Teleportation-assisted optical controlled-sign gates

Reliable entangling gates for qubits encoded in single-photon states represent a major challenge on the road to scalable quantum computing architectures based on linear optics. In this work, we present two approaches to develop high-fidelity, near-deterministic controlled-sign-shift gates based on the techniques of quantum gate teleportation. On the one hand, teleportation in a discrete-variable setting, i.e., for qubits, offers unit-fidelity operations but suffers from low success probabilities. Here, we apply recent results on advanced linear optical Bell measurements to reach a near-deterministic regime. On the other hand, in the setting of continuous variables, associated with coherent states, squeezing, and, typically, Gaussian states, teleportation can be performed in a deterministic fashion, but the finite amount of squeezing implies an inevitable deformation of a teleported single-mode state. Using a new generalized form of the nonlinear-sign-shift gate for gate teleportation, we are able to achieve fidelities of the resulting controlled-sign gate above $90\%$. A special focus is also put on a comparison of the two approaches, not only with respect to fidelity and success probability, but also in terms of resource consumption.

Figure 1

Nonlinear-sign-shift gate and linear-optics csign gate as proposed by KLM [1].

Figure 2

Gate teleportation of an unknown state $|\phi \rangle$.

Figure 3

Success probability vs number of required photons for the creation of the teleportation resource state $|\mathrm{res}\rangle$ via Bell state creation followed by csign (orange, dashed), via connecting three-qubit clusters with the standard Bell measurement (green, long dashed), or via connecting the cluster states with the advanced Bell measurement of Ref. [6] (blue, solid).

Figure 4

Success probability of the Bell measurement in Ref. [4] vs the number of required photons for the creation of the corresponding ancillary states. According to the Monte Carlo simulation (blue, solid), the rough estimate presented in the main text (orange, dashed) gives a number for the required photon sources that is too small by a factor of two (green, long dashed) for large Bell measurement success probabilities, while it is far too high for $p_{\text{BM}}<0.7$.

Figure 5

Continuous-variable teleportation. The input state $|{\psi }_{\text{in}}\rangle$ is combined with one half of a two-mode squeezed state (to which, in the case of gate teleportation, the gate $U$ is applied) at a beam splitter followed by homodyne detections. This is referred to as continuous-variable Bell measurement (highlighted area). Depending on the outcome $\beta$ of the CV Bell measurement (see the main text) a correction operation $C$ is applied. In the case of state teleportation, $C$ is a displacement $D\left(g\beta \right)$, where $g$ is the so-called gain-tuning parameter. In the case of gate teleportation, $C$ is usually chosen as $UD\left(g\beta \right)U^{-1}$.

Figure 6

Trade-off between success probability (dotted) and fidelity (dashed) when conditioning the teleportation on a CV Bell measurement result of $\left|\beta \right|\le B$. Here we assume a squeezing of $q=0.9$, corresponding to 12.8 dB, and we set $g=q$. The solid curves represent the quality $Q$ of the teleportation, as defined in the main text. From top to bottom (blue, yellow, green) the results for the Fock states $|0\rangle ,|1\rangle$, and $|2\rangle$ are shown. For superpositions of these states, the corresponding curves always lie between the two extrema given by $|0\rangle$ and $|2\rangle$. In particular, this implies that the fidelity obtained for $|2\rangle$ represents the worst-case fidelity of the teleportation, when restricting the input to the first three Fock states. In any case, the quality increases monotonically with increasing conditioning radius $B$. Thus, not conditioning at all yields the highest quality. (The value of $q$ is chosen for the sake of visual distinguishability. With a squeezing of $q=0.96$, corresponding to $\sim 17$ dB which lies in the regime of the current experimental record [22], fidelities above $90\%$ are achieved.)

Figure 7

Fidelity of the (unconditioned) quantum state teleportation of the lowest three Fock states [from top to bottom $|0\rangle$ (blue), $|1\rangle$ (yellow), and $|2\rangle$ (green)] as a function of the gain-tuning parameter $g$ (with a squeezing of $q=0.9$, as in Fig. 6, chosen for visual distinguishability). Only for the vacuum state the maximal fidelity is obtained for $g=q$. The vertical lines are placed at $g=q$ and $g=1$.

Figure 8

Optimal choice of the gain-tuning parameter $g$ for the two ${\mathrm{NSS}}_d$ gate teleportations in the csign gate as a function of the squeezing $q$. From bottom to top: $d=2,5,10,20,50$.

Figure 9

Worst-case fidelity of the csign gate as a function of the squeezing $q$. From bottom to top: $d=2,5,10,20,50,100$. Another version of this figure, i.e., Fig. 10, where the squeezing is represented in dB, can be found in Appendix pp3.

Figure 10

Worst-case fidelity of the csign gate as a function of the squeezing in dB. From bottom to top: $d=2,5,10,20,50,100$. This is another version of Fig. 9 in the main text, where the squeezing is represented by $q$.