Scalable method for demonstrating the Deutsch-Jozsa and Bernstein-Vazirani algorithms using cluster states

We show that fundamental versions of the Deutsch-Jozsa and Bernstein-Vazirani quantum algorithms can be performed using a small entangled cluster state resource of only six qubits. We then investigate the minimal resource states needed to demonstrate general $n$-qubit versions and a scalable method to produce them. For this purpose, we propose a versatile photonic on-chip setup.

Figure 1

BB circuits for the Deutsch-Jozsa algorithm and cluster state. (a) BBs (i) and (ii) [(iii)–(viii)] correspond to constant (balanced) functions. See Table for $f(x)$ in each case. (b) The six-qubit cluster state $|{\Phi }_C\rangle$ depicting the input-output logical qubits. The measurements and outcomes of qubits $1$, $3,$ and $6$ constitute the algorithm. The measurements of qubits 2, 4, and 5 (and their feed-forward operations) are carried out by the oracle, and thus boxes surround these qubits.

Figure 2

Photonic on-chip setup for demonstrating the algorithms. (a) The setup has three regions: (i) quantum dot sources (1–6) produce single photons; (ii) a network of silica-on-silicon waveguide PBSs (1–5) generate the cluster state via interference; and (iii) superconducting detectors detect the photons. The $\mathsf{H}$s are Hadamard gates implemented by LC polarization rotators. (b) PBS made of two waveguides with integrated LC.

Figure 3

Configuration for DJ${}_n$ and BV${}_n$. (a) Graph state depicting the in-out logical qubits. (b) Construction by PBS fusion ($[|\mathit{HH}\rangle \langle \mathit{HH}|+|\mathit{VV}\rangle \langle \mathit{VV}|]{}_{12}$ followed by ${\mathbb{1}}_1\otimes {\mathsf{H}}_2$) from one pair of photons already fused in the state $(\mathbb{1}\otimes \mathsf{H})1/\sqrt{2}[|\mathit{HH}\rangle +|\mathit{VV}\rangle ]{}_{23}$ and two single photons in $|+\rangle$.