Fault-tolerant quantum computation by hybrid qubits
with bosonic cat-code and single photons

We define a hybrid qubit by employing single photon (DV qubit) and cat-code encoded state (CV qubit), which we call the hybrid cat-code (H-cat) qubit. The cat-code is a bosonic QEC code designed to protect qubits against photon loss, in which qubits are encoded in the subspace of even photon number states so that a photon loss can be detected when the parity changes Leghtas2013 ; Mirrahimi2014 . Specifically, the code space is defined by even cat states $\left\{|\mathcal{C}_{\alpha}^{+}\rangle,|\mathcal{C}_{i\alpha}^{+}\rangle\right\}$, where $|\mathcal{C}_{\alpha}^{+}\rangle=\mathcal{N}_{\mathcal{C}}^{+}(|\alpha\rangle+|-\alpha\rangle)$ with the normalization factor $\mathcal{N}_{\mathcal{C}}^{+}\equiv 1/\sqrt{2(1+e^{-2\alpha^{2}})}$ and $|\alpha\rangle$ is a coherent state with amplitude $\alpha$. By combining DV and CV qubits, the basis of H-cat qubit can then be defined as

where $|\pm\rangle=(|H\rangle\pm|V\rangle)/\sqrt{2}$ is the polarization of single photon. Note that the logical basis are orthogonal to each other thanks to the DV part in contrast to the basis $\{|\mathcal{C}_{\alpha}^{+}\rangle,|\mathcal{C}_{i\alpha}^{+}\rangle\}$ in bosonic quantum computation which overlaps each other with finite $\alpha$ and causes inherent errors. We also consider another type of hybrid qubits LeeSW2013 for comparison, composed of single photon and coherent state in the basis $\{|+\rangle|\alpha\rangle,|-\rangle|-\alpha\rangle\}$, which we call here the hybrid coherent-state (H-coh) qubit. The hybrid qubits, i.e., H-cat and H-coh, are illustrated in Fig. 1(a).

Note that hybrid qubits can be generated in current photonic platforms. CV-DV hybrid entangled states have been experimentally demonstrated in optical systems Jeong2014 ; Morin2014 and successfully applied to quantum computing and communications in numerous experiments Huang2019 ; Guccione2020 ; Darras2023 . In those experiments, the CV basis is defined by coherent states i.e. $|\alpha\rangle$ and $|-\alpha\rangle$, constituting two-component cat states in CV part, so that the generated hybrid entangled states can be directly used as the H-coh qubit by simple modifications. We can also generate the H-cat qubit by using H-coh qubits by extending two-component cat states into four-component using $|\alpha\rangle$, $|i\alpha\rangle$, $|-\alpha\rangle$, and $|-i\alpha\rangle$ in CV part, e.g., using two H-coh qubits, a beam splitter and a photon number resolving detector as proposed in Ref. Hastrup2020 . We present the scheme for H-cat qubit generation in the Methods.

We stress that such CV-DV hybrid entanglement can be generated efficiently also in other platforms including superconducting and trapped-ion systems, which enables that our approach can be more generally implemented in any CV-DV hybrid platforms for quantum computing. We will discuss these further in the Discussion and the Methods.

We now introduce a hybrid fusion, i.e., a CV-DV hybrid entangling operation, which is performed by a joint work of Bell-state measurements on CV and DV qubits. A fusion measurement is typically applied on entangled states to generate larger size entangled states such as cluster states as prerequisites for measurement-based quantum computation (MBQC) Raussendorf2001 ; Menicucci2006 ; Larsen2021b , or also enables universal gate operations via teleportation in circuit-based quantum computation Gottesman1999 .

A hybrid fusion projects a two hybrid qubits on one of the hybrid Bell states $|\Psi_{L}^{\pm}\rangle=(|0_{L}\rangle|1_{L}\rangle\pm|1_{L}\rangle|0_{L}\rangle)/\sqrt{2}$, $|\Phi_{L}^{\pm}\rangle=(|0_{L}\rangle|0_{L}\rangle\pm|1_{L}\rangle|1_{L}\rangle)/\sqrt{2}$. The logical Bell states of hybrid qubits given in the basis of Eq. (1) can be rewritten by separating the CV and DV parts as (details are in the Appendix A.1)

where $|\Psi_{\mathcal{D}}^{\pm}\rangle=(|H\rangle|V\rangle\pm|V\rangle|H\rangle)/\sqrt{2}$, $|\Phi_{\mathcal{D}}^{\pm}\rangle=(|H\rangle|H\rangle\pm|V\rangle|V\rangle)/\sqrt{2}$ are the Bell states of DV qubits, and $|\tilde{\Psi}_{\mathcal{C}}^{\pm}\rangle=|\mathcal{C}_{\alpha}^{+}\rangle|\mathcal{C}_{i\alpha}^{+}\rangle\pm|\mathcal{C}_{i\alpha}^{+}\rangle|\mathcal{C}_{\alpha}^{+}\rangle$, $|\tilde{\Phi}_{\mathcal{C}}^{\pm}\rangle=|\mathcal{C}_{\alpha}^{+}\rangle|\mathcal{C}_{\alpha}^{+}\rangle\pm|\mathcal{C}_{i\alpha}^{+}\rangle|\mathcal{C}_{i\alpha}^{+}\rangle$ are unnormalized Bell states of CV qubits.

Therefore, the hybrid fusion can be implemented by applying CV and DV Bell-state measurements separately, denoted here as $\mathrm{B}_{\mathcal{C}}$ and $\mathrm{B}_{\mathcal{D}}$, respectively. Its logical outcome can then be discriminated by combining the results of $\mathrm{B}_{\mathcal{C}}$ and $\mathrm{B}_{\mathcal{D}}$ based on Eq. (2). $\mathrm{B}_{\mathcal{C}}$ can be implemented by linear optics and photon-number-resolving (PNR) detectors as illustrated in Fig. 1(b). Two schemes were recently proposed independently in Ref. Hastrup2022 and Ref. Su2022 , which we respectively refer to HA and SDR scheme. See Appendix A.2 for details. Note that $\mathrm{B}_{\mathcal{C}}$ cannot distinguish the logical letter $\Psi/\Phi$ with certainty even in an ideal case without loss, while the logical sign $+/-$ can be discriminated by counting the total number of photons detected. Ambiguity thus remains in discriminating between $|\Psi_{\mathcal{C}}^{\pm}\rangle$ and $|\Phi_{\mathcal{C}}^{\pm}\rangle$.

In our hybrid scheme, we can remove such an ambiguity by $\mathrm{B}_{\mathcal{D}}$. In the photon polarization encoding, $\mathrm{B}_{\mathcal{D}}$ can be chosen as a so called type II fusion Browne2005 that distinguishes two Bell states $|\Psi_{\mathcal{D}}^{\pm}\rangle$ out of four with linear optics Lut99 ; Calsamiglia2001 (see Appendix A.3). Once $\mathrm{B}_{\mathcal{D}}$ succeeds, it removes the ambiguity between the letter $\Psi/\Phi$. Remarkably, a hybrid fusion is thus able to distinguish hybrid Bell states with certainty even if only one of $\mathrm{B}_{\mathcal{C}}$ and $\mathrm{B}_{\mathcal{D}}$ succeeds. Otherwise, i.e., in the case that $\mathrm{B}_{\mathcal{C}}$ yields ambiguity and $\mathrm{B}_{\mathcal{D}}$ fails, the hybrid fusion induces an $X$ error with the rate $P_{X}$.

Moreover, in the presence of photon loss, $\mathrm{B}_{\mathcal{C}}$ can detect a single photon loss in CV part through detecting the parity change of the cat state to odd, i.e., when a total odd number of photons is detected. This is thanks to the bosonic cat-code error correction in the CV part. The hybrid fusion thus can yield the outcome successfully that contains $X$ error with a slightly changed rate from $P_{X}$ (see Appendix A.2). If two or more photons are lost, $Z$ errors remain undetected in the fusion measurement outcome, as we analyze in the Methods. The explicit expression of the $X$ error rate $P_{X}$ and the $Z$ error rate $P_{Z}$ are derived in Appendix C.

In Fig. 1(c), we present the $X$ and $Z$ error rates under loss for the hybrid fusions. Specifically, $P_{X}$ and $P_{Z}$ are plotted for the fusions of H-cat qubits with HA and SDR, and H-coh qubit used in Ref. LeeSW2013 ; Omkar2020 ; Omkar2021 by varying the amplitude $\alpha$ under a fixed loss rate $\eta=2\times 10^{-3}$. It shows that the effect of loss can be substantially suppressed in the hybrid fusion of H-cat compared to H-coh thanks to the bosonic cat-code error correction in the CV part. As common tendencies, $P_{X}$ error can be exponentially suppressed as $\alpha$ grows because the basis of CV part are more distinguishable, while $P_{Z}$ only increases linearly with $|\alpha|^{2}$ because its state becomes more fragile against photon loss. The result clearly shows that $P_{Z}$ is suppressed by employing the cat-code error correction. By taking larger encoding amplitude $\alpha$, we can always achieve much smaller $P_{X}$ and $P_{Z}$ in the fusions of H-cat qubits than H-coh qubits. By comparing two different types of the fusion schemes of H-cat qubits, i.e. HA and SDR, HA scheme exhibits lower error rates, whereas we show that the SDR scheme is advantageous when an unambiguous discrimination is required without error (see Appendix A.2).

We investigate the fault-tolerance of quantum computation with H-cat and H-coh qubits and compare the result with previous works. We here focus on analyzing and simulating the fault-tolerance of MBQC, which is suitable for photonic platforms. In MBQC, universal quantum computation requires to prepare an appropriate structure of cluster states. We construct the Raussendorf-Harrington-Goyal (RHG) lattice Raussendorf2006 ; Raussendorf2007 based on our hybrid qubits, which can embed the surface code and enables universal quantum computation. For the circuit-based model, we can employ the telecorrection Knill2005 and Steane code in simulation equivalently with the works using H-coh and CV qubits Lund2008 ; LeeSW2013 for the direct comparison, details of which are in Appendix D.

We depict the sturcture of a RHG lattice in Fig. 2(a) and describe the procedure for its construction in the Methods. In the RHG lattice, errors that occur in the hybrid fusion propagate to adjacent qubits (See Appendix C.3), so that a corresponding error rate is assigned to each individual qubit on the lattice. Based on assigned error rates, we can find the error pattern matching the syndrome measurement using the weighted minimum-weight perfect matching Fowler2012 in the PyMatching package Higgott2021 , and then count remaining error chains connecting two primal boundaries and determine whether a logical error occurs. We perform a Monte Carlo simulation to find the logical error rate $p_{L}$ for different code distances $d$, and then investigate whether the errors are accumulated, i.e., $p_{L}$ increases or not as increasing $d$. The fault-tolerance noise thresholds can then be determined as the maximum physical error rates by which the logical errors are not accumulated with $d$. See Appendix D for details.

We present the loss thresholds of photonic MBQC based on H-cat and H-coh qubits in Fig. 2(b) by changing the encoding amplitude $\alpha$ in CV part. Note that the previous estimation of H-coh in Ref. Omkar2020 employed a slightly different error model, and thus we resimulate the result under the same error model assigned to H-cat for fair comparison. We also depict the loss thresholds of the circuit-based model with hybrid and CV qubits estimated based on Steane codes for comparison.

It shows that hybrid MBQC with H-cat qubits achieves the highest loss thresholds over other approaches including MBQC with H-coh qubits as well as all hybrid and CV approach in circuit-based model. The loss threshold of quantum computing with H-cat 0.89% is about 4-times larger than the maximum 0.22% estimated with H-coh Omkar2020 . The hybrid fusion with HA scheme yields higher threshold than SDR scheme as the former exhibits lower error rates than the latter. The optimized encoding amplitude $\alpha$ of each hybrid fusions are marked as stars both in Fig. 1(c) and Fig. 2(b). The optimal encoding amplitude is $\alpha\approx 2.93$ for H-cat with HA scheme while it is 1.37 for H-coh qubits. As exhibited in Fig. 1(c), such an enhancement of loss thresholds with H-cat qubits is due to that $P_{X}$ can be significantly reduced further while suppressing $P_{Z}$ by enlarging the encoding amplitude $\alpha$ in CV part thanks to the inherent bosonic error correction in contrast to the case with H-coh qubits.

To investigate the resource overhead, we estimate the number of unit resources referred to as $\mathcal{N}_{p_{L}}$ to achieves the target logical error rate $p_{L}$. We choose H-coh pair as a unit resource for fair comparison with the resource estimation in previous works LeeSW2013 ; Omkar2020 . Note that H-cat pair can be created by consuming 8 H-coh pairs on average as we describe in the Methods section. In the Methods, we propose a scheme to generate H-cat pair consuming 8 H-coh pairs, and thus we here assume that an H-cat pair is 8 times as expensive as an H-coh pair.

We find that a target logical error rate can be achieved even with very small code distance $d$ with H-cat qubits. For example, to achieve $p_{L}=10^{-6}$, only $d=4$ is sufficient with H-cat qubits due to the suppressed errors by the bosonic cat-code in CV part of the physical level, while $d=15$ is required with H-coh qubits Omkar2020 . As a result, we can estimate that the resource overhead for the hybrid MBQC with H-cat qubits is $\mathcal{N}_{10^{-6}}=2.7\times 10^{4}$, while the overhead with H-coh qubits is $\mathcal{N}_{10^{-6}}=3.6\times 10^{5}$. Remarkably, employing H-cat qubits can reduce an order of magnitude resource cost compared to the approach with H-coh qubits.

In Fig. 2(c), we present the resource cost estimation of the proposed hybrid quantum computing schemes with H-cat and H-coh qubits. We also compare the resource efficiencies of proposed schemes with existing MBQC photonic schemes based on direct encoding Herrera2010 , repetition code Omkar2022 , and parity encoding of multiple photons LeeSH2023 as well as circuit-based photonic schemes Dawson2006 ; Lund2008 ; Cho2007 ; Hayes2010 ; LeeSW2015 . For resource analysis of other schemes, we refer to Refs. Omkar2021 ; Omkar2022 . It shows that our hybrid approach is at least an order of magnitude more resource-efficient compared to all the other photonic proposals with respect to the cost of the resources, while the direct comparison is not straightforward due to the different types of resource states. The improved resource efficiency is achieved in our scheme thanks to the deterministic nature of the proposed hybrid fusion and the loss-tolerance of the resource states by inherently encoded bosonic cat-code.

In the hybrid fusion, an $X$ error occurs when that $\mathrm{B}_{\mathcal{C}}$ yields ambiguity and $\mathrm{B}_{\mathcal{D}}$ fails. The resulting state after the fusion can be represented by $\mu|\Phi_{L}^{\pm}\rangle+\nu|\Psi_{L}^{\pm}\rangle$. So, when $|\mu|\geq|\nu|$ we can take $|\Phi_{L}^{\pm}\rangle$ as the outcome and assign $X$ error with rate $P_{X}=|\nu|^{2}/(|\mu|^{2}+|\nu|^{2})$, and vice versa. We calculate the $X$ error rate of hybrid fusion employing HA or SDR scheme as described in Appendix A.2.

If more than one photons are lost, errors can be modeled by applying higher power of $\hat{a}$. In CV part, the cat-code exhibits a cyclic behavior Bergmann2016 , that is, the state after $4k+l$ photon loss has the same form as the state after $l$ photon loss, where $k$ is a nonnegative integer and $l=0,1,2,3$. Since each photon loss induces a parity change, the CV qubit is in even(odd) space when $l$ is even(odd). $Z$ error may occur by multiple photon loss as $\mathrm{B}_{\mathcal{C}}$ only corrects single photon loss detected by the parity change. If two photons are lost ($l=2$) at one input port, a $Z$ error occurs as $\hat{a}^{2}(a|\mathcal{C}_{\alpha}^{+}\rangle+b|\mathcal{C}_{i\alpha}^{+}\rangle)=(a|\mathcal{C}_{\alpha}^{+}\rangle-b|\mathcal{C}_{i\alpha}^{+}\rangle)$ and it remains undetected because there is no parity change. When one photon is lost from each CV input port, the total photon number is even, but the parity change may be distinguishable by some measurement patterns (see Appendix A.2), which causes no error. If the parity change is not distinguishable, a $Z$ error remains undetected. If $l=3$, only single photon loss is corrected and thus a $Z$ error remains uncorrected as in the case of $l=2$. For the case of higher photon loss, we can make a similar error analysis, and we summarize it in Appendix C. On the other hand, if any of DV photon is lost, we lose the phase information, which causes a dephasing error. DV photon loss is locatable by counting the total number of photons detected in $\mathrm{B}_{\mathcal{D}}$. To sum up, we denote $P_{Z}$ to be the overall $Z$ error rate induced by photon loss.

Let us now design quantum computing architectures based on the hybrid qubits. We can consider both the circuit-based and MBQC models. In the circuit-based model, the gate operations for universal quantum computation can be chosen as the gate set $\{\hat{X},\hat{Z}_{\theta},\hat{H},\hat{CZ}\}$. The Pauli $X$ operation ($\hat{X}$) and arbitrary rotation along $Z$ axis ($\hat{Z}_{\theta}$) are implementable only using linear optical elements; $\hat{X}$ can be implemented by applying bit flip operations in both CV and DV parts, using a polarization rotator acting as $|+\rangle\leftrightarrow|-\rangle$ and a $\pi/2$ phase shifter acting as $|\mathcal{C}_{\alpha}^{+}\rangle\leftrightarrow|\mathcal{C}_{i\alpha}^{+}\rangle$. $\hat{Z}_{\theta}$ can be implemented by a $\theta$ phase shifter applied only as $|-\rangle\to e^{i\theta}|-\rangle$ on the DV part. The Hadamard ($H$) and Controlled-$Z$ ($CZ$) gates can be performed based on the gate teleportation technique Gottesman1999 , using the hybrid fusion scheme and entangled resource states. The resource states as the channel of teleportation for $H$ and $CZ$ are given respectively by $|\Phi_{H}\rangle\propto|0_{L},0_{L}\rangle+|0_{L},1_{L}\rangle+|1_{L},0_{L}\rangle-|1_{L},1_{L}\rangle$ and $|\Phi_{CZ}\rangle\propto|0_{L},0_{L},0_{L},0_{L}\rangle+|0_{L},0_{L},1_{L},1_{L}\rangle+|1_{L},1_{L},0_{L},0_{L}\rangle-|1_{L},1_{L},1_{L},1_{L}\rangle$. We present the scheme for resource state generation in the next section. The measurement on the $Z$ basis can be performed by the measurement in DV part using polarization measurement. See Appendix B for the details of the schemes of hybrid teleportation and measurement.

Constructing a fault-tolerant architecture in our hybrid approach requires to concatenate the hybrid qubits incorporating the bosonic cat-code and an outer DV quantum error correction code such as CSS Steane1996 and topological codes Raussendorf2006 ; Raussendorf2007 ; Fowler2009 . For a direct comparison with previous works Dawson2006 ; Herrera2010 ; Lund2008 ; LeeSW2013 ; Omkar2020 , we construct a circuit-based fault-tolerant model using the 7-qubit Steane code (see Appendix D), while we employ the surface code for MBQC as described below.

In MBQC, we construct a RHG lattice embedding the surface code. We first prepare 3-qubit micro cluster states by using hybrid qubits as element resources, as we describe in the next section, which we refer to as 3-qubit micro H-cluster states. We then merge such micro H-cluster states using hybrid fusion to construct a RHG lattice, as depicted in Fig. 2(a): In step 1, three micro H-cluster states are merged by two hybrid fusions to form a 5-qubit star H-cluster state with one central qubit and four side qubits. In step 2, we pile star H-cluster states in the form of RHG lattice, where the central hybrid qubits become vertices of cluster states and the side qubits are consumed by hybrid fusion to create edges between adjacent vertices. In the RHG lattice, each $xy$-plane is called a layer and $t$-axis represents the simulated time. The size of RHG lattice is characterized by the code distance $d$. A layer is primal (dual) if it is located at even (odd) $t$ and primal (dual) qubits reside on faces (edges) of the primal lattice. The stabilizer of a primal unit cell is given by the product of six Pauli $X$’s on primal qubits located on the faces. The error patterns matching the stabilizer measurement can be corrected and the remaining error chain connecting two primal boundaries causes a logical error. Note that we here only simulate $Z$ errors in primal lattices, because error corrections are done separately on the primal and dual lattice and qubits are measured in $X$ basis for stabilizer measurements.

We introduce a H-coh pair $(|+\rangle|\alpha\rangle+|-\rangle|-\alpha\rangle)/\sqrt{2}$ as a unit resource, which is employed as a logical qubit in previous hybrid approaches LeeSW2013 ; Omkar2020 . It can be prepared by the scheme of Refs. Morin2014 ; Guccione2020 ; Darras2023 , where DV-CV entanglement is heralded by the detection of single photon after a weak beamsplitter interaction between CV and DV modes. Since this scheme employs the CV basis using two coherent states $|\alpha\rangle$ and $|-\alpha\rangle$, we have to extend it to four-component cat code using $|\alpha\rangle$, $|i\alpha\rangle$, $|-\alpha\rangle$, and $|-i\alpha\rangle$. The generation of four-component cat state was proposed in Ref. Hastrup2020 , where a pair of two-component cat state in different phases are mixed by a beamsplitter interaction followed by photon number counting on one of the output modes. Here we propose the generation of a H-cat pair where the scheme is described in Fig. 3(a). We begin with two H-coh pairs, $(|+\rangle|\omega\alpha\rangle+|-\rangle|-\omega\alpha\rangle)\otimes(|+\rangle|-i\omega\alpha\rangle+|-\rangle|i\omega\alpha\rangle)$ where $\omega=e^{i\pi/4}$. For convenience, we omit the normalization factors. DV qubits are merged by type I fusion operation $\mathrm{B}_{\mathrm{I}}$ and coherent-state qubits are mixed by a 50/50 beamsplitter, resulting in the state $|H\rangle\left(|\alpha\rangle|i\alpha\rangle+|-\alpha\rangle|-i\alpha\rangle\right)+|V\rangle\left(|i\alpha\rangle|\alpha\rangle+|-i\alpha\rangle|-\alpha\rangle\right)$. We count the photon number on one of the CV modes and postselect even photon detection to rule out odd cat states. Then we have the state $i^{n}|H\rangle|\mathcal{C}_{\alpha}^{+}\rangle+|V\rangle|\mathcal{C}_{i\alpha}^{+}\rangle$ and the phase $i^{n}$ can be compensated by $Z$ gate when $n=4k+2$. Note the the DV basis can be freely interchanged between $H/V$ and $+/-$ using a waveplate. The success probability of $\mathrm{B}_{\mathrm{I}}$ is 50% and the probability to detect even photons are approximately 50% when $\alpha\gg 1$. Therefore, we need approximately 8 H-coh pairs on average to generate a H-cat pair. Note that any imperfections and loss during the process cause to cost few more H-cat qubits on average. If we also take into account such additional errors during the generation process of H-cat qubits, the loss thresholds can be slightly degraded in Fig. 2(b).

To generate cluster states of hybrid qubits, we employ ancilla qubits which will be merged by fusion operation. We here choose coherent-state qubits as ancilla, and therefore we prepare DV-cat-coherent triples $|\psi_{T}\rangle=|+\rangle|\mathcal{C}_{\alpha}^{+}\rangle|\beta\rangle+|-\rangle|\mathcal{C}_{i\alpha}^{+}\rangle|-\beta\rangle$ using the scheme shown in Fig. 3(b). The amplitude of coherent-state qubit $\beta$ does not need to be the same as the amplitude of cat-code qubit $\alpha$, but $\beta\gtrsim 1$ is required to suppress the failure probability of fusion operation. We merge a H-cat pair $|H\rangle|\mathcal{C}_{\alpha}^{+}\rangle+|V\rangle|\mathcal{C}_{i\alpha}^{+}\rangle$ with a H-coh pair $|H\rangle|\beta\rangle+|V\rangle|-\beta\rangle$ using $\mathrm{B}_{\mathrm{I}}$ and obtain the state $|\psi_{T}\rangle$. In the generation of $|\psi_{T}\rangle$, we need $2\times(8+1)=18$ H-coh pairs on average. In our scheme, we need resource states which are equivalent to cluster states with Hadamard gates applied on some qubits. For that purpose, we prepare a different type of triple state $|\psi_{T}^{\prime}\rangle=|+\rangle|\mathcal{C}_{\alpha}^{+}\rangle(|\beta\rangle+|-\beta\rangle)+|-\rangle|\mathcal{C}_{i\alpha}^{+}\rangle(|\beta\rangle-|-\beta\rangle)$, which can be obtained by rotating the DV basis of H-coh pair from $H/V$ to $+/-$. Coherent-state qubits are consumed by fusion operation when generating offline resource states. The fusion operation is performed by the Bell measurement $\mathrm{B}_{\alpha}$, which can be implemented using one beamsplitter and two PNR detectors LeeSW2013 ; Jeong2001 . It fails if both PNR detectors click no photon, which occurs with the probability is $p_{\alpha}=\exp(-2|\beta|^{2})$.

In the construction of RHG lattice, we use two kinds of 3-qubit micro H-cluster states, $|C_{3}^{\mathrm{c}}\rangle$ and $|C_{3}^{\mathrm{s}}\rangle$. Typical 3-qubit cluster states can be represented as $|C_{3}\rangle_{123}=CZ_{12}CZ_{23}|+_{1},+_{2},+_{3}\rangle$. Micro cluster states used in our scheme differ by Hadamard gates from typical cluster states. A central micro cluster state is given by $|C_{3}^{\mathrm{c}}\rangle_{102}=H_{1}H_{2}|C_{3}\rangle_{102}=(|0_{1},0_{0},0_{2}\rangle+|1_{1},1_{0},1_{2}\rangle)/\sqrt{2}$ and a side micro cluster state is given by $|C_{3}^{\mathrm{s}}\rangle=H_{1}|C_{3}\rangle_{102}=(|0_{1},0_{0},0_{2}\rangle+|1_{1},1_{0},0_{2}\rangle+|0_{1},0_{0},1_{2}\rangle-|1_{1},1_{0},1_{2}\rangle)/2$. The subscript 0 represents a data qubit which will be the vertex of RHG lattice, while the subscripts 1 and 2 are for side qubits which will be consumed by hybrid fusion. The generation of $|C_{3}^{\mathrm{c}}\rangle$ is described in Fig. 3(c). We prepare three hybrid triples $|\psi_{T}\rangle$, where the second one has two coherent-state qubits which can be obtained by splitting larger coherent-state qubit with the amplitude $\sqrt{2}\beta$. Then we perform Bell measurements $\mathrm{B}_{\alpha}$ on coherent-state qubits to merge three hybrid qubits. We postselect on successful Bell measurements only so that no error is induced on off-line resource states. The desired state $|C_{3}^{\mathrm{c}}\rangle$ is obtained after applying the appropriate Pauli corrections according to the measurement outcome of $\mathrm{B}_{\alpha}$. The generation scheme of $|C_{3}^{\mathrm{s}}\rangle$ is almost the same, but one needs to replace the last triple state with $|\psi_{T}^{\prime}\rangle$. The average number of unit resources to generate a three-qubit cluster state is $54(1-p_{\alpha})^{-2}$.

The generation of resource states for gate teleportation, $|\Phi_{H}\rangle$ and $|\Phi_{CZ}\rangle$, is described in Fig. 3(d) and (e), respectively. The Hadamard resource $|\Phi_{H}\rangle$ can be generated by merging $|\psi_{T}\rangle$ and $|\psi_{T}^{\prime}\rangle$ using $\mathrm{B}_{\alpha}$. For the controlled-Z resource $|\Phi_{CZ}\rangle$, we need to prepare another type of resource which has two coherent-state ancillae with different basis. To prepare such a state, we merge a triple $|H\rangle|\mathcal{C}_{\alpha}^{+}\rangle(|\beta\rangle+|-\beta\rangle)+|V\rangle|\mathcal{C}_{i\alpha}^{+}\rangle(|\beta\rangle-|-\beta\rangle)$ and a H-coh pair $|H\rangle|\beta\rangle+|V\rangle|-\beta\rangle$ using $\mathrm{B}_{\mathrm{I}}$. Once $\mathrm{B}_{\mathrm{I}}$ is successful, we merge the state with three $|\psi_{T}\rangle$’s using $\mathrm{B}_{\alpha}$. The average number of unit resources to generate $|\Phi_{H}\rangle$ and $|\Phi_{CZ}\rangle$ is, respectively, $36(1-p_{\alpha})^{-1}$ and $(18\times 3+(18+1)\times 2)(1-p_{\alpha})^{-3}=92(1-p_{\alpha})^{-3}$.

A DV-CV hybrid entangled pair is employed in our approach as the basic building block of hybrid quantum computation. In CV part, a bosonic cat-code state needs to be implemented and entangled with a DV qubit such as single photon. While we focus on all-optical implementation in our analysis, we also emphasize that our hybrid approach is applicable to other platforms. CV qubits in cat states have been successfully demonstrated in superconducting circuits Leghtas2015 ; Ofek2016 ; Lescanne2020 ; Grimm2020 ; Vlastakis2013 ; Wang2016 ; Wang2022 ; He2023 ; Pan2023 , ion traps Gan2020 ; Eickbusch2022 ; Monroe2021 ; Heeres2017 ; Blais2021 ; Monroe1996 ; Kienzler2016 ; Johnson2017 ; Jeon2023 , and circuit acoustic devices Chu2018 ; Hann2019 ; Bild2023 . Especially, the error correction using cat code has been demonstrated in a superconducting circuit system Ofek2016 . Another line of efforts in the superconducting circuit system is a stabilized cat qubit by two-photon driven dissipation Leghtas2015 ; Touzard2018 ; Lescanne2020 ; Gertler2023 or Kerr nonlinear interaction Puri2017 ; Puri2019 ; Puri2020 ; Grimm2020 ; Putterman2022 ; Gravian2023 . By stabilizing the cat-code subspace, a bit-flip error rate is exponentially suppressed so that it suffices to correct phase-flip error using simple error correction code with low overhead Guillaud2019 ; Guillaud2021 ; Chamberland2022 ; Regent2023 ; Gouzien2023 . In those schemes, the implementation of universal gate set preserving the error bias enables hardware-efficient fault-tolerant quantum error correction Guillaud2019 ; Chamberland2022 ; Puri2020 . In the presence of CV qubits, any interaction strong enough to entangle the CV qubits with other DV qubits can produce the hybrid qubits. For example, H-cat qubits can be generated by applying a cross-Kerr interaction between DV and CV qubits, acting as $(|+\rangle+|-\rangle)|\mathcal{C}_{\alpha}^{+}\rangle\to|+\rangle|\mathcal{C}_{\alpha}^{+}\rangle+|-\rangle|\mathcal{C}_{i\alpha}^{+}\rangle=|+_{L}\rangle$.

While the required nonlinearity is also within the reach of the current photonic technology Sagona2020 ; Cui2022 , in other platforms, strong nonlinear interactions between DV and CV qubits are readily available. In trapped-ion systems, a variety of hybrid gate operations have been demonstrated, coupling the internal states of atoms and the phonon modes Monroe2021 ; Gan2020 ; Chen2023 ; Katz2023 . In superconducting circuit QED, universal control of the oscillator mode in a microwave cavity was demonstrated using nonlinear interaction between the oscillator mode and the transmon qubit Eickbusch2022 ; Heeres2017 ; Blais2021 ; Kudra2022 . Though those works have restricted the role of DV mode to the control qubit for manipulating CV mode, the coupling between CV mode and reliable DV qubit will provides us a resource for the hybrid quantum computation.