Correcting finite squeezing errors in continuous-variable cluster states

We introduce an efficient scheme to correct errors due to the finite squeezing effects in continuous-variable cluster states. Specifically, we consider the typical situation where the class of algorithms consists of input states that are known. By using the knowledge of the input states, we correct the errors and significantly improve the fidelity in the context of continuous-variable cluster states. We illustrate the error correction scheme for single- and two-mode unitaries implemented by spatial continuous-variable cluster states. We show that there is no resource advantage to error correcting multimode unitaries implemented by spatial cluster states. However, the generalization to multimode unitaries implemented by temporal continuous-variable cluster states shows significant practical advantages since it costs only a small fixed number of optical elements (squeezer, beam splitter, etc.) for an arbitrary number of modes.

Figure 1

Measurement-based single-mode unitary implemented via a spatial two-mode CV cluster state. To implement a single-mode unitary, the input mode (mode $a$, with input state ${|\psi \rangle }_a$) is coupled to one (mode $c$) of the two modes of the CV cluster state via a beam splitter $B(\pi /4)$. Two output modes are detected by homodyne detectors with measurement angles ${\theta }_1$ and ${\theta }_3$, respectively. After the homodyne measurements, one has to displace the state in mode $c^{\prime }$ (not shown in the figure) according to the measurement outcomes $m_1$ and $m_3$.

Figure 2

Amount of squeezing required in correcting single-mode errors versus the input state squeezing for a spatial CV cluster state with squeezing 4 dB (green dotted line), 6 dB (orange dashed line), and 10 dB (blue solid line).

Figure 3

Measurement-based two-mode unitary implemented via a spatial four-mode CV cluster state. Two input modes $a$ and $b$ with input states ${|\psi \rangle }_a$ and ${|\psi \rangle }_b$ pass through a beam splitter $B(\pi /4)$, the output states of which act as the input states of two measurement-based single-mode unitary circuits (two green shaded boxes). The top one is exactly the same as that shown in Fig. 1 and implements a unitary ${\widehat{V}}_a({\theta }_1,{\theta }_3)$; the bottom one is similar but with measurement quadratures ${\widehat{b}}_{{\theta }_2}$ and ${\widehat{b}}_{{\theta }_4}$, and implements a unitary ${\widehat{V}}_b({\theta }_2,{\theta }_4)$. The two output modes of these two measurement-based single-mode unitary circuits pass through another beam splitter $B(-\pi /4)$, giving the final outputs. The implemented two-mode unitary is completely determined by the measurement angles ${\theta }_i$ ($i=1,\cdots ,4$). Note that displacements depending on measurement outcomes $m_i$ have to be applied, which are not shown in the figure.

Figure 4

Amount of squeezing required in correcting two-mode errors versus the input state squeezing for a spatial CV cluster state with squeezing 4 dB (green dotted line), 6 dB (orange dashed line), and 10 dB (blue solid line).

Figure 5

Fidelity between the target state ${\sigma }_t$ and the actual output state $\sigma$ (orange dashed line), and the error correcting state ${\sigma }_{\mathrm{ec}}$ (blue solid line). $cos\theta$ is the transmission coefficient of the beam splitter which is used to generate the target state. The minimum variances of the cluster state and the target state are chosen to be the same: $\epsilon =s^2=0.5$ ($\sim 3$ dB).

Figure 6

Comparison of fidelities with (blue solid line) and without (black dashed line) error correction. The minimum variance of the target state is chosen to be $s^2=0.5$ ($\sim 3$ dB) and the minimum variance of the cluster state goes down from 10 dB ($\epsilon =0.1$) to 3 dB ($\epsilon =0.5$).

Figure 7

The error correction circuit (blue shaded box) for multimode Gaussian unitaries implemented via a temporal CV cluster state consists of two beam splitters, $B({\theta }_1,{\varphi }_1)$ and $B({\theta }_2,{\varphi }_2)$, and two single-mode squeezers, $S(r_1,{\phi }_1)$ and $S(r_2,{\phi }_2)$ (note that we have absorbed the phase shifts into the beam splitters and squeezers to simplify the circuit). The parameters of these beam splitters and squeezers are adjusted to correct errors imposed on implemented two-mode unitaries. The remaining part of the optical setup generates two-dimensional CV cluster states [29] and implements universal quantum computation with addition of non-Gaussian gates [38], e.g., cubic-phase gates. $B_i(i=1,\cdots ,6)$ are six 50:50 beam splitters, $D_i(i=1,2,3,4)$ are four homodyne detectors. Four single-mode squeezed pulses (leftmost) are generated continuously with time interval $\mathrm{\Delta }t$ and are injected into the circuit. A pair of two-mode squeezed states are produced after the beam splitters $B_1$ and $B_2$. One pulse of the top two-mode squeezed state is delayed by $\mathrm{\Delta }t$, while one pulse of the bottom two-mode squeezed state is delayed by $M\mathrm{\Delta }t$ (we assume the time passing through the error correction circuit is much shorter than $\mathrm{\Delta }t$), where $M$ is an integer and can be thought of as the depth of the circuit. If the error correction circuit is absent and no homodyne detection is performed, a temporal two-dimensional cluster state is generated after the beam splitters $B_3, B_4, B_5$, and $B_6$ [29]. To do quantum computation, two switches (green boxes) have to be introduced to couple the input states into the cluster state or to readout the output states after the computation [39]. At the beginning, the switches are turned on to let the input states couple into the cluster state and then turned off. Homodyne measurements are performed to implement a two-mode unitary each time, depending on the measurement quadratures of the homodyne detection. Error correction follows the homodyne measurements to correct the error induced by the finite squeezing in the input squeezed pulses (note that displacements are also required, which are not shown in the figure). After all computations are done, the switches are turned on such that one can readout or detect the output states.