Quantum computation using weak nonlinearities: Robustness against decoherence

We investigate decoherence effects in the recently suggested quantum-computation scheme using weak nonlinearities, strong probe coherent fields, detection, and feedforward methods. It is shown that in the weak-nonlinearity-based quantum gates, decoherence in nonlinear media can be made arbitrarily small simply by using arbitrarily strong probe fields, if photon-number-resolving detection is used. On the contrary, we find that homodyne detection with feedforward is not appropriate for this scheme because in this case decoherence rapidly increases as the probe field gets larger.

Figure 1

A schematic of the parity gate using weak nonlinearities that entangles two qubits. The two-mode nonlinear interactions $\theta$ and $-\theta$ occur only for the horizontally polarized qubit state $\mid H⟩$.

Figure 2

(Color online) Geometric diagram for the weak-nonlinearity-based two-qubit parity gate using (a) homodyne and (b) photon-number-resolving detection. (a) As the initial amplitude becomes large, the “travel path” $\alpha \theta$ of the coherent state in the phase space should increase to maintain the distance $d_{HD}$, i.e., ${\alpha }_2{\theta }_2>{\alpha }_1{\theta }_1$. This causes increase of decoherence effects for large amplitudes. (b) Regardless of the initial amplitude, the travel path $\alpha \theta$ of the coherent state of the same order can maintain the distance $d_{PD}$, i.e., ${\alpha }_2{\theta }_2\approx {\alpha }_1{\theta }_1$. This reduces the decoherence effects because the interaction time $t$ in a nonlinear medium becomes shorter as the initial amplitude increases.

Figure 3

The amplitude parameter $\mathcal{A}$ (solid line) and the absolute coherence parameter $\mid \mathcal{C}\mid$ (dashed line) against the initial amplitude $\alpha$ for homodyne and photon-number detection. The two-qubit parity gate works when both $\mathcal{A}$ and $\mid \mathcal{C}\mid$ are large. (a) When homodyne detection is used, it is obvious that the condition of both $\mathcal{A}$ and $\mid \mathcal{C}\mid$ being large cannot be met. (b) When photon-number-resolving detection is used, the condition is satisfied for large $\alpha$.

Figure 4

The absolute coherence parameter $\mid \mathcal{C}\mid$ against the energy decay rate $\gamma$. The nonlinear strength is assumed to be $\chi =0.01$. Note that the amplitude parameter $\mathcal{A}$ will be always close to 1 in this regime since $\gamma t$ is very small. The solid line corresponds to $\alpha ={10}^3$ so that ${\theta }_{HD}=\chi t\approx 0.13$. The dashed line corresponds to $\alpha ={10}^4$ and ${\theta }_{HD}=\chi t\approx 0.04$. The coherence parameter $\mid \mathcal{C}\mid$ rapidly decreases for a small increase of $\gamma$.