Figure 1

Scheme for near-deterministic quantum teleportation for a hybrid qubit using linear optics and photon detection. An unknown hybrid qubit, $|\varphi \rangle =a|0_L\rangle +b|1_L\rangle$, is teleported through channel $|{\Psi }_C\rangle \propto |0_L\rangle |0_L\rangle +|1_L\rangle |1_L\rangle$. ${\mathrm{B}}_{\alpha }$ and ${\mathrm{B}}_{\mathrm{II}}$ are performed on coherent-state modes and photon modes, respectively, between the qubit and one party of the channel state. All possible outcomes and corresponding feed-forward operations are presented in the table. A failure occurs when both ${\mathrm{B}}_{\alpha }$ and ${\mathrm{B}}_{\mathrm{II}}$ fail. The failure probability is found to be $P_f=\exp (-2{\alpha }^2)/2$. In order to perform Hadamard and CZ gates, entangled states $|Z\rangle$ and $|Z^{\prime }\rangle$ should be used, respectively, instead of $|{\Psi }_C\rangle$.Reuse & Permissions

Figure 2

Schemes to prepare entangled channels. (a) A maximally entangled state, $|{\Psi }_C\rangle$, is generated using a ${\mathrm{B}}_{\mathrm{I}}$ operation out of a two-photon pair and a hybrid pair of $\sqrt{2}\alpha$. In a single-photon mode, $\pm$ and $H|V$ denote bases $\left\{\right|+\rangle ,|-\rangle \}$ and $\left\{\right|H\rangle$,$|V\rangle \}$, respectively, which can be modified by a polarization rotation before performing the ${\mathrm{B}}_{\mathrm{I}}$ operation. A 50:50 beam splitter, used to split the coherent-state part with amplitude $\sqrt{2}\alpha$ into two modes, is omitted in the figure. (b) Two schemes, ${\mathrm{G}}_{\mathrm{I}}$ and ${\mathrm{G}}_{\alpha }$, to generate $|Z\rangle$. In ${\mathrm{G}}_{\mathrm{I}}$, two ${\mathrm{B}}_{\mathrm{I}}$ operations are performed on one two-photon pair and two hybrid pairs so as to link them. The other method, ${\mathrm{G}}_{\alpha }$, requires three hybrid pairs with ${\mathrm{B}}_{\mathrm{I}}$ and ${\mathrm{B}}_{\alpha }$ operations. One of those hybrid pairs has amplitude $\sqrt{2}\alpha$ to obtain a three-mode state $|+\rangle |\alpha \rangle |\alpha \rangle +|-\rangle |-\alpha \rangle |-\alpha \rangle$ by a 50:50 beam splitter (omitted in the figure). Appropriate feedforwards with Pauli operations are necessary for all ${\mathrm{B}}_{\mathrm{I}}$ and ${\mathrm{B}}_{\alpha }$ operations dependent on the measurement outcome.Reuse & Permissions

Figure 3

Noise thresholds and resource requirements. (a) Noise thresholds based on two generation schemes, ${\mathrm{G}}_{\mathrm{I}}$ and ${\mathrm{G}}_{\alpha }$, obtained using the seven-qubit steane code [33]. (b) Resource requirements estimated for one round of error correction based on the telecorrection protocol. For ${\mathrm{G}}_{\mathrm{I}}$, a constant number (28 780) of resources (two-photon and hybrid pairs) are required irrespective of amplitude $\alpha$, while for ${\mathrm{G}}_{\alpha }$ the required number of resource states tend to decrease rapidly to 4182 with increasing $\alpha$. Two curves intersect at $\alpha \approx 0.59$. We can take the lower curve between them by choosing ${\mathrm{G}}_{\alpha }$ for $\alpha >0.59$ and otherwise ${\mathrm{G}}_{\mathrm{I}}$. It shows a remarkable improvement compared to pLOQC (about $1.8\times {10}^5$ [15]) and CSQC (about ${10}^4$ [12]). (c) Ralph-Pryde diagram [4] for the comparison with pLOQC and CSQC. The hybrid approach using ${\mathrm{G}}_{\alpha }$ presented in this paper is shown to outperform pLOQC and CSQC.Reuse & Permissions

Figure 4

Three Bell-type measurement elements used for our scheme. A coherent-state Bell measurement, ${\mathrm{B}}_{\alpha }$, is implemented by a 50:50 BS and two photon number parity detectors (PNPD) [8]. ${\mathrm{B}}_{\alpha }$ unambiguously discriminates between all four Bell states and the success probability is $1-\exp (-2|\alpha {|}^2)$. It fails only when no photon is detected at both the detectors. A type I fusion operation [17], ${\mathrm{B}}_{\mathrm{I}}$, is implemented by polarizing beam splitters (PBSs), wave plates, and photon detectors. It effectively performs $|+\rangle \langle H|\langle H|\pm |-\rangle \langle V|\langle V|$ with a success probability $1/2$ when only one photon is detected at either detector [17]. A nondeterministic Bell measurement or modified version of the typeII fusion operation, ${\mathrm{B}}_{\mathrm{II}}$, identifies only two of the Bell states, $|H\rangle |V\rangle \pm |V\rangle |H\rangle$, with success probability $1/2$. It succeeds only when one detector from the upper two and another from the lower two click at the same time.Reuse & Permissions

Figure 5

Two schemes, ${\mathrm{G}}_{\mathrm{I}}$ and ${\mathrm{G}}_{\alpha }$, for generating $|Z\rangle$ and $|Z^{\prime }\rangle$ states. In a single-photon mode, $\pm$ and $H|V$ denote bases $\left\{\right|+\rangle ,|-\rangle \}$ and $\left\{\right|H\rangle$,$|V\rangle \}$, respectively, which can be modified by a polarization rotation. A Pauli $Z$ operation (omitted in figure) is performed on the output qubit of ${\mathrm{B}}_{\mathrm{I}}$ when the measurement outcome of ${\mathrm{B}}_{\mathrm{I}}$ is ($H$). Likewise, for ${\mathrm{B}}_{\alpha }$ appropriate Pauli operations are performed on the remaining qubit (denoted by dotted circle) according to the measurement results as shown in Table .Reuse & Permissions