Quantum search by parallel eigenvalue adiabatic passage

We propose a strategy to implement the Grover search algorithm by adiabatic passage in a very efficient way. An adiabatic process can be characterized by the instantaneous eigenvalues of the pertaining Hamiltonian, some of which form a gap. The key to the efficiency is based on the use of parallel eigenvalues. This allows us to obtain nonadiabatic losses that are exponentially small, independently of the number of items in the database in which the search is performed.

Figure 1

(Color online) Dynamics of the local strategy search for $N=20$ and $\epsilon =1∕11$ (leading to $\alpha T_{\text{local}}\approx 96$). Top frame: $a$ and $b$ (in units of $1∕T_{\text{local}}$). Middle frame: The eigenvalues ${\lambda }_{\pm }$ (in units of $1∕T_{\text{local}}$) exhibiting an avoided crossing. Bottom frame: Populations $P_u={\mathbf{\mid }⟨u\mid \varphi ⟩\mathbf{\mid }}^2$ and $P_m={\mathbf{\mid }⟨m\mid \varphi ⟩\mathbf{\mid }}^2$. The search is achieved with probability 0.995.

Figure 2

(Color online) Dynamics of the parallel strategy search for $F_{T_{\parallel }}\left(t\right)=\tanh (t∕T_{\parallel })$, $N=20$, and $\beta T_{\parallel }=4.7$. Top frame: $a$ and $b$ (in units of $1∕T_{\parallel }$). Middle frame: The parallel eigenvalues ${\lambda }_{\pm }$ (in units of $1∕T_{\parallel }$). Bottom frame: Populations $P_u={\mathbf{\mid }⟨u\mid \varphi ⟩\mathbf{\mid }}^2$ and $P_m={\mathbf{\mid }⟨m\mid \varphi ⟩\mathbf{\mid }}^2$. The search is achieved with probability 0.995.

Figure 3

(Color online) Final losses (in logarithmic scale) as a function of $1∕\gamma =T_{\parallel }∕\sqrt{N}$ (dimensionless) for $N=20$ with $F\left(t\right)=\tanh (t∕T_{\parallel })$ (oscillating full line), its asymptotic value given by (33) (nonoscillating full line, almost indistinguishable from the oscillating full line), and truncated time domains $r=12$ and 8 (dashed lines).

Figure 4

(Color online) Final losses as a function of $N$ for $\epsilon =1∕11$ for the local strategy and for the parallel strategy with $F\left(t\right)=\tanh (t∕T_{\parallel })$ with a truncated time intervals ($r=12$ and 8) and $\gamma$ given by Eq. (36) to have the same cost.