All-photonic architecture for scalable quantum computing with Greenberger-Horne-Zeilinger states

Out of the many physical platforms available for quantum computing, optical platforms facilitate quicker gate operations compared to the decoherence time, fast readouts and efficient qubit transfer [1, 2]. These features make them one of the strongest contenders for realizing scalable quantum computing. Linear optical quantum computing [1, 3, 4] uses only beam splitters, phase shifters and photon detectors to process the quantum information encoded in light. Besides this apparent simplicity, the ability to operate at room temperature makes this approach attractive. Measurement-based quantum-computing [5] is particularly suitable for optical platforms [6] in terms of practical implementation. In this approach, a particular class of multi-qubit entangled states, known as cluster states, are first generated, and single-qubit-measurements are then performed on them to realize a universal set of gate operations. In optical platforms, linear optical elements are sufficient for implementing the entangling operations for the generation of cluster states as well as single-qubit-measurements.

However, one major shortcoming that besets linear optical quantum computing is the fact that a direct Bell-state measurement (BSM), which is an entangling operation used to form a larger entangled state from smaller ones, is not deterministic [2, 7]. Adding to the shortcoming, photon loss is ubiquitous in all optical platforms and specifically in integrated optics [8], which not only causes optical qubit states to leak out of the computational basis but also introduces dephasing or depolarizing noise into qubits, gate operations and readouts (measurements). Thus, the bare-bone measurement-based quantum-computing scheme in Ref. [5] is not tolerant against the aforementioned practical issues, and additional enhancements are necessary to ensure its successful execution in real experiments. To overcome indeterminism of BSM and quantum noise, we need fault-tolerant architectures that employ quantum error correction (QEC) [9, 10]. With QEC, it is possible to achieve scalable linear optical quantum computing using lossy optical components provided the photon-loss level is below a certain threshold. This threshold value is dependent on the QEC codes and noise model considered in the fault-tolerant architecture [11, 12, 13].

Various kinds of linear-optical platforms have been proposed for quantum computing depending on the nature of variables used for encoding quantum information. Discrete-variable (DV) platforms manipulate level-structure properties of photons like polarization to encode quantum states. A BSM on DV qubits using linear optics is probabilistic with a success rate of $50\%$ without additional resources [7, 14]. Fault-tolerant architectures for linear optical quantum computing in Refs. [6, 13, 15, 16, 17, 18] overcome indeterminism of entangling operations by the repeat-until-success strategy. However, if we wish to carry out, say, $m$ simultaneous and identical entangling operations successfully, the average resource overhead is $O(1/p_{\mathrm{s}}^{m})$, which will result in an increase in overhead by a factor of $O(\Delta^{m})$ that is exponential in $m$ when the success rate $p_{\mathrm{s}}$ of each entangling operation decreases by a factor of $\Delta$ [15]. In our protocol, which we detail in Sec. II, the strategy with low success rates of entangling operations is used only at a certain stage unlike other mentioned architectures. This makes our protocol competitive in terms of resource overheads. Furthermore, because of such probabilistic entangling operations, these schemes would require optical switches to pass the successfully entangled states to the next step, which is known to contribute significant photon loss [19]. Alternatively, continuous-variable (CV) platforms that employ coherent states described by a continuous parameter (amplitude) [20, 21, 22, 23] offers BSMs that can be nearly deterministic [24]. Here, the success rate of a BSM grows with the coherent-state amplitude. The corresponding architectures for linear optical quantum computing require lower resource overheads, but are sensitive to photon loss and can only tolerate smaller thresholds [22]. Also, there has been significant developments in using optical Gottesman-Kitaev-Preskill states for fault-tolerant quantum computing [25, 26]. These schemes need very large squeezing strengths ($>10$ dB) to implement gates with high precision.

Recently, there have been efforts to combine the DV and CV approaches for quantum computing [27, 28, 29, 30, 31]. It was demonstrated that by using optical hybrid qubits, which are entangled states in the DV and CV domains, near-deterministic entangling operations can be implemented [27, 28]. Moreover, many shortcomings faced individually by CV and DV qubit-based schemes are overcome. Importantly, quantum computing with hybrid qubits reduces resource overheads and also improves the photon-loss tolerance [27, 28, 29]. By increasing the amplitude of the CV part, incurred resources can also be reduced. However, if the coherent amplitude of hybrid qubits is too large, the dephasing noise level in the presence of photon loss will also be commensurately too high [32]. Reference [29] also supports the logic that quantum computing on a special cluster state known as Raussendorf–Harrington–Goyal (RHG) lattice [33, 34] built with only DV qubits could tolerate higher photon loss, albeit at higher resource costs. Larger cluster states like the RHG lattice is built by performing BSMs on smaller entangled states.

Besides probabilistic BSMs and photon loss, practical implementation of linear optical quantum computing is greatly hindered by indeterministic generation of multiphoton entangled states, such as the GHZ and cluster states. There are theoretical proposals for on-demand generations of such multiphoton entangled states [35, 36]. For instance, various multi-photon entangled states can be generated by coupling a multi-level ancillary system to a two-level photon-emitting source and performing sequential unitary operation on both the ancillary system and source [35]. Here, the type of entangled states generated depends on the chosen unitary operation. In the experimental work [37], both the ancillary system and photon source are transmons, and are coupled via a flux-tunable superconducting quantum interference device. Furthermore, by interacting the single-photon sources, such as quantum dots, with spin-1/2 states, the sources can be made to emit multi-photon entangled states [36]. Recent experimental realizations [38, 39] alternatively make use of entanglement between the electron spin and polarization of photons emitted from optical excitations. An interesting point is that experimentally-realized three-photon and four-photon GHZ states respectively possess fidelities 0.9 and 0.82 [37]. We can also observe that as the number of photons in the GHZ state increases, the fidelity drops.

Motivated by the state-of-the-art techniques in deterministic generation of multiphoton entangled states, in this work we propose a multiphoton-qubit-based topological quantum computing protocol (MTQC) that uses multiphoton GHZ polarization states from deterministic sources to build RHG lattice. Furthermore, we demonstrate that our protocol provides an exceptionally high tolerance against photon-loss that reaches $11.5\%$. Considering GHZ states as the raw ingredients, we also show that the protocol is resource-efficient than all known [17, 13, 40, 41, 42, 22, 27] non-hybrid qubit-based schemes. We further encode each qubit of the RHG lattice with a multiqubit repetition code [9]—a concatenation of two QEC codes—to improve the photon-loss tolerance. Another salient and practically favorable feature of our protocol is that it requires only on-off detectors, unlike Ref. [18] that demands detectors that can resolve photon numbers and are thus more difficult to implement with competitive accuracy. Also photon number resolving nano-wire [43] and transition-edge [44] based detectors cannot operate at room temperatures. However, on-off detectors can operate at room temperatures [45], which allows our MTQC protocol pave the way for scalable all-photonic quantum computing.

The rest of the article is organized as follows. In Sec. II, we describe our MTQC in detail. Next, in Sec. III, we discuss the noise model we consider throughout the work. Section IV shall touch on the employment of concatenated QEC methods and their effects on further raising the threshold photon-loss rate that can be tolerated with our all-photonic architecture, where as the numerical simulation procedure of QEC is separately outlined in Appendix A. In Sec. V, we present our results on the photon-loss thresholds, and the details concerning resource estimation, specifically the average number of three-qubit GHZ states consumed, are provided in Sec. VI. In Sec. VII, we compare the results of our MTQC with those of other linear optical quantum computing schemes. Finally, some pertinent discussion and conclusion are presented in Sec. IX.

The primary aim of MTQC is to build an RHG lattice for fault-tolerant quantum computing using multiphoton GHZ polarization states from deterministic sources and processing them with passive linear-optical elements like polarizing beam splitters, phase shifters, optical delay lines, optical switches and on-off detectors. To begin, multiphoton GHZ states are entangled by BSMs in an efficient manner to form two kinds of resource states. We point out that like other DV protocols, the repeat-until-success strategy with low-success-rate entangling operations is employed only at this stage to generate these resource states. After which, we perform a collective BSM [41] on multiphoton resource states, which is a near-deterministic entangling operation.

The collective BSM (described in Sec. II.1), is a crucial ingredient of our protocol. This is because the near-deterministic entangling operation requires only on-off detectors to boost its success rate of entangling arbitrarily close to 100%. This is unlike some other BSMs [46, 47] that demand photon-number resolving detectors that can distinguish up to four photons to boost the success rate to 75%, and the ability to resolve higher photon numbers is essential for further enhancements. Another salient feature of the collective BSM is that it needs no optical switching to entangle two optical qubits like the other DV protocols when using repeat-until-success strategy. These features make MTQC practically more attractive. It is also important to note that once we start using collective BSMs in our protocol, the repeat-until-success strategy, even if employed, shall not drastically increase resource overheads as the success rate of a collective BSM is very high. This makes our protocol competitive in terms of resource overheads.

The RHG lattice built using BSMs with a boosted success rate of 75% still cannot support fault-tolerant quantum computing as the failure rate must be lesser than 14.5% [48]. To overcome this shortfall, there exists a proposal to purify the RHG lattice [49] by which the effective failure rate of BSMs can be brought down to 7% from 25%. While the purified RHG lattice can support fault-tolerant quantum computing, this approach has the disadvantage of reducing the effective size of the RHG lattice which will contribute to a large resource overhead. Also, in this situation the RHG lattice is less robust against dephasing errors. There is also an attempt to build the lattices with CV-based qubits in Ref. [23]. But this demands average photon numbers of CV qubits over 100 to build an RHG lattice that is of a sufficiently high fidelity for fault-tolerant computation. Such high average photon numbers are not achievable and lattices built under practical values [50] (average photon number of 2) are far from fault-tolerant. Recently, there has been progress in the generation of two-dimensional CV cluster-states without BSM [51, 52]. Another proposal [18] aims to build RHG lattices using BSMs with a boosted success rate of 75% (or higher) by adding single photons or Bell pairs and employing photon number resolving detectors. Here, the 14.5% failure-rate mark is overcome by the repeat-until-success strategy to create entanglement between qubits similar to Ref. [13]. Involvement of tree clusters [53, 54] render the scheme fault-tolerant against BSM failure, but this comes at the cost of feed-forward measurements. Depending on the failure or success of BSM remaining qubits of tree cluster are measured in appropriate basis [18]. These feed-forward operations form the bottleneck in linear optical quantum computing.

The state $|\mathcal{C}_{\mathcal{L}}\rangle$ of an RHG lattice is a unique 3D cluster state [34] where qubits are entangled to their nearest neighbors represented by the edges of the RHG lattice. In general, a cluster state $|\mathcal{C}\rangle$ over a collection of qubits $\mathcal{C}$ is a state stabilized by the operators $X_{a}\bigotimes_{b\in{\rm nh}(a)}Z_{b}$, where $a,b\in\mathcal{C}$, $Z_{i}$ and $X_{i}$ are the Pauli operators on the $i$th qubit, and nh($a$) denotes the adjacent neighborhood of qubit $a\in\mathcal{C}$:

$$|\mathcal{C}\rangle=\prod_{\begin{subarray}{c}a\in\mathcal{C}\\ a<b\in\rm{nh}(a)\end{subarray}}\!\!\!\!\!\!\!\!\textrm{CZ}_{a,b}\,\bigotimes_{a^{\prime}\in\mathcal{C}}|+\rangle_{a^{\prime}}\,,$$

(1)

where $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$ are eigenkets of $X$, while $|0\rangle,|1\rangle$ are those of $Z$. The controlled-$Z$ unitary gate ${\rm CZ}_{a,b}$, which is an entangling operation, applies $Z$ on the target qubit $b$ if the source qubit $a$ is in the state $|1\rangle$. The successful action of ${\rm CZ}_{a,b}$ on the lattice qubits on sites $a$ and $b$ is represented by an edge in the lattice. The state $|\mathcal{C}_{\mathcal{L}}\rangle$ is currently the best available choice, to the best of our knowledge, to make linear-optical platform fault-tolerant; it can tolerate qubit loss [55, 56], probabilistic entangling operations [15, 48], dephasing and depolarizing noises [33, 34, 57], all of which are peculiar to the platform. Furthermore, QEC and gate operations on $|\mathcal{C}_{\mathcal{L}}\rangle$ is topological in nature and thus offers the highest fault tolerance in the platforms where interactions are restricted to those of the nearest neighbors [58].

In MTQC, the logical basis for an $l$-photon qubit is

$$|0_{l}\rangle\equiv|\textsc{h}\rangle^{\otimes l},~{}~{}|1_{l}\rangle\equiv|\textsc{v}\rangle^{\otimes l}.$$

(2)

where $|\textsc{h}\rangle$, $|\textsc{v}\rangle$ are the discrete orthonormal polarizations and are eigenkets of the $Z$ Pauli operator. An $r$-photon GHZ ket has the form $|{\rm GHZ}_{r}\rangle\propto|\textsc{h}\rangle^{\otimes r}+|\textsc{v}\rangle^{\otimes r}$ (up to normalization). Once there is a continuous and reliable supply of $|{\rm GHZ}_{r}\rangle$ from deterministic sources, the following prescription forms the stages of our protocol to create $|\mathcal{C}_{\mathcal{L}}\rangle$ and perform topological fault-tolerant quantum computing on it:

1. 1.

   The foremost step is to create multiphoton three-qubit resource states using $|{\rm GHZ}_{r}\rangle$ and BSMs.

2. 2.

   Near-deterministic collective BSM are performed on these resource states to form star cluster states.

3. 3.

   The star clusters then undergo collective BSM to form layers of $|\mathcal{C}_{\mathcal{L}}\rangle$.

4. 4.

   Finally, the qubits are measured layer by layer in a suitable basis to effect both QEC and Clifford-gate operations on the logical states of $|\mathcal{C}_{\mathcal{L}}\rangle$. Initialization of $|\mathcal{C}_{\mathcal{L}}\rangle$ to certain logical states and magic-state distillation are also possible by measurements which complete the universal set of gates for quantum computing.

Certain aspects of MTQC do bear some resemblance with the protocol in Ref. [28], which uses hybrid qubits as basic ingredients. However, this resemblance is only superficial. Since we use GHZ states as basic ingredients in MTQC, the very process of generating three-qubit resource states is different. The kind of entangling operations and apparatus employed are also very different. Here, we need on-off detectors whereas the protocol in Ref. [28] requires photon-number-parity detectors. Additionally, MTQC exploits the concatenation of error-correcting codes to improve the tolerance against photon loss. In what follows, all stages of the MTQC scheme are discussed in detail.

Apart from photon loss being a major source of errors [2], the lattice $|\mathcal{C}_{\mathcal{L}}\rangle$ built with linear optics suffer from missing edges due to probabilistic entangling operations. In this section, we study the effect of photon loss on multiphoton qubits, success rate of entangling operation and consequently on the MTQC protocol. Suppose that the overall photon-loss rate suffered by each photon due to imperfect optical components is $\eta$ and the initial state of an $l$-photon qubit is defined by $|\Psi^{l}\rangle=\alpha|\textsc{h}\rangle^{\otimes l}+\beta|\textsc{v}\rangle^{\otimes l}$. When $\eta$ is nonzero, the state of the multiphoton qubit is [41]

	$\displaystyle\widetilde{\rho}_{l}=$	$\displaystyle\,|\Psi^{l}\rangle(1-\eta)^{l}\langle\Psi^{l}|+\frac{1}{2}\sum^{l}_{q=1}\eta^{q}(1-\eta)^{l-q}$
		$\displaystyle\,\times\mathcal{P}_{l}[|\Psi^{l-q}\rangle\langle\Psi^{l-q}|+|\Psi^{l-q}_{-}\rangle\langle\Psi^{l-q}_{-}|,|\textsc{vac}_{q}\rangle\langle\textsc{vac}_{q}|]\,,$		(5)

where $|\Psi^{l-q}_{-}\rangle=\alpha|\textsc{h}\rangle^{\otimes l-q}-\beta|\textsc{v}\rangle^{\otimes l-q}$ and $|\textsc{vac}_{q}\rangle$ is a vacuum state. The probability of an $l$-photon qubit losing a photon or more is $1-(1-\eta)^{l}$. When photons are lost, with probability 0.5 the state possesses the component $|\Psi^{l}_{-}\rangle\langle\Psi^{l}_{-}|$, that is, undergoes dephasing.

If $0\leq\eta\leq 1$ is the overall photon-loss rate due to imperfect GHZ source ($\eta^{\rm soc}$), delay lines ($\eta^{\rm dly}$), optical switching network ($\eta^{\rm swc}$) and detectors ($\eta^{\rm det}$), the rate of dephasing on the multiphoton lattice qubits due to photon loss is

$\displaystyle p_{Z}=\frac{1-(1-\eta)^{l}}{2}.$

(6)

If photon losses in different components of quantum computing are statistically independent, we have the relation

$$\eta=1-(1-\eta^{\rm soc})(1-\eta^{\rm dly})(1-\eta^{\rm swc})(1-\eta^{\rm det})$$

(7)

that associate the overall photon-loss rate to the relevant individual component loss rates. For the delay lines $(1-\eta^{\rm dly})=\mathrm{e}^{\mbox{\footnotesize$-c\,\tau_{0}\,\kappa/L_{0}$}}$, where $L_{0}=22$ km is attenuation length of optical fiber for the standard telecom wavelength of 1550 nm [59], $c=2\times 10^{5}$ km/s is the speed of light in, say, acrylic (PMMA) optical fibers, $\tau_{0}=150\times 10^{-9}$ s [2] is the time duration to complete one BSM, which can be made smaller using electro-optical modulators, and $\kappa$ is the total time steps (in units of $\tau_{0}$) a photon of lattice-qubit; that is, central qubit of $|\mathcal{C}_{\ast}\rangle$ spends before being measured during fault tolerant quantum computing. Also, each photon has to pass through a network of $\kappa$ switches. Therefore, $\eta^{\rm swc}=1-(1-\eta^{\rm s})^{\kappa}$, where $\eta^{\rm s}$ is photon-loss rate through one optical switch. It is important to note from Eq. (6) that photon-loss introduces larger rate of dephasing on qubits with larger number of photons.

The ${\rm B_{S}}$ operates on the lossy states and when input photons are lost the failure rate increases to $1-(1-\eta)^{2}/2$. An $n$-BSM would fail only when all the constituent ${\rm B_{S}}$’s fail. Therefore, the failure rate of an $n$-BSM due to photon loss is

$$p_{\mathrm{f}}=\left[1-\dfrac{(1-\eta)^{2}}{2}\right]^{n}\underbrace{\cong}_{\eta\ll 1}\left(\dfrac{1}{2}+\eta\right)^{n}\,.$$

(8)

From the above expression, we observe that when $\eta$ is $O(10^{-2})$ and $n$ is large, $p_{\rm f}$ is not very sensitive to photon loss.

We point out that like other DV optical schemes [17], photon loss does not necessarily imply lattice-qubit loss. The probability that photon loss leads to lattice-qubit loss, $\eta^{m}$, is much smaller than $p_{\rm f}$ for $\eta\sim 10^{-2}$, $n\geq 5$ and $m\geq 2$ considered in this work. Therefore, $n$-BSM failure has a dominant effect over qubit loss when calculating logical error rate on $|\mathcal{C}\rangle_{\mathcal{L}}$ and the latter can be neglected. However, having large $m$ is not favorable as it invites stronger dephasing as inferred from Eq. (6). To mitigate this issue we set $m=2$ through out the work, which also sufficient to neglect the effect of qubit loss.

The MTQC is operated at the same photon-loss rate $\eta$ on all qubits and the number of photons in lattice qubits is $m=2$. In this case $\kappa<k$ and it is crucial to note that the lattice qubits always spend $\kappa=3~{}(4)$ time steps in MTQC-1 (MTQC-2) before being measured. Now one can appreciate that our protocol to generate $|\mathcal{C}_{3^{\prime}}\rangle_{n,2,n}$ makes sure that the photons of lattice qubits passes through least number of optical components. Therefore, for a fixed $\eta$ component-wise photon-loss tolerence of MTQC is enhanced. Also, note that before performing $n$-BSMs to form $|\mathcal{C}_{\mathcal{L}}\rangle$, the surrounding qubits of a $|\mathcal{C}_{\ast}\rangle$ has to pass through a larger number of lossy components as $n>m$ and therefore have a larger photon-loss rate. Moreover, qubits that are part of the $n$-BSM between the current and future layers spend an extra time step and thus suffer from stronger losses. However, this time delay depends on the physical architecture that runs measurement-based quantum computation along with other technological details; in principle, different layers can be generated simultaneously. Therefore, we hereby neglect the influence of time delays on $\eta$, which is reasonable if the time delay is sufficiently shorter than $-\left(L_{0}/c\right)\ln\eta$.

For the star cluster states, qubit dephasing owing to photon loss happens locally on the central and other qubits. In addition, the surrounding qubits are consumed during an $n$-BSM and noise owing to photon loss can be dealt with by suitably encoding these qubits. Investigation of this procedure is a subject matter of our future work that is beyond the scope of this article. In this work, we consider only the dephasing noise due to photon loss on the central qubits. One can also consider other kinds of amplitude-damping noise [60, 61, 62, 63] to evaluate the performance our MTQC protocols.

As described in the previous section, photon-loss leads to dephasing on the qubits. We observe from Eq. (6) that if the degree of dephasing can be reduced, MTQC can have a larger photon-loss threshold $\eta_{\rm th}$. To reduce the effect of dephasing, one intuitive approach would be to encode the qubits of $|\mathcal{C}_{\mathcal{L}}\rangle$ with a multiqubit repetition code. This can be achieved by encoding the $m$-photon qubits of resource ket $|\mathcal{C}_{3^{\prime}}\rangle_{n,m,n}$ with an $N$-qubit repetition code; one would replace $|0_{m}\rangle$ with $\left(|0_{m}\rangle+|1_{m}\rangle\right)^{\otimes N}$, and $|1_{m}\rangle$ with $\left(|0_{m}\rangle-|1_{m}\rangle\right)^{\otimes N}$ (up to normalization), where $N$ is the repetition number. This gives the following encoded resource ket

	$\displaystyle|\mathcal{C}_{3^{\prime}}\rangle_{\rm enc}$	$\displaystyle=$	$\displaystyle|0_{n}\rangle\left(|0_{m}\rangle+|1_{m}\rangle\right)^{\otimes N}|0_{n}\rangle$		(9)
			$\displaystyle~{}~{}+|1_{n}\rangle\left(|0_{m}\rangle-|1_{m}\rangle\right)^{\otimes N}|1_{n}\rangle.$

Note that the extreme qubits, each holding $n$ photons, are not encoded as they shall be consumed by $n$-BSMs anyway.

As an example for demonstrating that we can increase $\eta_{\rm th}$, let us consider $N=3$. The generation of the encoded $|\mathcal{C}_{3^{\prime}}\rangle$ kets using $|\rm GHZ_{3}\rangle$’s and ${\rm B_{S}}$ is explained in Appendix C. The QEC procedure using this three-qubit repetition code employs the majority voting strategy, which fails when two of more qubits undergo dephasing [9]. Thus, the effective dephasing rate due to photon loss on the encoded lattice qubits is

$$p_{Z,\mathrm{enc}}=3p_{Z}^{2}(1-p_{Z})+p_{Z}^{3}\,,$$

(10)

where $p_{Z}$ is the unencoded dephasing error from Eq. (6). It is clear that $p_{Z,\mathrm{enc}}<p_{Z}$ when $p_{Z}<0.5$. Hence, such a repetition encoding can suppress the dephasing rate, which would result in an improvement on $\eta_{\rm th}$. The new tolerable photon-loss rate, $\eta_{\rm th}^{\rm enc}$ is deduced by inverting the expression for $p_{Z,\mathrm{enc}}$. In principle, we can make $\eta_{\rm th}^{\rm enc}$ arbitrarily close to 1 by increasing the value of $N$. This encoding strategy would evidently require more GHZ ingredient states to generate encoded (star-)cluster states. However, as we shall see in Sec. VI, numerical simulations show that using encoded states also reduces the effective dephasing rate (implying a larger tolerable photon-loss rate) to the extent that outweighs the additional GHZ states needed for encoding, such that smaller values of $d$ would suffice to reach $p_{\rm L}^{\rm targ}$.

We suppose that now, the lattice qubit is encoded with a finite $N$-qubit repetition code [9]. Using majority voting, the effective dephasing rate on the encoded lattice qubits is

$$p_{Z,\mathrm{enc}}=\sum^{N}_{q=\lceil(N+1)/2\rceil}\binom{N}{q}p_{Z}^{q}(1-p_{Z})^{N-q}\,,$$

(11)

where $p_{Z}$ is the dephasing rate on un-encoded lattice qubits. The task is to reveal the influence of $N$ on the value of the photon-loss threshold rate $\eta_{\mathrm{th}}$.

We make use of the convenient approximation

$$\binom{N}{q}p_{Z}^{q}(1-p_{Z})^{N-q}\cong\dfrac{\exp\!\left(-\frac{(q-Np_{Z})^{2}}{2\,Np_{Z}(1-p_{Z})}\right)}{\sqrt{2\pi Np_{Z}(1-p_{Z})}}$$

(12)

that is valid for sufficiently large $N$ owing to the central limit theorem. After a variable substitution with $x=q/N$, this allows us to convert Eq. (11) into an integral,

	$\displaystyle p_{Z,\mathrm{enc}}\cong$	$\displaystyle\,\int^{\infty}_{1/2}\mathrm{d}x\,\dfrac{\exp\!\left(-\frac{(x-p_{Z})^{2}}{2\,p_{Z}(1-p_{Z})/N}\right)}{\sqrt{2\pi p_{Z}(1-p_{Z})/N}}$
	$\displaystyle=$	$\displaystyle\,\dfrac{1}{2}-\dfrac{1}{2}\,\mathrm{erf}\!\left(\dfrac{1-2p_{Z}}{2\sqrt{2p_{Z}(1-p_{Z})/N}}\right)\,,$		(13)

which involves the error function $\mathrm{erf}\!\left(\cdot\right)$. The infinite upper limit of the integral is justified by the extremely narrow width of the Gaussian integrand for large $N$. Since for a large argument $z$,

$$\mathrm{erf}\!\left(z\right)\cong 1-\dfrac{\mathrm{e}^{\mbox{\footnotesize$-z^{2}$}}}{\sqrt{\pi}\,z}\,,$$

(14)

we get $y\,\mathrm{e}^{\mbox{\footnotesize$y^{2}/8$}}\cong\sqrt{2/\pi}/p_{Z,\mathrm{enc}}$, where $y=\dfrac{1-2p_{Z}}{\sqrt{p_{Z}(1-p_{Z})/N}}$. Squaring this relation allows us to write

$$y^{2}\cong 4\,\mathrm{W}\!\left(1/\left(2\pi p^{2}_{Z,\mathrm{enc}}\right)\right)\,,$$

(15)

which expresses the solution as a Lambert function $\mathrm{W}\!\left(\cdot\right)$. This leads to the physical solution

$$p_{Z}\cong\dfrac{1}{2}-\dfrac{1}{2\sqrt{1+4\,t}}\,,\quad t=\dfrac{N}{4\,\mathrm{W}\!\left(1/\left(2\pi p^{2}_{Z,\mathrm{enc}}\right)\right)}\,.$$

(16)

We may now immediately identify $(1-\eta_{\mathrm{th}}^{\rm enc})^{m}\cong(1+4\,t)^{-1/2}$, with $m$ being the number of photons of each qubit in the repetition code, which finally yields

$$\eta_{\mathrm{th}}^{\rm enc}\cong 1-\left[1+\dfrac{N}{\mathrm{W}\!\left(1/\left(2\pi p^{2}_{Z,\mathrm{enc}}\right)\right)}\right]^{-1/(2m)}\,.$$

(17)

As $N$ increases, we find that $\eta_{\mathrm{th}}^{\rm enc}\cong 1-O(1/N^{1/(2m)})\rightarrow 1$ for any designated value of $p_{Z,\mathrm{enc}}<p_{Z}$ and $m$. On the other hand, the function $\mathrm{W}\!\left(1/\left(2\pi p^{2}_{Z,\mathrm{enc}}\right)\right)$ itself is a slowly increasing function of $p_{Z,\mathrm{enc}}$ as $p_{Z,\mathrm{enc}}$ decreases, originating from the large-argument expansion

$\displaystyle\mathrm{W}\!\left(x\right)\cong$

$\displaystyle\,\ln x-\ln\ln x+\frac{\ln\ln x}{\ln x}+\frac{(\ln\ln x-2)\ln\ln x}{2(\ln x)^{2}}\,,$

(18)

so that within the typical range $0.001\leq p_{Z,\mathrm{enc}}\leq 0.1$ of interest, the order of magnitude for $\mathrm{W}\!\left(1/\left(2\pi p^{2}_{Z,\mathrm{enc}}\right)\right)$ does not change. For a repetition code of fixed $N$ and a given $p_{Z,\mathrm{enc}}$, increasing $m$ reduces $\eta_{\mathrm{th}}^{\rm enc}$.

When carrying out QEC on $|\mathcal{C}_{\mathcal{L}}\rangle$, the logical error rate $p_{\rm L}$ is determined against $p_{Z}$ for various code distances, $d$ via simulations. This procedure is repeated for various values of $n$, which determine the respective $p_{\rm f}$ of $n$-BSMs. The intersection point of the curves corresponding to various $d$’s is the threshold dephasing rate $p_{Z,\rm th}$ as marked in Fig. 5. The photon-loss threshold, $\eta_{\rm th}$ is determined using Eq. (6) by replacing $p_{Z}$ with $p_{Z,\rm th}$.

For example, from Fig. 5 which corresponds to $n=8$, we have $p_{Z,{\rm th}}\approx 2.9\times 10^{-2}~{}(3.2\times 10^{-2})$ for MTQC-1 (MTQC-2). Considering that MTQC is operated below the threshold value (see Tab. 1) at $\eta=0.01$, according to Eq. (8), we then have the associated value of $p_{\rm f}=4.58\times 10^{-3}$. Replacing $p_{Z}$ by $p_{Z,{\rm th}}$ in Eq. (6) we find that the total tolerable photon loss rate, $\eta_{\rm th}$ is $2.9\times 10^{-2}~{}(3.2\times 10^{-2})$ for MTQC-1 (MTQC-2). However, the photon loss rate tolerable by individual components can be deduced as follows. The total time steps spent by lattice-qubits before being measured is $\kappa=3~{}(4)$ for MTQC-1 (MTQC-2). Therefore, we have $\eta_{\rm dly}=4.1\times 10^{-3}(5.4\times 10^{-3})$. Further, by inserting the value of $\eta_{\rm dly}$ in Eq. (7) ($m=2$), we have $\eta^{\rm soc}_{\rm th}=\eta^{\rm swc}_{\rm th}=\eta^{\rm det}_{\rm th}=8.6\times 10^{-3}~{}(8.8\times 10^{-3})$ for MTQC-1 (MTQC-2). This implies that equal amount of photon-loss can be tolerated in GHZ source, optical switches and measurement. In this case each switch in the network can tolerate photon-loss rate of $\eta^{\rm s}=2.8\times 10^{-3}(1.7\times 10^{-3})$. While the $\eta_{\rm th}$ in MTQC-2 is higher both subvariants offer comparable component-wise photon-loss thresholds. However, MTQC-2 imposes more stringent restriction on allowed photon-loss rate in switches. We lastly note that the time delays between consecutive layers can be neglected if they are sufficiently shorter than $-\left(L_{0}/c\right)\ln\eta_{\mathrm{th}}\approx 4\times 10^{-4}\text{ s}$ for both MTQC-1 and MTQC-2, as discussed in Sec. III.

The $\eta^{\rm soc}$, $\eta^{\rm swc}$ and $\eta^{\rm det}$ are complementary in nature as the overall tolerable photon-loss rate of the MTQC is fixed. Therefore, if the GHZ source and detectors are operated at lower loss levels, the MTQC protocol can tolerate a much higher photon-loss rate in the optical switches. Similarly, we have repeated calculation for $n=5$ through 9 and the values of $\eta_{\rm th}$ are tabulated in Tab. 1. We know from the Ref. [48] that when $p_{\rm f}=0.145$ in MTQC-2, the $|\mathcal{C}_{\mathcal{L}}\rangle$ cannot tolerate any dephasing, and hence any photon loss. This threshold $p_{\rm f}$ is overcome when $n\geq 3$. However, the situation is different for MTQC-1 and is explained in the following.

In MTQC-1, distorted $|\mathcal{C}_{\ast}\rangle$’s are allowed and this gives rise to $|\mathcal{C}_{\mathcal{L}}\rangle$ with diagonal edges (refer also to Fig. 2(c) of Ref. [28]). This is overcome by removing the qubits at the ends of a diagonal edge during QEC. Therefore, the probability that a qubit survives on $|\mathcal{C}_{\mathcal{L}}\rangle$ is $1-p_{\rm f}$. Note also that each qubit in $|\mathcal{C}_{\mathcal{L}}\rangle$ is susceptible to losses from diagonal edged connected to four $|\mathcal{C}_{*}\rangle$’s: two in the layer containing the qubit, and another two coming from different layers. Additionally, failure of an $n$-BSM that connects two $|\mathcal{C}_{\ast}\rangle$’s would leave an edge between qubits missing. This situation is handled by removing one of the qubits associated with such a missing edge [48]. The survival probability of a lattice qubit in this case is $1-p_{\rm f}/2$, where four $n$-BSMs are involved. It follows that the total probability of loosing a qubit on $|\mathcal{C}_{\mathcal{L}}\rangle$ is $1-(1-p_{\rm f})^{4}(1-p_{\rm f}/2)^{4}$ and this number should not exceed $0.249$ [64, 55] if $|\mathcal{C}_{\mathcal{L}}\rangle$ should be useful for quantum computing. Therefore, the threshold value for $p_{\rm f}$ is 0.047, which is determined by solving the equation $1-(1-p_{\rm f})^{4}(1-p_{\rm f}/2)^{4}=0.249$. So, MTQC-1 is possible only when $n\geq 5$. Note that lattices other than the RHG type have different values for threshold failure rates of entangling operation [65, 66]. Accordingly, the probability of missing qubits in MTQC-2 is $1-(1-p_{\rm f})^{4}$. For a given value of $p_{\rm f}$, the resulting lattice in MTQC-1 has more missing qubits and is therefore of a relatively poorer quality.

The tolerable photon-loss thresholds for MTQC naturally increase with the usage of repetition codes in the manner discussed in Sec. IV. Let us consider an example of encoded lattice-qubit with $n=8$, $m=2$ and $N=3$. In this encoded situation, The value for the overall photon-loss threshold with a 3-qubit repetition code, obtained by using Eq. (10), is $\eta_{\rm th}^{\rm enc}=10.7\%~{}(11.1\%)$. $\kappa=$6 (7) for MTQC-1 (MTQC-2), following which we have $\eta_{\rm dly}=8.1\times 10^{-3}(9.5\times 10^{-3})$. The component wise tolerence would be $3.2\times 10^{-2}~{}(3.4\times 10^{-2})$. In other words, encoding the lattice qubits with a three-qubit repetition code increases the $\eta_{\rm th}$ by nearly four times. Also, each switch can now tolerate a higher $\eta^{\rm s}=5.4\times 10^{-3}~{}(4.9\times 10^{-3})$. The results for other values of $n$ are available in Tab. 1. Additionally, a sufficiently large set of stabilizer measurements on encoded qubits can be used to reconstruct the underlying noisy quantum process acting on these qubits [67, 68].

In this work, we consider GHZ states as the raw ingredients for constructing $|\mathcal{C}\rangle_{\mathcal{L}}$, on which fault-tolerant gate operations are performed via single-qubit measurements. Therefore, the resource overhead, $\mathcal{N}$, is the average number of $|{\rm GHZ}\rangle$’s consumed to build $|\mathcal{C}\rangle_{\mathcal{L}}$ of required size. Other components used in MTQC, such as delay lines, detectors, optical switches and beam splitters scale proportionately to $\mathcal{N}$. Alternatively, Ref. [18] considers detectors to be resources. Logical gate operations are possible by passing defects through $|\mathcal{C}\rangle_{\mathcal{L}}$ [33, 34]. As logical errors occur when a chain of $Z$ errors connects two defects or encircles a defect, error-free operations would demand these defects be separated by a distance $d$ and also have a perimeter of $d$. When noise is below threshold, by increasing the value of $d$, the logical error rate $p_{\rm L}$ can be reduced arbitrarily. If $|\mathcal{C}\rangle_{\mathcal{L}}$ has sides of length $l=5d/4$, it can accommodate a defect of perimeter $d$ and other defects placed a distance $d$ apart from each other (refer to Fig. 8 of Ref. [29]).

The time for simulation of QEC on $|\mathcal{C}\rangle_{\mathcal{L}}$ increases drastically with $d$, rendering the estimation of (a very small) $p_{\rm L}$ for arbitrarily large $d$ unfeasible. So, the value of $d$ at which an extremely small target $p_{\rm L}$ ($p^{\mathrm{targ}}_{\rm L}$) is achieved can be estimated by extrapolating known values of $p_{\rm L}$. We can determine the target $d$ required to achieve $p^{\mathrm{targ}}_{\rm L}=10^{-6}$ and $10^{-15}$ using the following expression [56]

$$p^{\mathrm{targ}}_{\rm L}=b\,\left(\dfrac{a}{b}\right)^{\displaystyle-(d-d_{b})/2},$$

(19)

where $a$ and $b$ are the values of $p_{\rm L}$ corresponding to the second largest $d_{a}$ and the largest code distance $d_{b}$, respectively considered in our simulations. For example, as seen from the Fig. 5 when $p_{Z}=0.01$ we have $d_{a}=11$ and $d_{b}=13$.

As a practice, for $n>5$, we operate MTQC at $\eta=0.01$, which is below the photon-loss threshold value. This corresponds to $p_{Z}\approx 0.01$ at $m=2$ for the unencoded case and $p_{Z,\mathrm{enc}}\approx 3\times 10^{-4}$ for the encoded one. However, unencoded MTQC-1 with $n=5$ yields a threshold rate of $\eta_{\rm th}\approx 6\times 10^{-3}$ and is thus operated at $\eta=3\times 10^{-3}$. Therefore, the $\mathcal{N}_{p_{\rm L}^{\rm targ}}$ is also determined at the operation point. In Eq. (19), $a$ and $b$ also correspond to the operation point. The reason for choosing the operation point away from the threshold is as follows: It is known empirically that $p_{\rm L}\propto(p_{Z}/p_{Z,{\rm th}})^{(d+1)/2}$ when the minimum weight perfect matching decoder is used [69]. If we operate closer to the threshold, a larger $d$ is essential to reach some pre-chosen $p_{\rm L}^{\rm targ}$. Thus, the operation point is chosen away from the threshold. Also, sufficiently away from the threshold point the ratio $a/b$ is reasonably constant and the estimation with Eq. (19) is reliable [56]. Once $d$ for achieving $p^{\mathrm{targ}}_{\rm L}$ is determined, $\mathcal{N}$ can be estimated by counting the average number of GHZ states required to build $|\mathcal{C}\rangle_{\mathcal{L}}$ of side $l=5d/4$. Only the central qubit of a $|\mathcal{C}_{\ast}\rangle$ stays in the lattice and rest of them are consumed by $n$-BSMs. A $|\mathcal{C}\rangle_{\mathcal{L}}$ of sides $l$ would have $6l^{3}$ qubits. Therefore, we need $6l^{3}$ $|\mathcal{C}_{\ast}\rangle$’s per fault-tolerant gate operation.

As explained in Sec. II.3, to create a $|\mathcal{C}_{\ast}\rangle$ we need two $|\mathcal{C}_{3}\rangle_{n,n,n}$’s and one $|\mathcal{C}_{3^{\prime}}\rangle_{n,m,n}$. Based on the noise model in Sec. III, in order to minimize dephasing effects on the central qubit, we set $m=2$. According to Fig. 2, the creation of a $|\mathcal{C}_{3}\rangle_{n,n,n}$ necessitates the entanglement of two $|{\rm GHZ}_{n+1}\rangle$ and a $|{\rm GHZ}_{n+2}\rangle$ using two ${\rm B_{S}}$’s. As the failure rate of one ${\rm B_{S}}$ is $(1+2\eta)/2$ given a photon-loss rate $\eta$ [obtained by setting $n=1$ in Eq. (8)], one needs $8(1-2\eta)^{-2}$ copies of $|{\rm GHZ}_{n+1}\rangle$ and $4(1-2\eta)^{-2}$ copies of $|{\rm GHZ}_{n+2}\rangle$, on average. Similarly, to create a $|\mathcal{C}_{3^{\prime}}\rangle_{n,m,n}$, one needs on average of $8(1-2\eta)^{-2}$ $|{\rm GHZ}_{n+1}\rangle$ and $4(1-2\eta)^{-2}$ $|{\rm GHZ}_{m+2}\rangle$. Therefore,

$$\mathcal{N}_{\ast}=\dfrac{4(6N_{n+1}+2N_{n+2}+N_{m+2})}{(1-2\eta)^{2}}$$

(20)

$|{\rm GHZ}_{3}\rangle$’s consumed to create a $|\mathcal{C}_{\ast}\rangle$ in MTQC-1, on average, where $N_{r}$ is the average number of $|{\rm GHZ}_{3}\rangle$’s consumed to create a $|{\rm GHZ}_{r}\rangle$, example values of which can be read from Tab. 2 in Appendix B. On the other hand, one needs

$$\mathcal{N}_{\ast}^{\prime}=\dfrac{4(6N_{n+1}+2N_{n+2}+N_{m+2})}{(1-2\eta)^{2}(1-p_{\rm f})^{2}}$$

(21)

$|{\rm GHZ}_{3}\rangle$’s, on average, in MTQC-2. For example, consider the case when $n=8$ and $m=2$, and MTQC is operated at $\eta=0.01$. Looking at Tab. 2 and inserting the values of $N_{10}$, $N_{9}$ and $N_{4}$ in to Eq. (20) one can estimate that $4(6\times 55.16+2\times 68.00+4.08)(1-2\times 0.01)^{-2}\approx 1962$ $|{\rm GHZ}_{3}\rangle$’s, on average, are consumed in MTQC-1 to create a $|\mathcal{C}_{\ast}\rangle$. Similarly, using Eq. (21) we estimate that approximately $1980$ $|{\rm GHZ}_{3}\rangle$’s are needed in MTQC-2.