Process-fidelity estimation of a linear optical quantum-controlled-$Z$ gate: A comparative study

We present a systematic comparison of different methods of fidelity estimation of a linear optical quantum-controlled-$Z$ gate implemented by two-photon interference on a partially polarizing beam splitter. We utilize a linear fidelity estimator based on the Monte Carlo sampling technique as well as a nonlinear estimator based on maximum likelihood reconstruction of a full quantum-process matrix. In addition, we evaluate the lower bound on quantum-gate fidelity determined by average quantum-state fidelities for two mutually unbiased bases. In order to probe various regimes of operation of the gate we introduce a tunable time delay between the two photons. This allows us to move from high-fidelity operation to a regime where the photons become distinguishable and the success probability of the scheme significantly depends on the input state. We discuss in detail possible systematic effects that could influence the gate-fidelity estimation.

Figure 1

Experimental setup: PPBS denotes partially polarizing beam splitter; PBS, polarizing beam splitter; HWP, half-wave plate; QWP, quarter-wave pate; D, single-photon detector; and DL, tunable temporal delay line.

Figure 2

Dependence of the gate fidelity $F_{\chi }$ (solid red line), lower bound on gate fidelity $F_H$ (blue dashed line), and lower bound valid for deterministic operations $F_D$ (green dot-dashed line) on two-photon interference visibility $\mathcal{V}$.

Figure 3

(a)–(c) quantum-process matrices $\chi$ of a linear optical cz gate determined by maximum-likelihood reconstruction from experimental data and (d)–(f) theoretical process matrices determined from the model presented in Sec. 4. The results are shown for three values of two-photon interference visibility (a) and (d) $\mathcal{V}=0.953$, (b) and (e) $\mathcal{V}=0.50$, and (c) and (f) $\mathcal{V}=0.022$. Imaginary parts of reconstructed $\chi$ represent a small noise background and are not plotted. To facilitate comparison, all matrices are normalized such that $\mathrm{Tr}\left[\chi \right]=4$.

Figure 4

(a)–(f) Normalized coincidences $C_{j,j^{\prime }}^k/S_j^k$ and (g)–(i) relative success probabilities $P_{j,k}$ used for determination of the Hofmann bound on the quantum-process fidelity. The results are shown for the three visibilities (a), (d), and (g) $\mathcal{V}=0.953$, (b), (e), and (h) $\mathcal{V}=0.500$, and (c), (f), and (i) $\mathcal{V}=0.022$.

Figure 5

Reference coincidences $D_{jk}$ plotted as a function of time for the three considered two-photon visibilities $\mathcal{V}=0.953$ (solid blue line), $\mathcal{V}=0.50$ (red dotted line), and $\mathcal{V}=0.022$ (green dashed line).