Macroscopic quantum computation using Bose-Einstein condensates

We analyze quantum computation using quantum information stored on two component Bose-Einstein condensates (BECs). We construct a general framework for quantum algorithms to be executed using the collective states of the BECs. The use of BECs allows for an increase of energy scales via bosonic enhancement, resulting in two-qubit gate operations that can be performed at a time reduced by a factor of $N$, where $N$ is the number of bosons in the BEC. We illustrate the scheme by an application to Grover's algorithm, and discuss possible experimental implementations and decoherence effects.

Figure 1

A schematic representation of the entangled state (7).

Figure 2

(a) Schematic energy-level structure of the Grover Hamiltonian. Rabi oscillations take place between the initial $x$ eigenstate $|X\rangle$ and the solution state $|\text{ANS}\rangle$. (b) Rabi oscillations executed by the Grover Hamiltonian for $M=3$ for various boson numbers as shown. The dotted line shows the mean-field result corresponding to the $N\to \infty$ limit, calculated by evolving the Heisenberg equations of motion for products of spin operators up to linear in $S_n^i$. The dashed lines show the Grover evolution including decoherence of the form given by (12) with ${\Gamma }_z=0.2$. The solid lines show results with ${\Gamma }_z=0$.

Figure 3

Two bosonic qubits mediated by a quantum bus. The quantum bus couples transitions between levels $b$ and $c$ with energy $G$. Individual pulses coupling levels $a$ and $c$ with energy $g$ create an adiabatic passage between levels $a$ and $b$.