Enhanced Quantum State Detection Efficiency through Quantum Information Processing

We investigate theoretically and experimentally how quantum state-detection efficiency is improved by the use of quantum information processing (QIP). Experimentally, we encode the state of one ion qubit with one additional ancilla qubit. By measuring both qubits, we reduce the state-detection error in the presence of noise. The deviation from the theoretically allowed reduction is due to infidelities of the QIP operations. Applying this general scheme to more ancilla qubits suggests that error in the individual qubit measurements need not be a limit to scalable quantum computation.

Figure 1

Simulated Poissonian distributions $D$ of photon counts for the detection of fluorescence in the case of (left to right) zero, one, or two qubits in the $|\downarrow ⟩$ state (the state that scatters photons). Background noise counts are responsible for the finite photon counts in $D_{\uparrow }$. For the plot shown, the rate of background noise counts $⟨{\dot{n}}_{\mathrm{b}\mathrm{k}\mathrm{g}}⟩=0.125⟨{\dot{n}}_{\downarrow }⟩$, where $⟨{\dot{n}}_{\downarrow }⟩=\frac{1}{2}⟨{\dot{n}}_{\downarrow \downarrow }⟩=2.5\times {10}^4{\mathrm{s}}^{-1}$ is the count rate for the $|\downarrow ⟩$ state and where the detection duration is taken to be $324\mu \mathrm{s}$ (the parameters are chosen to correspond to the actual experiment). The state determination is ambiguous for detected counts $n_{\mathrm{d}\mathrm{e}\mathrm{t}}$ in the region where the $D_{\uparrow }$ and the $D_{\downarrow }$ distributions overlap (see magnified inset), leading to errors. In the case of one- qubit, a threshold $n_1$ must be determined to distinguish the two states. For two qubits being either in the $|\downarrow ⟩|\downarrow ⟩$ state with distribution $D_{\downarrow \downarrow }$ or in the $|\uparrow ⟩|\uparrow ⟩$ state with distribution $D_{\uparrow \uparrow }$, the corresponding threshold $n_2$ always provides a smaller overlap of the distributions, thus a smaller readout error.

Figure 2

Theoretical error in state identification as a function of the detection duration at a given average count rate $⟨{\dot{n}}_{\downarrow }⟩=2.5\times {10}^4{\mathrm{s}}^{-1}$, for two levels of background noise. The one-qubit cases (without encoding) are represented by the dotted lines, and the one-ancilla encoded case by the solid lines. The shaded areas emphasize the reduction of the error by the encoded detection scheme. We assume a background count rate of $⟨{\dot{n}}_{\mathrm{b}\mathrm{k}\mathrm{g}}⟩=0.125⟨{\dot{n}}_{\downarrow }⟩$ in the lower pair of curves (a) and of $⟨{\dot{n}}_{\mathrm{b}\mathrm{k}\mathrm{g}}⟩=1.5⟨{\dot{n}}_{\downarrow }⟩$ in the upper pair of curves (b). Since the photon number thresholds (see Fig. 1) are integers, the curves can show steps where $n_1$ and $n_2$ change.

Figure 3

Experimental error in state identification as a function of the duration of the detection at a given average count rate $⟨{\dot{n}}_{\downarrow }⟩=2.5\times {10}^4{\mathrm{s}}^{-1}$. As for the simulated results shown in Fig. 2, we examine two cases, with (a) $⟨{\dot{n}}_{\mathrm{b}\mathrm{k}\mathrm{g}}⟩=0.125⟨{\dot{n}}_{\downarrow }⟩$ and (b) $⟨{\dot{n}}_{\mathrm{b}\mathrm{k}\mathrm{g}}⟩=1.5⟨{\dot{n}}_{\downarrow }⟩$. The differences between Figs. 2, 3, notably the leveling off of the error probability for long detection times in the encoding cases, are primarily due to the infidelity in the experimental gates required for state preparation and implementing the enhancement protocol (see the text).