Effective spin models of Kerr-nonlinear parametric oscillators for quantum annealing

A method of quantum annealing (QA) using Kerr-nonlinear parametric oscillators (KPOs) was proposed. This method is described by bosonic operators and has different characteristics from QA based on the transverse-field Ising model. As the first step to describe this method in the conventional framework of QA, we propose effective spin models of KPOs. The spin models are obtained via a variant of the Holstein-Primakoff transformation and are described by spin-$s$ operators. The terms for detuning, coherent driving, parametric driving, and the Kerr effect are mapped to the transverse field, longitudinal field, and nonlinear terms for the $z$ and $x$ components of spins, respectively. By analyzing the spin models corresponding to KPOs in several settings, we demonstrate that the present spin models for a rather large $s$ and tuned parameters qualitatively reproduce the behavior of KPOs.

Figure 1

The photon number $\langle {\widehat{a}}^{†}\widehat{a}\rangle$ of the ground state of a KPO as a function of $\tilde{p}$ and the corresponding function $f_{\text{sp}}=s-\langle {\widehat{s}}^x\rangle$ of the spin models for (a) $\alpha =0$, (b) $\alpha ={\alpha }_c$, and [inset of (b)] ${\alpha }^2$ equal to the exact $\langle {\widehat{a}}^{†}\widehat{a}\rangle$. Both the two panels display the same photon number (solid). The curves for the spin models are for $s=1$ (dashed double dotted), 2 (dotted), 4 (dashed dotted), and 10 (dashed). All the systems are for $\tilde{\mathrm{\Delta }}=1$ and $\tilde{\epsilon }=0$.

Figure 2

The Wigner function of the ground state of (a) a KPO and the corresponding function of the spin models for (b) $(\alpha ,s)=(0,1)$, (c) $(0,4)$, (d) $(0,10)$, (e) $({\alpha }_c,1)$, (f) $({\alpha }_c,4)$, and (g) $({\alpha }_c,10)$. $x$ and $y$ denote $\sqrt{2}\text{Re}\left(\alpha \right)$ and $\sqrt{2}\text{Im}\left(\alpha \right)$ for a coherent state $|\alpha \rangle$, respectively. (h) The overlap $|\langle {\psi }_{\text{b}}|{\psi }_{\text{s}}\rangle |$ of bosonic and spin ground states as a function of $s$ for $\alpha =0$ (circle) and ${\alpha }_c$ (square). All the systems are for $\tilde{\mathrm{\Delta }}=1$, $\tilde{p}=2$, and $\tilde{\epsilon }=0$.

Figure 3

The quadrature amplitude $\langle {\widehat{a}}^{†}+\widehat{a}\rangle /2$ of the ground state of a KPO as a function of $\tilde{p}$ and the corresponding function $f_{\text{sq}}={(2s-{\alpha }^2)}^{-1/2}\langle {\widehat{s}}^z\rangle$ of the spin model [Eq. (18)] for (a) $\alpha =0$, (b) $\alpha ={\alpha }_c$, and [inset of (b)] ${\alpha }^2$ equal to the exact $\langle {\widehat{a}}^{†}\widehat{a}\rangle$. Both the two panels display the same quadrature amplitude (solid). The curves for the spin models are for $s=1$ (dashed double dotted), 2 (dotted), 4 (dashed dotted), and 10 (dashed). All the systems are for $\tilde{\mathrm{\Delta }}=0$ and $\tilde{\epsilon }=0.1$.

Figure 4

(a) The photon number $\langle {\widehat{a}}^{†}\widehat{a}\rangle$ and (b) quadrature amplitude $\langle {\widehat{a}}^{†}+\widehat{a}\rangle /2$ of the ground state of a KPO as a function of $\tilde{p}$ and the corresponding functions (a) $f_{\text{sp}}^{\left(2\right)}=s-\langle {\widehat{s}}^x\rangle$ and (b) $f_{\text{sq}}^{\left(2\right)}$ [Eq. (A3)] of the spin models including the second order terms [Eq. (A7)] for $\alpha ={\alpha }_c$. The solid curves show the results of the KPO. The curves for the spin models are for $s=1$ (dashed double dotted), 2 (dotted), 4 (dashed dotted), and 10 (dashed). $\tilde{\mathrm{\Delta }}$ and $\tilde{\epsilon }$ are set to the same as those for (a) Fig. 1 and (b) Fig. 3, i.e., (a) $(\tilde{\mathrm{\Delta }},\tilde{\epsilon })=(1,0)$ and (b) (0, 0.1), respectively.

Figure 5

The correlation $C_{\text{b}}=\langle ({\widehat{a}}_1^{†}+{\widehat{a}}_1)({\widehat{a}}_2^{†}+{\widehat{a}}_2)\rangle /4$ of quadrature amplitude of the ground state of two KPOs as a function of $\tilde{p}$ and the corresponding function $C_{\text{s}}=\langle {\widehat{s}}_1^z{\widehat{s}}_2^z\rangle /(2s-{\alpha }_{c0}^2)$ of the spin models for $\alpha ={\alpha }_{c0}$. $C_{\text{s}}$ shown in (a) and (b) is calculated for ${\widehat{H}}_{\text{s}}^{\left(2\right)}$, and in (c) and (d) is for ${\widehat{H}}_{\text{s}}^{\prime \left(2\right)}$. $({\tilde{\epsilon }}_1,{\tilde{\epsilon }}_2,\tilde{J})$ is set to (a) $(0.1,-0.1,0.08)$, (b) $(0.1,-0.1,0.12)$, (c) $(0.1,-0.1,0.08)$, and (d) $(0.1,-0.1,0.12)$. The curves for the spin models are for $s=1$ (dashed double dotted), 2 (dotted), 4 (dashed dotted), and 10 (dashed). $\tilde{\mathrm{\Delta }}=0$ in all the systems.

Figure 6

(a) The expectation value $x$ of quadrature amplitude of the ground state of the mean-field model for KPOs as a function of $\tilde{p}$ (solid) and the corresponding function $\overline{m^z}$ of the mean-field spin models for $\alpha ={\alpha }_{c0}$ and $s=1$ (dashed double dotted), 2 (dotted), 4 (dashed dotted), and 10 (dashed). The close-ups of the onset of (b) $x$ and (c) $\overline{m^z}$ for $s=1$ are also shown. All the systems are for $\tilde{\mathrm{\Delta }}=0.4$, $\tilde{\epsilon }=0$, and $\tilde{J}=0.2$.

Figure 7

The critical pump amplitude ${\tilde{p}}_c$ of the ground state of the mean-field model for KPOs as a function of $\tilde{\mathrm{\Delta }}$ (solid) and the corresponding function ${\tilde{p}}_c^{\text{spin}}$ of the mean-field spin models for $\alpha ={\alpha }_{c0}$ and $s=1$ (dashed double dotted), 2 (dotted), 4 (dashed dotted), and 10 (dashed). The curves correspond to the phase boundaries between the paramagnetic (P) and ferromagnetic (F) phases. All the systems are for $\tilde{\epsilon }=0$ and $\tilde{J}=0.2$.