Experimental Realization of Programmable Quantum Gate Array for Directly Probing Commutation Relations of Pauli Operators

We experimentally demonstrate an advanced linear-optical programmable quantum processor that combines two elementary single-qubit programmable quantum gates. We show that this scheme enables direct experimental probing of quantum commutation relations for Pauli operators acting on polarization states of single photons. Depending on a state of two-qubit program register, we can probe either commutation or anticommutation relations. Very good agreement between theory and experiment is observed, indicating high-quality performance of the implemented quantum processor.

Figure 1

Experimental setup. Femtosecond laser pulses (394 nm, 150 fs, 76 MHz) pass through two main BBO crystals (2 mm) to produce two pairs of entangled photons. The mutual delays between paths 2, 3, 4 are controlled by movable prisms $\Delta d_1$ and $\Delta d_2$. We incorporate after each PBS a compensator (Comp.) to counter the additional phase shifts of the PBS. The photon in mode 2 serves as signal photon and the photon pair in modes 3 and 4 represents the two-qubit program. The inset shows a conceptual diagram of the scheme involving two programmable quantum gates (PQG) interleaved with another transformation $U$ on the signal qubit.

Figure 2

Reconstructed completely positive maps $\chi$ characterizing the implemented commutators (a) and anticommutators (b). For each case both real and imaginary part of the $4\times 4$ matrix $\chi$ are plotted in the Pauli operators basis. The matrices are normalized according to the experimentally evaluated $K$ factors given in Tables  (see main text).

Figure 3

Four-photon coincidence dips. We plot the dependence of the four-photon coincidence counts on transformation $U$ parameterized by the rotation angle of HWP. The dips directly demonstrate the anticommutativity of $Z$ and $X$ (a) and commutativity of $Z$ operator with itself (b). The coincidence counts were measured in 10 minutes for six input states $|H⟩$, $|V⟩$, $|D⟩$, $|A⟩$, $|R⟩$, $|L⟩$ and summed up. The squares are experimental data and the solid line represents best sinusoidal fit. The error bars are determined from Poissonian statistics.