Dissipative environment may improve the quantum annealing performances of the ferromagnetic $p$-spin model

We investigate the quantum annealing of the ferromagnetic $p$-spin model in a dissipative environment ($p=5$ and $p=7$). This model, in the large-$p$ limit, codifies Grover's algorithm for searching in an unsorted database [L. K. Grover, Proceedings of the 28th Annual ACM Symposium on Theory of Computing (ACM, New York, 1996), pp. 212–219]. The dissipative environment is described by a phonon bath in thermal equilibrium at finite temperature. The dynamics is studied in the framework of a Lindblad master equation for the reduced density matrix describing only the spins. Exploiting the symmetries of our model Hamiltonian, we can describe many spins and extrapolate expected trends for large $N$ and $p$. While at weak system-bath coupling the dissipative environment has detrimental effects on the annealing results, we show that in the intermediate-coupling regime, the phonon bath seems to speed up the annealing at low temperatures. This improvement in the performance is likely not due to thermal fluctuation but rather arises from a correlated spin-bath state and persists even at zero temperature. This result may pave the way to a new scenario in which, by appropriately engineering the system-bath coupling, one may optimize quantum annealing performances below either the purely quantum or the classical limit.

Figure 1

Residual energy in units $\mathrm{\Gamma }$ as a function of the annealing time $t_{\text{f}}$ in units $\tau$, for the Hamiltonian (3) with $p=2$ (bilogarithmic scale). Three different regimes can be observed: a constant beginning region, an intermediate LZ region, and the final power-law tail proportional to $1/t_{\text{f}}^2$.

Figure 2

Residual energy in units $\mathrm{\Gamma }$ as a function of the annealing time $t_{\text{f}}$ in units $\tau$, for the Hamiltonian (3) with $p=5$ (bilogarithmic scale). The power-law adiabatic tail is not visible when $N>16$ in the analyzed range of annealing times.

Figure 3

Residual energy in units $\mathrm{\Gamma }$ for an open or closed quantum system as a function of the annealing time $t_{\text{f}}$ in units $\tau$, for $N=8$ and $p=5$, with (a) $\beta =2$ and (b) $\beta =10$. The scale is bilogarithmic. Different system-bath couplings are shown. When the temperature is lower, the driving force of the bath towards the ground state is more evident.

Figure 4

Bilogarithmic plot of the residual energy in units $\mathrm{\Gamma }$ as a function of $t_{\text{f}}$ in units $\tau$, for $N=8, p=5$, and $\beta \to \infty$. The speed-up in the quantum annealing of the open $p$-spin model is observed also at $T=0$; it is most likely a quantum effect.

Figure 5

Comparison of the SA residual energy of a chain of $N=8$ qubits with that at the end of a quantum annealing, for $\beta =1/T_{\text{f}}=10$ and (a) $p=5$ or (b) $p=7$. The scale is bilogarithmic. Several system-bath couplings are shown for the QA dynamics; a cross-over $t_{\text{f}}^{*}$ is present, when thermal annealing starts to outperform QA. Interestingly, $t_{\text{f}}^{*}$ seems to become longer with increasing $p$.