Implementation of multipartite unitary operations with limited resources

A general method for implementing multipartite unitary operations that are close to the identity using a small amount of entanglement and classical communication is presented. For cases where methods were previously known, our approach requires less entanglement. In addition, it is shown that, via compression, the average communication required from all parties except one may be reduced to the same as the entanglement. This improves upon previous methods which required a large amount of communication that did not scale down with the strength of the interaction.

Figure 1

The average entanglement consumed to implement the operation $U\left(\alpha \right)$ as a function of $\alpha$. The case where the resource states have been numerically optimized is shown as the solid line, and the case for the scheme of Ref. 3 is shown as the dashed line. The communication from party $N$ for the numerically optimized scheme is shown as the dotted line. The numerically optimized scheme improves on the entanglement consumption for the scheme of Ref. 3 for all nontrivial values of $\alpha$.

Figure 2

The upper limit on $\mathcal{E}\left(\alpha \right)∕\alpha$ [the RHS of Eq. (11)] minus 5.6418 as a function of $A$.

Figure 3

A diagram of the ancilla qubits held by the $N$ parties. The qubits are represented by circles, and the horizontal rectangles show the qubits held by each party. The columns correspond to each resource state $\cos \left({\beta }_l\right){\mid 0⟩}^{\otimes N}+i\sin \left({\beta }_l\right){\mid 1⟩}^{\otimes N}$ used for independent implementations of $U\left(\alpha \right)$. The dashed ellipses show the qubits upon which each party performs their joint measurement in the basis (15).

Figure 4

Results for the scheme optimized for minimum classical communication from the last party. This communication is shown as the dotted line, and the corresponding entanglement required is shown as the dashed line. The entanglement for the entanglement-optimized scheme is shown as the solid line for comparison.

Figure 5

The ratio of the communication from the final party to the interaction strength $\alpha$.