing with quantum Monte Carlo updates, an algorithm that samples from the instantaneous Gibbs state of the device Hamiltonian. Finally, we use the quantum adiabatic master equation, which cannot be efficiently simulated classically and which has previously been used to successfully capture the open-system quantum dynamics of the device. We find that only the master equation is able to reproduce the features of the tunneling spectroscopy experiment, while both the classical rotor model and simulated quantum annealing fail to reproduce the experimental results. We argue that this bolsters the evidence for the reported entanglement.

Figure 1

The annealing schedules used in the experiment [18]. The functional form for $A_{\mathrm{S}}\left(s\right)$ and $A_{\mathrm{P}}\left(s\right)$ in Eq. (11) is identical to the function $A\left(s\right)$ shown. The dashed line corresponds to the experimental temperature of $12.5$ mK.

Figure 2

Depiction of the ($2+1$)-qubit Ising problem. Qubits are displayed as (labeled) green disks, with their local field value above them. Couplings are shown as solid lines connecting qubits with their value above the line. Values are picked according to the experiment in Ref. [18], i.e., $J_{\mathrm{S}}=-2.5, J_{1\mathrm{P}}=-1.8$, and $h_{\mathrm{P}}$ is varied.

Figure 3

Depiction of the ($8+1$)-spin Ising problem. The local fields are zero, and we take the couplings to be $J_{\mathrm{S}}=-2.5$. The probe qubit couples as in Fig. 2.

Figure 4

Tunneling rate calculated using the ME, SSSV, and SQA for $s^{*}=0.339$ and $g_{\mathrm{P}}=10g_{\mathrm{S}}$ compared with the experimental results in Ref. [18] [see the center figure in their Fig. 8], shifted to the left by $14.5$ GHz to align the first peak with the ME results (a constant term in the Hamiltonian present in the experimental setup but not in the ME accounts for this shift). For the experimental results, $1\sigma$ error bars are shown. The two dominant peaks correspond to resonances with the ground state and the first excited state; the magnitude of the peak grows as $g_{\mathrm{P}}\to g_{\mathrm{S}}$ (shown in Appendix pp2), but the peak position remains the same regardless of the value of $g_{\mathrm{P}}$. For SSSV, we show the results with $T=12.5$ mK and ${\tau }_1=5$. For each $h_{\mathrm{P}}$ value, ${10}^6$ runs were performed, and the population of the $|\mathbf{1}\rangle$ state was determined by counting the number of times it occurred in these runs. For SQA, we show the results with $T=12.5$ mK and ${\tau }_1=5$. We performed ${10}^6$ runs for each $h_{\mathrm{P}}$ value, and the population of the $|\mathbf{1}\rangle$ state was determined by counting the number of times it occurred in these runs. The tunneling rate for the ME is measured in $\mu {\text{s}}^{-1}$, while for SSSV and SQA it is in inverse sweeps.

Figure 5

ME results for the ($2+1$)-qubit problem. (a) Energy levels calculated using the tunneling rate peaks via the ME with $g_{\mathrm{P}}=10g_{\mathrm{S}}$. $E$ on the vertical axis label is $E_2, E_3$, or $E_4$. The solid curves correspond to the theoretical result from diagonalizing $H_{\mathrm{S}}$. Error bars (only visible at a few points) represent the 95% confidence interval for the fit of the mean for the Gaussian used to estimate the position of the peak in the tunneling rate (see Fig. 4). (b) Population fraction as calculated using Eq. (7) via the ME with $g_{\mathrm{P}}=10g_{\mathrm{S}}, \tau =1.5$ ms, and a temperature of $12.5$ mK. The solid curves correspond to calculating the Gibbs state associated with $H_{\mathrm{S}}$. We do not show the population of the higher states since they are $<{10}^{-5}$.

Figure 6

The population of the eight computational states for SQA with $N_{\tau }=128$ Trotter slices at 1000 sweeps for ${10}^6$ repetitions compared to their population in the quantum Gibbs state. Both are evaluated at $s^{*}=0.339$ and $h_P=1.03$, which is very close to the resonance condition for the states $|E_1\rangle \otimes |0\rangle$ and $|{\psi }_0\rangle \otimes |1\rangle$.

Figure 7

Tunneling rate calculated using the ME, SSSV, and SQA for $s^{*}=0.284$ as well as the DW experimental results (shifted to align with the first ME peak) on the ($8+1$)-qubit problem. For the experimental results, $1\sigma$ error bars are shown. We show the results with $T=12.5$ mK for all simulation methods. Otherwise, the parameters for all simulation methods are as for the ($2+1$)-qubit case. The tunneling rate for the ME is measured in $\mu {\text{s}}^{-1}$, while for SSSV and SQA it is in inverse sweeps.

Figure 8

Gap and populations ME vs Gibbs state results for the ($8+1$)-qubit problem. (a) The gap to the first excited state calculated using the tunneling rate peaks via the ME with $g_{\mathrm{P}}=10g_{\mathrm{S}}$. The solid curves correspond to the theoretical result from diagonalizing $H_{\mathrm{S}}$. Error bars (barely visible) represent the 95% confidence interval for the fit of the mean for the Gaussian used to estimate the position of the peak in the tunneling rate (see Fig. 4). (b) Population fraction as calculated using Eq. (7) via the ME with $g_{\mathrm{P}}=10g_{\mathrm{S}}, \tau =1.5$ ms, and $T=12.5$ mK. The solid curves correspond to calculating the Gibbs state associated with $H_{\mathrm{S}}$. We do not show the population of the higher states since they are $<{10}^{-5}$. In our simulations, we truncated the energy spectrum to the lowest 16 energy eigenstates to keep the computational time tractable. We kept track of the populations along the evolution in order to make sure that this truncation does not lead to a loss of population. We checked that the results do not change when additional energy levels are included.

Figure 9

Histogram of the negativity of the Gibbs state associated with the two-qubit Hamiltonian of Eq. (1) when a noise term of the form given in Eq. (A1) is included. The red vertical line at $0.211$ is the negativity for the noiseless case. A total of ${10}^5$ noise instances are shown.

Figure 10

Tunneling rate for the ($2+1$)-qubit problem calculated using the ME for $s^{*}=0.339$ and variable $g_{\mathrm{P}}/g_{\mathrm{S}}$ compared with the experimental results in Ref. [18].

Figure 11

Tunneling rate for the ($2+1$)-qubit problem calculated using the ME for $s^{*}=0.339$ and $g_{\mathrm{P}}=10g_{\mathrm{S}}$ with $\sigma =0.05$ and $\sigma =0.1$ ICE on only the probe qubit and without ICE compared with the experimental results in Ref. [18].

Figure 12

Tunneling rate for the ($2+1$)-qubit problem calculated using the ME for $s^{*}=0.339$ and $g_{\mathrm{P}}=10g_{\mathrm{S}}$ with $\sigma =0.05$ and $\sigma =0.1$ ICE on all qubits and without ICE compared with the experimental results in Ref. [18].

Figure 13

Tunneling rate for the ($2+1$)-qubit problem calculated using the ME for $s^{*}=0.339$ and $g_{\mathrm{P}}=10g_{\mathrm{S}}$ averaged over $\alpha$ [see Eq. (B2)] compared with the experimental results in Ref. [18].

Figure 14

The modifications due to the addition of telegraph noise [Eq. (B3)] to the purely Ohmic spectral density function $\gamma \left(\omega \right)$.

Figure 15

Tunneling rate for the three-qubit problem calculated using the ME for $s^{*}=0.339$ and $g_{\mathrm{P}}=10g_{\mathrm{S}}$ using purely Ohmic and Ohmic plus telegraph spectral noise compared with the experimental results in Ref. [18].

Figure 16

Results for the spin Langevin model described in Eq. (C1) for the three-qubit problem. Simulation parameters are $\chi ={10}^{-3}$ and $T=12.5$ mK. One thousand runs were performed at each $h_{\mathrm{P}}$ value. $\langle M_{\mathrm{P}}^z\rangle$ denotes the average over these 1000 runs for the $z$ component of the magnetization of the penalty qubit.