Counterfactual distributed controlled-phase gate for quantum-dot spin qubits in double-sided optical microcavities

The existing distributed quantum gates required physical particles to be transmitted between two distant nodes in the quantum network. We here demonstrate the possibility to implement distributed quantum computation without transmitting any particles. We propose a scheme for a distributed controlled-phase gate between two distant quantum-dot electron-spin qubits in optical microcavities. The two quantum-dot-microcavity systems are linked by a nested Michelson-type interferometer. A single photon acting as ancillary resource is sent in the interferometer to complete the distributed controlled-phase gate, but it never enters the transmission channel between the two nodes. Moreover, we numerically analyze the effect of experimental imperfections and show that the present scheme can be implemented with high fidelity in the ideal asymptotic limit. The scheme provides further evidence of quantum counterfactuality and opens promising possibilities for distributed quantum computation.

Figure 1

Relevant energy levels and optical selection rules for the optical transition of negatively charged exciton $X^{-}$. The superscript arrows of the photon states indicate their propagation direction along or against the $z$ axis.

Figure 2

Schematic of counterfactual distributed cphase gate for two spin qubits. Spin 1(2) denotes the QD spin coupled with two optical microcavity. $C_k$ and $M_k$ $\left(k=1,2,3...\right)$ are optical circulator and normal mirror, respectively. ${\mathrm{SM}}_{1\left(2\right)}$: switchable mirror. ${\mathrm{SPR}}_{1\left(2\right)}$: switchable polarization rotator, which rotates the polarization by an angle $\vartheta \left(\theta \right)$. $c\text{−}{\mathrm{PBS}}_k$ $\left(k=1,2,3...\right)$: polarizing beam splitter in the circular basis. $D_k$ $\left(k=1,2,R,L\right)$: conventional photon detector. PS is the phase shifter used to perform the transformation $|R\rangle \leftrightarrow -|R\rangle$.

Figure 3

The parameters $x_M-z_M,x_M+z_M$, and $y_M$ in Eq. (15) vs the different values of $N$ and $M$. (a) $x_M-z_M$ is close to $-1$ for large $N$ and appropriate $M$. (b) $x_M+z_M$ approaches 1 for appropriate values of $N$ and $M$. (c) $y_M$ is close to 1 with the increase of $M$ and does not change with $N$.

Figure 4

The success probability of counterfactual distributed chpase gate vs different values of outer and inner cycles $M$ and $N$.

Figure 5

The average fidelity of the counterfactual distributed cphase gate vs the loss probability $\gamma$ for the different values of $M$ and $N$.

Figure 6

The fidelity of counterfactual distributed chpase gate for the case $N=500$ and $M=60$ vs the side leakage rate ${\kappa }_s/\kappa$ and the normalized coupling strength $g/\kappa$. Here we have set $\gamma =0.1\kappa$, which is experimentally achievable.