Resource-Efficient Topological Fault-Tolerant Quantum Computation with Hybrid Entanglement of Light

We propose an all-linear-optical scheme to ballistically generate a cluster state for measurement-based topological fault-tolerant quantum computation using hybrid photonic qubits entangled in a continuous-discrete domain. Availability of near-deterministic Bell-state measurements on hybrid qubits is exploited for this purpose. In the presence of photon losses, we show that our scheme leads to a significant enhancement in both tolerable photon-loss rate and resource overheads. More specifically, we report a photon-loss threshold of $\sim 3.3\times {10}^{-3}$, which is higher than those of known optical schemes under a reasonable error model. Furthermore, resource overheads to achieve logical error rate of ${10}^{-6}({10}^{-15})$ is estimated to be $\sim 8.5\times {10}^5(1.7\times {10}^7)$, which is significantly less by multiple orders of magnitude compared to other reported values in the literature.

Figure 1

(a) $B_{\alpha }$ acts on CV modes and fails when neither of the two PNPDs click. The failure rate of a $B_{\alpha }$ on the hybrid qubits is $e^{-2{\alpha }^2}$. $B_s$ acts on DV modes and is successful with probability $1/2$ only when both the PDs click. (b) The three-hybrid-qubit cluster with one unfilled circle represents $|{\mathcal{C}}_3⟩$, while that with two represents $|{\mathcal{C}}_{3^{\prime }}⟩$ in Eq. (1). An unfilled circle means a difference by a Hadamard transform from the original three-qubit cluster (see the Supplemental Material [33]). Success of both HBSMs creates a star cluster $|{\mathcal{C}}_{*}⟩$ and other cases lead to distorted star clusters as shown.

Figure 2

(a) When connecting $|{\mathcal{C}}_{*}⟩$s, a successful HBSM creates an edge between hybrid qubits whereas a failed HBSM leaves the edge missing. (b) 3D illustration of building two layers of $|{\mathcal{C}}_{\mathcal{L}}⟩$ for practical HTQC with $|{\mathcal{C}}_{*}⟩\mathrm{s}$ and HBSMs to connect them. (c) A diagonal edge is created due to failure of an HBSM corresponding to $|{\mathcal{C}}_{*}⟩$, and a missing edge is due to failure of an HBSM while connecting them. A single layer of $|{\mathcal{C}}_{\mathcal{L}}⟩$ is shown for convenience, and $M_z$ is measurement on a $Z$ basis.

Figure 3

(a) Logical error rate $p_L$ is plotted against the dephasing rate $p_Z$ for coherent-state amplitude ${\alpha }_{\mathrm{opt}}=1.247$ and code distances $d=3$, 5, 7. The intersecting point of these curves corresponds to the threshold dephasing rate $p_{Z,\mathrm{th}}$. (b) The tolerable photon-loss rate ${\eta }_{\mathrm{th}}$ is plotted against coherent-state amplitude $\alpha$. The behavior of the curve is due to the trade-off between the success rate of HBSM and dephasing rate $p_Z$ with growing $\alpha$. As we increase $\alpha$, both the success rate and $p_Z$ increase; but the former dominates and leads to an increase in ${\eta }_{\mathrm{th}}$. When $\alpha >1.247$, $p_Z$ dominates and causes ${\eta }_{\mathrm{th}}$ to decrease. Compared to the nontopological HQQC [25], HTQC has an order of higher value for ${\eta }_{\mathrm{th}}$.

Figure 4

(a) Optimal photon-loss threshold ${\eta }_{\mathrm{th}}$ for various QC schemes. It should be noted that ${\eta }_{\mathrm{th}}\mathrm{s}$ of OCQC, PLOQC, EDQC, and TPQC (dashed borders) are valid only for zero computational error, which is physically unachievable. Other schemes evaluate optimal ${\eta }_{\mathrm{th}}$ at nonzero computational errors naturally related to $\eta$. (b) Resource overhead $N$ to achieve logical error rate $p_L\sim {10}^{-6}(\text{blue shorter bars})$ and $p_L\sim {10}^{-15}(\text{orange taller bars})$ in terms of the average numbers of hybrid qubits (HTQC), entangled photon pairs (OCQC and EDQC), coherent-state superpositions (CSQC) from our analysis and published data in [13, 23, 36]. For CSQC data only for $p_L\sim {10}^{-6}$ is available [23]. Obviously, HTQC is practically favorable for large scale QC both in terms of ${\eta }_{\mathrm{th}}$ and $N$. See the Supplemental Material [33] for more details of comparisons.