Nested quantum annealing correction at finite temperature: $p$-spin models

Quantum annealing in a real device is necessarily susceptible to errors due to diabatic transitions and thermal noise. Nested quantum annealing correction is a method to suppress errors by using an all-to-all penalty coupling among a set of physical qubits representing a logical qubit. We show analytically that nested quantum annealing correction can suppress errors effectively in ferromagnetic and antiferromagnetic Ising models with infinite-range interactions. Our analysis reveals that the nesting structure can significantly weaken or even remove first-order phase transitions, in which the energy gap closes exponentially. The nesting structure also suppresses thermal fluctuations by reducing the effective temperature.

Figure 1

Saddle point equation (16) with $m=m_i$ at $J=1$, $\lambda =1.5$, $T/C=0.02$ with various values of $\mathrm{\Gamma }$. For $\mathrm{\Gamma }/C<5$ there are three intersections (ferromagnetic phase), and above $\mathrm{\Gamma }/C=5$ there is only one, $m=0$ (paramagnetic phase).

Figure 2

The critical line in the $(\mathrm{\Gamma }/C,T/C)$ plane for $J=1$ and $\lambda =0.5$, 1.0, and 1.5. The ordered (disordered) phase is below (above) the lines.

Figure 3

Free energy at the transition point normalized by $C^4$ as a function of $m$ for $p=q=4$. Parameters are chosen to be $J=1,\lambda =1,\mathrm{\Gamma }/C^3=2.37,T/C^4=0.01.$

Figure 4

Free energy at the transition point, normalized by $C^4$, as a function of $m$, for $p=4$, $q=2$, and $J=1,T/C^4=0.01.$ The red dotted line is for $\lambda /C^2=0.01$ and $\mathrm{\Gamma }/C^3=1.2$, the blue dashed line for $\lambda /C^2=0.1$ and $\mathrm{\Gamma }/C^3=1.3$, the green solid line for $\lambda /C^2=0.6$ and $\mathrm{\Gamma }/C^3=2.0$, and the orange dotted line for $\lambda /C^2=3.3$ and $\mathrm{\Gamma }/C^3=6.7$.

Figure 5

The height of the barrier $\mathrm{\Delta }F$ and the width $\mathrm{\Delta }m$ at the transition point, for the same parameters as in Fig. 4. The dotted line (red) represents $\mathrm{\Delta }F/C^4$ and the dashed line (blue) represents $\mathrm{\Delta }m$.

Figure 6

The critical value ${\lambda }_c$ at which the phase transition changes from first order to second order for $p=4$ and $q=2$. The region with $\lambda >{\lambda }_c$ exhibits a second-order phase transition, while the region with $\lambda <{\lambda }_c$ exhibits a first-order phase transition. Above $T/C^4\simeq 12.5$ the ferromagnetic phase disappears due to the strong thermal fluctuations.

Figure 7

$C$ dependence of the potential width at the phase transition. The same data as in Fig. 5. The red lines (UE) represents the unencoded case.

Figure 8

$C$ dependence of the potential height at the phase transition. The same data as in Fig. 5. UE represents the unencoded case.

Figure 9

The $p=3$ case. The difference of the critical values of the transverse field $({\mathrm{\Gamma }}_{c_1}-{\mathrm{\Gamma }}_{c_2})/C^2$ as a function of $\lambda /C$. Since ${\mathrm{\Gamma }}_{c_1}>{\mathrm{\Gamma }}_{c_2}$, there is only a first-order phase transition.

Figure 10

Free energy around the critical point for $\lambda /C^2=4$, $\mathrm{\Gamma }/C^3=8$ for $p=4$.

Figure 11

The critical values ${\mathrm{\Gamma }}_{c_1}/C^4$ and ${\mathrm{\Gamma }}_{c_2}/C^4$ as a function of $\lambda /C^3$ for $p=5$. For $\lambda /C^4<2.078$, ${\mathrm{\Gamma }}_{c_1}>{\mathrm{\Gamma }}_{c_2}$. Therefore, there is a first-order phase transition. For $2.078<\lambda /C^4<2.5$ the first-order phase transition and the second-order phase transition coexist. For $2.5<\lambda /C^4$, there is only a second-order phase transition.

Figure 12

The order of the phase transitions for various values of $p$ and $q=2$. For $p=3$ there is a unique first-order phase transition. For $p=4$, there is a critical value of $\lambda$ above which the phase transition becomes of second order. For $p\ge 5$, there is an intermediate region where the first and second phase transitions coexist. For $p\ge 4$, all the phase transitions turn into second-order phase transitions for sufficiently large $\lambda$.

Figure 13

The overlap of the $m=m_0=0$ and $m=m_{\mathrm{c}}$ eigenstates of the effective $T=0$ Hamiltonian (29), as a function of the effective penalty $\lambda$ for $p=4,q=2$. This overlap determines the closing of the gap, as per Eq. (30).

Figure 14

Comparison of $\langle {\sigma }_{c_i}^z\rangle$ (green dashed line), $\langle {\sigma }_{c_i}^z{\sigma }_{c_j}^z\rangle$ (red dotted line), and the correlation function $Z_{ij}$ (blue solid line) for a single encoded qubit using $Z$-encoded and fully encoded NQAC. (a) $Z$-encoded case [Eq. (31)]: the correlation is small everywhere, and finite only in a small region. (b) Fully encoded case [Eq. (32)]: the correlation is large when $t$ is small, implying that mean-field theory is not valid. The result is independent of the number $C$ (not shown). Both (a) and (b) are for $C=6$. (c) Correlation for the $Z$-encoded case: the correlation decreases as the system size $C$ increases. This is consistent with the fact that the mean-field theory is valid for the $Z$-encoded case.

Figure 15

(a) $w_1$ and $w_2$ for the ground state $k/N=0$, $k/N=0.03$ and highly excited state $k/N=0.48$, as a function of $\mathrm{\Gamma }/C$ at $T/C=0.03$, $\lambda =1$. (b) $F_k$ for $\mathrm{\Gamma }/C=0.03$. The solid purple line represents the ground state free energy contribution $F_0/C^2$.

Figure 16

(a) The borders between the existence and the absence of the metastable states for $k/N=0.1$ and $p=4$. (b) Penalty coupling $\lambda /C^2$ as a function of temperature $T/C^3$ for $\mathrm{\Gamma }=0$ and a function of $\mathrm{\Gamma }$ for $T/C^3=4\times {10}^4$ and $p=4$.

Figure 17

(a) Normalized free energy $F(w_1=1,w_2)-F(w_1=1,w_2=1)$, for $k/N=0.1$, $T/C^3=3.7\times {10}^{-4}$, $\mathrm{\Gamma }=0.1$. The potential barrier between the metastable state $(w_1,w_2)\simeq (-1,1)$ and the true ground state $(w_1,w_2)=(1,1)$ becomes larger as $\lambda /C^2$ increases. (b) The free energy difference $\mathrm{\Delta }F\left(MS\right)=F(1,w_2^{\text{max}})-F(1,-1)$, $\mathrm{\Delta }F\left(GS\right)=F(1,w_2^{\text{max}})-F(1,1)$ between the local maximum $(w_2=w_2^{\text{max}})$, and the local minimum $(w_2=-1)$ or the ground state $(w_2=1)$ for $w_1=1$ seen in (a) (normalized by $C^4$). Other parameters: $k/N=0.1T/C^3=3.7\times {10}^{-4}$, $\mathrm{\Gamma }/C^3=3.7\times {10}^{-3}$.

Figure 18

Free energy barrier in the antiferromagnetic case between two phases at a first-order phase transition, at $T/C=0.03$ and 3.3, for $p=q=4$. Contrast with Fig. 5 for the ferromagnetic case.

Figure 19

Critical values of the penalty strength $\eta /C^3$ and $\lambda /C^2$ for $T=0,p=4,q=2,J=1$, for the hybrid NQAC-PQAC strategy and ferromagnetic coupling. The phase transition is of first order below the blue dotted line and of second order above it. Since the second-order phase transition region is preferred, increasing $\eta$ in the hybrid NQAC-PQAC strategy allows for a lower value of $\lambda$ than in the pure NQAC case.

Figure 20

Solutions to Eqs. (F1a) and (F1b) for $k/N=0.09,J=1,T/C=0,\gamma =0.9$. Red dots are local and global minima. (a) $\mathrm{\Gamma }/C=3.8$, (b) $\mathrm{\Gamma }/C=3.3$, (c) $\mathrm{\Gamma }/C=0.07$, and (d) $\mathrm{\Gamma }/C=0$.

Figure 21

Phase diagrams in the $k\text{−}\mathrm{\Gamma }$ plane and $k\text{−}T$ planes. (a) $k/N$ vs $\mathrm{\Gamma }/C$ at $T=0$. for $\gamma =0.9$ and $\gamma =1.1$, the phase transition line is determined only by the ratio $k/N$: $k/N=0.05$ is the lowest excited state which can have a metastable state for $\gamma =0.9$. (b) $k/N$ vs $T/C$ at $\mathrm{\Gamma }=0$. for $\gamma =0.9$ and $\gamma =1.1$. the phase transition line is determined only by the ratio $k/N$. $T/C=1.28$ is the highest temperature which can have a metastable state for $\gamma =0.9$. Above this, metastable state disappears.

Figure 22

(a) Disappearance of lower excited states for $k/N=0.1,0.2$ in the ferromagnetic case for two values of $\lambda$: metastable states exist only below the transition lines (toward the bottom left). (b) Critical $\lambda$ as a function of $k/N$ in the ferromagnetic case. The metastable states are first excited states. Here $T/C=0.03$, $\mathrm{\Gamma }/C=0.03$, and $J=1$.

Figure 23

Existence of metastable excited states in the antiferromagnetic case. (a) $T/C=0$, (b) $T/C=0.33$. Red dot-dashed, green dotted, and blue dashed lines represent $k/N=0.1$, 0.2, and 0.3, respectively. To the right of these lines the corresponding excited states exist as metastable states (MS), while to the left the corresponding excited states do not exist as metastable states (No MS). Above the black solid line, only the paramagnetic ground state (PM GS) exists and the free energy does not have a local minimum.

Figure 24

Phase diagram in the $T\text{−}\mathrm{\Gamma }$ plane between the phases with $m=0$ and $m\ne 0$, for $\lambda =1.5$ and $\lambda =1$.

Figure 25

$w_1$ (starts from 1 at $\mathrm{\Lambda }=0$) and $w_2$ (starts from $-1$ at $\mathrm{\Lambda }=0$) for a low energy state $k/N=0.01$ and highly excited state $k/N=0.46$. Here $T/C=0.03$, $\lambda =1$ (b) $w_1$ (ends at 1 at $\gamma =1.0$) and $w_2$ (ends at $-1$ at $\gamma =1.0$) for $\mathrm{\Gamma }/C=0.67$ and $T/C=0.03$.