Validating a two-qubit nonstoquastic Hamiltonian in quantum annealing

We propose a two-qubit experiment for validating tunable antiferromagnetic XX interactions in quantum annealing. Such interactions allow the time-dependent Hamiltonian to be nonstoquastic, and the instantaneous ground state can have negative amplitudes in the computational basis. Our construction relies on how the degeneracy of the Ising Hamiltonian's ground states is broken away from the end point of the anneal: above a certain value of the antiferromagnetic XX interaction strength, the perturbative ground state at the end of the anneal changes from a symmetric to an antisymmetric state. This change is associated with a suppression of one of the Ising ground states, which can then be detected using solely computational basis measurements. We show that a semiclassical approximation of the annealing protocol fails to reproduce this feature, making it a candidate “quantum signature” of the evolution.

Figure 1

Properties of the perturbative ground state calculated from first-order perturbation theory for various $\beta$ values. (a) Expectation value of the SWAP operator. (b) Population in the state $|00\rangle$.

Figure 2

(a) Population in the $|00\rangle$ state at the end of the anneal for a system evolving according to the weak-coupling master equation simulations with $\alpha =2,\beta =0.05$, dimensionless system-bath coupling ${\kappa }^2$, energy scale $k_{B}T/\hslash \omega =1.57$, and varying total annealing time $t_f$. (b) Computational basis populations of the evolved state at the end of the anneal for $\beta =0.05$, $\omega t_f={10}^4$, ${\kappa }^2=0$, $\alpha =2$ with Gaussian noise $\mathcal{N}\sim (0,{\sigma }^2)$ on the Ising parameters. Results are for the mean value of ${10}^3$ independent noise realization, and the error bars correspond to the $95\%$ confidence interval calculated using a bootstrap.

Figure 3

(a) Spin-vector dynamics results for $\alpha =2$ and $\beta =0.05$. (b) Cut through the semiclassical potential for fixed ${\varphi }_i=0$ for the system parameters $\alpha =2,\beta =0.05,s=0.35$.

Figure 4

Spin-vector Monte Carlo results for varying temperatures at $\alpha =2,\beta =0.05$. (a) Average magnetization along $x$ axis. (b) Average magnetization along $z$ axis. The simulations use ${10}^6$ sweeps for each of the ${10}^4$ independent runs performed. The error bars correspond to $2\sigma$ confidence intervals generated by ${10}^3$ bootstraps over the independent runs.

Figure 5

Computational basis populations for (a) spin-vector dynamics and (b) spin-vector Monte Carlo with $\alpha =2$ and $\beta =0.05$. The results here are equivalent to those of Figs. 3 and 4. For (b), the simulations use ${10}^6$ sweeps for each of the ${10}^4$ independent runs performed. The error bars correspond to $2\sigma$ confidence intervals generated by ${10}^3$ bootstraps over the independent runs.

Figure 6

Negativity at fixed $\beta =0.05$ for (a) ${\kappa }^2=0$ and varying $t_f$, and (b) $\omega t_f={10}^4$ and varying ${\kappa }^2$. Also shown is the ground-state value as calculated from perturbation theory (denoted “GS”).

Figure 7

Instantaneous ground-state population along the anneal according to the weak-coupling master equation simulations with $\alpha =2,\beta =0.05$, dimensionless system-bath coupling ${\kappa }^2$, energy scale $k_{B}T/\hslash \omega =1.57$ and (a) $\omega t_f={10}^3$ and (b) $\omega t_f=5\times {10}^3$.