Quantum-annealing correction at finite temperature: Ferromagnetic $p$-spin models

The performance of open-system quantum annealing is adversely affected by thermal excitations out of the ground state. While the presence of energy gaps between the ground and excited states suppresses such excitations, error correction techniques are required to ensure full scalability of quantum annealing. Quantum annealing correction (QAC) is a method that aims to improve the performance of quantum annealers when control over only the problem (final) Hamiltonian is possible, along with decoding. Building on our earlier work [S. Matsuura et al., Phys. Rev. Lett. 116, 220501 (2016)], we study QAC using analytical tools of statistical physics by considering the effects of temperature and a transverse field on the penalty qubits in the ferromagnetic $p$-body infinite-range transverse-field Ising model. We analyze the effect of QAC on second ($p=2$) and first ($p\ge 3$) order phase transitions, and construct the phase diagram as a function of temperature and penalty strength. Our analysis reveals that for sufficiently low temperatures and in the absence of a transverse field on the penalty qubit, QAC breaks up a single, large free-energy barrier into multiple smaller ones. We find theoretical evidence for an optimal penalty strength in the case of a transverse field on the penalty qubit, a feature observed in QAC experiments. Our results provide further compelling evidence that QAC provides an advantage over unencoded quantum annealing.

Figure 1

Plot of Eq. (11) for $C=3,p=2,\gamma =0.1,\beta =100$, and $\mathrm{\Gamma }=0.5, \mathrm{\Gamma }=4$. Since $\beta <\infty , m=0$ is always a solution. The number of solutions is determined by the slope of the curves at $m=0$: If the slope is greater than 1, there are three solutions ($m_1<0,m_2=0,m_3>0$), and the state is in the symmetry-broken phase. If the slope is less than 1, there is only one solution at $m=0$, and the system is in the symmetric phase.

Figure 2

Phase diagram $(\mathrm{\Gamma },\gamma )$ at $p=2$ for various values of temperature $T$, and for $C=3$. Each second-order phase transition line separates the symmetry-broken phase $m\ne 0$ (above the lines) from the symmetric phase $m=0$ (below the lines). For $T<T_1=1$, the phase transition lines merge at $\mathrm{\Gamma }=2$ as $\gamma \to 0$. The $\mathrm{\Gamma }(\gamma \to 0)$ point takes smaller values for $T_1<T<T_2$, and at $T=T_2=2$ it is zero. Above $T_2$, the phase transition lines merge on finite $\gamma$ in $\mathrm{\Gamma }\to 0$ limit. This means that even in the absence of quantum fluctuations, the spin fluctuations due to the temperature are large enough to have $m=0$ always be the global minimum if $\gamma$ is not too large.

Figure 3

$(\mathrm{\Gamma },T)$ phase diagrams for $p=4,5$ and different penalty values. The space to the right of the curves corresponds to the paramagnetic or symmetric phase, labeled by PM, while the space to the left of the curves corresponds to the ferromagnetic or symmetry-broken phase, labeled by FM. (a) For $p=4$ and $\gamma =0.5$, 0.7, and 1.5. At zero temperature, there is a first-order phase transition. The lower branch ($\mathrm{\Gamma }=2.1,T<0.07$) of the phase boundary connects to the phase transition at zero temperature. This line separates the state characterized by $m_{\text{large}}$ from $m_{\text{small}}$. Another line ($\mathrm{\Gamma }>2.1,T<0.07$) is for the phase transition between $m_{\text{small}}$ and $m=0$. There is no corresponding phase transition at $T=0$ and this transition effectively disappears ($m_{\text{small}}$ goes to zero) as $T$ approaches zero. For $\gamma =1.5$, there is no corresponding phase transition at $T=0$. Therefore there is no branch that reaches $T=0$. (b) A closeup around the branching point in (a). The branching temperature increases as $\gamma$ increases. (c) The phase diagram for $p=4,\gamma =0.5$ and for $p=5,\gamma =0.8$.

Figure 4

Free energy at the branching point for $\gamma =0.7$. At $T=0.078$ and $\mathrm{\Gamma }=2.096$, the free energy possesses three minima at $m=0, m=m_{\text{small}}$, and $m=m_{\text{large}}$.

Figure 5

Free energy as a function of the order parameter $m$ along the phase transition lines, for $p=4, \gamma =0.7$. (a) Transition between $m=0$ and $m=m_{\text{small}} (T<T_{\text{branch}},\mathrm{\Gamma }>{\mathrm{\Gamma }}_{\text{branch}})$. (b) Transition between $m=m_{\text{small}}$ and $m=m_{\text{large}} (T<T_{\text{branch}},\mathrm{\Gamma }={\mathrm{\Gamma }}_{\text{branch}})$. (c) Transition between $m=0$ and $m=m_{\text{large}} (T>T_{\text{branch}},\mathrm{\Gamma }<{\mathrm{\Gamma }}_{\text{branch}})$.

Figure 6

(a) Phase diagram in the presence of a penalty transverse field. Shown are the $p=2$ phase transition lines with $C=3$ for different values of $\gamma$ and $\epsilon$. (b) Comparison of $p=2$ phase transition lines with $C=2$ and $C=3$ for $\gamma =0.5$ and $\epsilon =0$ and 1. The space to the right of the curves corresponds to the paramagnetic or symmetric phase, while the space to the left of the curves corresponds to the ferromagnetic or symmetry-broken phase.

Figure 7

Phase diagram for $p=4$ in the presence of a penalty transverse field. The space to the right of the curves corresponds to the paramagnetic or symmetric phase, while the space to the left of the curves corresponds to the ferromagnetic or symmetry-broken phase. The penalty coupling is chosen to be $\gamma =0.5$ for all curves.

Figure 8

Free energies $F$ with and without the penalty transverse field at the critical points. Parameters are chosen to be $p=4,\gamma =0.5,T=0.03$. The blue line is for $\epsilon =0,\mathrm{\Gamma }=1.85$, and the red line is for $\epsilon =0.1,\mathrm{\Gamma }=1.76$.

Figure 9

(a) Critical value of $\mathrm{\Gamma }$ as a function of the penalty coupling $\gamma$. (b) $m_{\text{large}}$ as a function of $\gamma$. (c) The barrier height $\mathrm{\Delta }F$ as a function of $\gamma$. In all panels $p=4, T=0.025$, and $\epsilon =1$.

Figure 10

Illustration of instanton trajectories. (a) The four-instanton solution. If the transition between two minima happens very quickly, the Euclidean action can be approximated by the sum of the zero instanton action shown in panel (b) and four times the transition action shown in panel (c). The partition function is multiplied by ${\beta }^4/4!$ due to the possible locations of ${\tau }_i, i\in \{1,2,3,4\}$.

Figure 11

A comparison of the gap scaling coefficient $\mathrm{\Delta }=e^{-\alpha N}$ estimated using the sharp instanton and using the area under the free-energy potential barrier for the ferromagnetic $p$ model. (a) The area of the potential barrier (shaded region). (b) The comparison for $\beta =20$. (c) The comparison for $\beta =30$. The solid blue curve corresponds to the area under the free-energy barrier [Eq. (B28)], while the dashed orange curve is the sharp instanton coefficient $-log\left|\langle {\lambda }_{+}\left(m_1\right)|{\lambda }_{+}\left(m_2\right)\rangle \right|$.

Figure 12

Free energy as a function of the order parameter $m$ for $p=4$. $T=0.025$ in panels (a)–(f) with $\gamma =0.5$ in panels (a)–(c), and $\gamma =0.8$ in panels (d)–(f). (a) Four different $\mathrm{\Gamma }$ values for $\gamma =0.5$, with two separated for clarity in panels (b) and (c); (b) $\mathrm{\Gamma }=1.91$, (c) $\mathrm{\Gamma }=1.846$. (d) Four different $\mathrm{\Gamma }$ values for $\gamma =0.8$, with two separated for clarity in panels (e) and (f); (e) $\mathrm{\Gamma }=2.95$, (f) $\mathrm{\Gamma }=2.22$. In panels (g) and (h) $T=0.1, \gamma =0.7$. (g) Four different $\mathrm{\Gamma }$ values, with $\mathrm{\Gamma }=2.08$ separated for clarity in panel (h). (i) Again $T=0.025$. The green solid line is for $\gamma =0.5$ [Fig. 12] and the red dotted line is for $\gamma =0$. The potential barrier for $\gamma =0$ is much larger than that of $\gamma =0.5$. See text for analysis.