Error of an arbitrary single-mode Gaussian transformation on a weighted cluster state using a cubic phase gate

In this paper we propose two strategies for decreasing the error of arbitrary single-mode Gaussian transformations implemented using one-way quantum computation on a four-node linear cluster state. We show that it is possible to minimize the error of the arbitrary single-mode Gaussian transformation by a proper choice of the weight coefficients of the cluster state. We modify the computation scheme by adding a non-Gaussian state obtained using a cubic phase gate as one of the nodes of the cluster. This further decreases the computation error. We evaluate the efficiencies of the proposed optimization schemes comparing the probabilities of the error correction of the quantum computations with and without optimizations. We show that for some transformations, the error probability can be reduced by up to 900 times.

Figure 1

(a) Configuration of the cluster state used as a resource for computation. (b) Scheme for implementing the arbitrary single-mode Gaussian operation on a linear weighted four-node cluster state. (c) Scheme of implementation of arbitrary single-mode Gaussian operation on a pair of two-node cluster states. In the diagram, In is the input state, ${\mathrm{S}}_j$ are squeezed states, ${\widehat{C}}_z\left(g_j\right)$ is the cz transformation with weight coefficient $g_j$, ${\theta }_j$ are the phases of the local oscillators employed for a balanced homodyne detection, and $X$ and $Y$ are operations that displace the corresponding quadratures of the fields in the channel, depending on the detection results.

Figure 2

Implementation of in-line squeezing.

Figure 3

Graph of the dependence of the value of the cz gate weight coefficient on the variance of the squeezed oscillator used to implement this transformation. On the graph, the blue solid line indicates the dependence of the weight coefficient on the variance of the squeezed state and the red dotted line indicates the squeezing limit experimentally implemented to date.

Figure 4

Distribution of errors $\parallel \langle \delta {\widehat{\mathbf{e}}}_0^2\rangle {\parallel }_{\infty }/\langle \delta {\widehat{y}}_s^2\rangle$ depending on the implemented single-mode transformation, i.e., depending on $b$ and $d$ [see Eq. (15)]. The graph shows two error surfaces corresponding to computations on two cluster states. The lower surface corresponds to the case of computations on a weighted optimized cluster state ($g_1=g_2=5$ and $g_3=g_4=4$). The upper surface corresponds to the computational errors on the unweighted cluster state ($g_1=g_2=g_3=g_4=1$).

Figure 5

Distribution of single-mode transformation errors $\parallel \langle \delta {\widehat{\mathbf{e}}}_0^2\rangle {\parallel }_{\infty }/\langle \delta {\widehat{y}}_s^2\rangle$ depending on $b$ and $d$ [see Eq. (15)]. (a) Transformations performed on the cluster state with optimized weight coefficients. In the diagram, the upper surface corresponds to the case when the phase of the homodyne detector is kept constant ${\theta }_4^{\prime }=\pi /2$. The lower surface corresponds to the case when the phase is optimized for the performed transformation (for $b$ and $d$). (b) In the diagram, the upper surface corresponds to computations on the cluster state with weight coefficients $g_1=g_2=g_3=g_4=5$ and the phase ${\theta }_4^{\prime }=\pi /2$. The lower surface corresponds to computations on the cluster state with optimized weight coefficients $g_1=g_2=5$ and $g_3=g_4=4$ and optimized homodyne detector phase ${\theta }_4^{\prime }=f(b,d)$.

Figure 6

(a) Configuration of the cluster state used as a resource for computation: Gaussian nodes are shown in green and non-Gaussian nodes are shown in blue. (b) Scheme of non-Gaussian resource state preparation. (c) Scheme for implementing the arbitrary single-mode Gaussian operation on a linear weighted four-node cluster state using a cubic phase gate. (d) Scheme of implementation of arbitrary single-mode Gaussian operation on a pair of two-node cluster states using a cubic phase gate. In the diagram $Y\left(\alpha \right)$ is the operation that displaces $y$ quadrature on a real value $\alpha$ and $\widehat{Q}\left(\gamma \right)$ is the cubic phase gate with nonlinearity $\gamma$.

Figure 7

Distribution of single-mode transformation errors $\parallel \langle \delta {\widehat{\mathbf{e}}}_0^2\rangle {\parallel }_{\infty }/\langle \delta {\widehat{y}}_s^2\rangle$. The graph demonstrates three error distributions depending on the implemented operation, i.e., depending on $b$ and $d$ [see Eq. (15)]. All three distributions are calculated for optimized weight coefficients ($g_1=g_2=5$ and $g_3=g_4=4$). The orange (upper) and blue (middle) surfaces correspond to errors for the scheme without a cubic phase gate ($\gamma =0$): The orange (upper) surface corresponds to the case of a fixed phase value ${\theta }_4^{\prime }=\pi /2$ and the blue (middle) surface corresponds to the case when the angle ${\theta }_4^{\prime }$ optimization is performed (${\theta }_4^{\prime }=\tilde{f}(b,d)$). The green (lower) surface corresponds to the error of the scheme using the cubic phase gate ($\gamma \ne 0$) with ${\theta }_4^{\prime }$ being optimized.

Figure 8

Ratios of the error probabilities $P_{\text{err}}(x_{\text{err}},y_{\text{err}})/P_{\text{err}}(x_{\text{err}}^{\text{opt}},y_{\text{err}}^{\text{opt}})$, depending on $b$ and $d$ [see Eq. (15)]. Here $P_{\text{err}}(x_{\text{err}},y_{\text{err}})$ is the error probability of nonoptimized computations and $P_{\text{err}}(x_{\text{err}}^{\text{opt}},y_{\text{err}}^{\text{opt}})$ is the error probability of optimize computations. In the diagram, (a) optimization is performed by weight coefficients ($g_4=g_3=4$, $g_2=g_1=5$, and ${\theta }_4^{\prime }=\pi /2$), (b) optimization is performed by weight coefficients and the phase of the homodyne detector, and (c) full optimization is performed and the cubic phase gate is used.