Quantum annealing and the Schrödinger-Langevin-Kostin equation

We show, in the context of quantum combinatorial optimization, or quantum annealing, how the nonlinear Schrödinger-Langevin-Kostin equation can dynamically drive the system toward its ground state. We illustrate, moreover, how a frictional force of Kostin type can prevent the appearance of genuinely quantum problems such as Bloch oscillations and Anderson localization which would hinder an exhaustive search.

Figure 1

(Color online) (a) The probability density ${\mid \psi (t,x)\mid }^2$ at the initial time $t=0$ and final $t=t_{\mathit{max}}=50$. The potential $V\left(x\right)$ is drawn in arbitrary scale for expository purposes. (b) The expected value $⟨\psi \left(t\right)\mid H_{\nu }\mid \psi \left(t\right)⟩$ of the Hamiltonian operator ($\nu =1$, $\beta =0.5$); inset: the overlap ${\mid ⟨\psi \left(t\right)\mid {\varphi }_{\nu }⟩\mid }^2$ between the ground state of $H_{\nu }$ and the state of the system at time $t$, for $0⩽t⩽t_{\mathit{max}}=50$.

Figure 2

(Color online) (a) The probability density ${\mid \psi (t,x)\mid }^2$ at the initial time $t=0$ and final $t=t_{\mathit{max}}=50$. The potential $V\left(x\right)$, corresponding to the choice $a_{+}=2.25$, $a_{-}=1.75$, $V_0=1$, $\delta =0.1$ of the parameters of Eq. (3), is drawn in arbitrary scale for expository purposes. (b) The expected value $⟨\psi \left(t\right)\mid H_{\nu }\mid \psi \left(t\right)⟩$ of the Hamiltonian operator ($(\nu =1$, $\beta =0.3$); inset: the potential profile $W(t,x)$ at the initial and final times; the potential profile at $t=t_{\mathit{max}}=50$ completely overlaps $V\left(x\right)$.

Figure 3

(Color online) $s=100$, $ϵ=17$, $k=(ϵ-1)∕2$, $g_0=2∕s$, and $0⩽t⩽4s$. (a) Free evolution $(g=0,\beta =0)$; (b) $g=3g_0$, $\beta =0$: Bloch oscillations prevent the wave packet from exploring most of the solution space; (c) $g=3g_0$, $\beta =4g_0$: Bloch oscillations no longer appear, thus allowing a complete exploration of the solution space and convergence to the ground state; and (d) the probabilities, as a function of time, of reaching the $\delta =2ϵ$ rightmost sites of ${\Lambda }_s$ in the free case (thin dashed line) and in the forced and damped case $g=3g_0$, $\beta =4g_0$ (solid thick line).

Figure 4

(Color online) $s=100$, $ϵ=17$, $k=(ϵ-1)∕2$, $g_0=2∕s$, ${\sigma }_0={(10∕s)}^{3∕2}$, and $0⩽t⩽20s$. (a) $g=0$, $\beta =0$, $\sigma =2{\sigma }_0$: the wave packet gets confined in the first half of ${\Lambda }_s$; (b) for $g=0$, $\beta =0$, $\sigma =2{\sigma }_0$ the probability of ever reaching the $\delta =2ϵ$ rightmost sites is negligible (Anderson localization); (c) $g=3g_0$, $\beta =4g_0$, $\sigma =2{\sigma }_0$: viscous friction allows the particle to drift to the right, by successive sojourns (the vertical strips) around successive minima of the Anderson potential; the ensuing slow transfer of the probability mass to the right of ${\Lambda }_s$ is shown in (d).