Experimental Realization of Deutsch’s Algorithm in a One-Way Quantum Computer

We report the first experimental demonstration of an all-optical one-way implementation of Deutsch’s quantum algorithm on a four-qubit cluster state. All the possible configurations of a balanced or constant function acting on a two-qubit register are realized within the measurement-based model for quantum computation. The experimental results are in excellent agreement with the theoretical model, therefore demonstrating the successful performance of the algorithm.

Figure 1

Network diagrams for the black boxes in Deutsch’s algorithm. We have $\mathrm{BB}(\mathrm{i})=\mathbb{1}\otimes \mathbb{1}$, $\mathrm{BB}(\mathrm{ii})=\mathbb{1}\otimes {\sigma }_x$, $\mathrm{BB}(\mathrm{iii})=\mathrm{CNOT}$, and $\mathrm{BB}(\mathrm{iv})=(\mathbb{1}\otimes {\sigma }_x)\mathrm{CNOT}$ (CNOT denotes a Control-NOT gate).

Figure 2

(a) Experimental setup. An ultraviolet pump laser performs two passages through a nonlinear beta-barium-borate crystal (BBO) aligned to produce entangled photon pairs of the form $(|00⟩-|11⟩{)}_{ab}/\sqrt{2}$ and $(|00⟩+|11⟩{)}_{cd}/\sqrt{2}$. Compensators (Comp) are half-wave plates (HWP) and BBO crystals used in order to counteract walk-off effects at the BBO. By considering the possibility of obtaining a double-pair emission into the same pair of modes and the action of the polarizing-beam splitters (PBS’s), the four terms entering $|{\Phi }_c⟩$ are obtained and their amplitudes and respective signs adjusted [7] with an additional HWP in mode (a). The algorithm is executed by using quarter-wave plates (QWPs), HWP’s, PBS’s, and photocounter pairs $\{{\mathrm{D}}_j,{\mathrm{D}}_{j^{\prime }}\}$ for the performance of polarization measurements in arbitrary bases of the photons in mode $j$. (b) Sketch of the cluster-state configuration. Qubit 1 embodies the logical input for $|x⟩$ and its output. Qubit 4 (3) is the logical input (output) for $|y⟩$, which is always found to be $|-{⟩}_3$. (c) and (d): Real and imaginary plots, respectively, of the reconstructed experimental density matrix $\varrho$.

Figure 3

The output density matrices for cluster qubits 1 and 3 when BB(i) and BB(iii) are implemented. Panels (a) and (c) show the real parts of the two-qubit density matrix elements as obtained from a maximum-likelihood reconstruction for the no-FF cases of BB(i) and BB(iii), respectively. Panels (b) and (d) show the corresponding plots for the FF case, due to the randomness of measurement outcomes for qubits 2 and 4. In all four cases the imaginary parts are zero in theory and negligible in the experiment (average values $<0.02$).