Experimental Realization of High-Efficiency Counterfactual Computation

Counterfactual computation (CFC) exemplifies the fascinating quantum process by which the result of a computation may be learned without actually running the computer. In previous experimental studies, the counterfactual efficiency is limited to below 50%. Here we report an experimental realization of the generalized CFC protocol, in which the counterfactual efficiency can break the 50% limit and even approach unity in principle. The experiment is performed with the spins of a negatively charged nitrogen-vacancy color center in diamond. Taking advantage of the quantum Zeno effect, the computer can remain in the not-running subspace due to the frequent projection by the environment, while the computation result can be revealed by final detection. The counterfactual efficiency up to 85% has been demonstrated in our experiment, which opens the possibility of many exciting applications of CFC, such as high-efficiency quantum integration and imaging.

Figure 1

Generalized CFC. (a) Quantum circuit of the generalized CFC scheme, with $K$ possible black-box unitaries as defined in Eq. (1). (b) Photonic analog of the $K=2$ quantum circuit, with a photon passing through a series of cascaded three-arm interferometers (blue ovals) for $T_{\theta }$ operation. Different configurations of transparent pathways (empty boxes) and absorbers (black boxes) are associated with different choices of $U_r$. (c) For small $\theta =(\pi /20)$, the simulated populations of ${|0⟩}_S{|0⟩}_R$ (red circle), ${|1⟩}_S{|0⟩}_R$ (black square), and ${|2⟩}_S{|0⟩}_R$ (blue triangle) as a function of the repetition number $n$, for $r=1,2$. The pink dotted line indicates the optimized repetition number $N=\pi /\theta$, where the highest counterfactual efficiency is achieved. (d) The counterfactual efficiency $\eta$ versus the optimized repetition number $N$, with $\theta =\pi /N$. For large $N$, $\eta$ approaches 100%.

Figure 2

Experimental system. (a) Structure and energy levels of the NV center. (b) Relevant energy levels of the NV electron spin with $\{|0_e⟩,|-1_e⟩\}$ for the two-level register $\{{|0⟩}_R,{|1⟩}_R\}$ and the nuclear spin with $\{|0_n⟩,|1_n⟩,|-1_n⟩\}$ for the three-level quantum switch $\{{|0⟩}_S,{|1⟩}_S,{|2⟩}_S\}$. (c) Pulse sequence to control the quantum switch. rf1 and rf2 pulses represent resonant transitions associated with $|0_n⟩\leftrightarrow |\pm 1_n⟩$, respectively. (d) rf-induced Rabi oscillation of the nuclear spin. The red circle, black square, and blue triangle represent $|0_n⟩,|1_n⟩$, and $|-1_n⟩$, respectively. Solid curves are the fittings. Error bars ($\pm 1$ s.d.) induced by the photon shot noise are smaller than the symbols, as the sequence is repeated several million times to get enough photons.

Figure 3

Counterfactual efficiency. (a) Pulse sequences for the generalized CFC scheme, with $U_r$ implemented by a selective MW $\pi$ pulse resonant with the transition conditioned on the switch (nuclear spin) state ${|2⟩}_S$ (i.e., $|-1_n⟩$) if $r=1$ or ${|1⟩}_S$ (i.e., $|1_n⟩$) if $r=2$. (b) For fixed $\theta =\pi /20$, measured populations of ${|0⟩}_S{|0⟩}_R$ (red circle), ${|1⟩}_S{|0⟩}_R$ (black square), and ${|2⟩}_S{|0⟩}_R$ (blue triangle) as a function of the repetition number $n$. Solid curves show the simulated results. The pink dotted line indicates the highest counterfactual efficiency achieved with optimized repetition number $N=\pi /\theta =20$. Error bars ($\pm 1$ s.d.) are smaller than the symbols. (c) For varying $\theta =\pi /N$, the measured counterfactual efficiency $\eta$ versus the optimized repetition number $N$. The green solid curve shows the simulated efficiency with practical imperfections, while the dotted curve shows the ideal efficiency the same as Fig. 1. The dashed line shows the 50% limit. All error bars ($\pm 1$ s.d.) are induced by the photon shot noise.