Abstract
A central challenge for many quantum technologies concerns the generation of large entangled states of individually addressable quantum memories. Here, we show that percolation theory allows the rapid generation of arbitrarily large graph states by heralding the entanglement in a lattice of atomic memories with singlephoton detection. This approach greatly reduces the time required to produce large cluster states for quantum information processing including universal oneway quantum computing. This reduction puts our architecture in an operational regime where demonstrated coupling, collection, detection efficiencies, and coherence time are sufficient. The approach also dispenses the need for timeconsuming feedforward, high cooperativity interfaces and ancilla single photons, and can tolerate a high rate of site imperfections. We derive the minimum coherence time to scalably create large cluster states, as a function of photoncollection efficiency. We also propose a variant of the architecture with longrange connections, which is even more resilient to site yields. We analyze our architecture for nitrogen vacancy (NV) centers in diamond, but the approach applies to any atomic or atomlike systems.
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Introduction
The past years have seen rapid advances in controlling small groups of qubits encoded in atomic or atomlike quantum memories. An important question now concerns the development of architectures to efficiently combine these memories into largescale systems capable of generalpurpose quantum computing,^{1,2,3} quantum simulation,^{4} and metrology near the quantum limit.^{5} A promising approach is entangling the atomic qubits with optical links to generate cluster states, which can perform generalpurpose quantum computing with adaptive singlequbit measurements.^{6} A key challenge is to produce this cluster state fast enough to allow computation and error correction within a coherence time of memories.
Here, we show that percolation of heralded entanglement allows us to create arbitrarily large cluster states. Our cluster statebased architecture does not require reconfiguration of physical qubits by the result of the entanglement success unlike conventional approaches.^{7} Instead, the optical network reconfigures the connectivity. Meanwhile, the concept of percolation greatly relaxes the high success probability required in the previous proposals.^{1,2,3} The process is fast enough for implementation with the device parameters demonstrated; one does not need high cooperativity cavities and ancilla single photons. It requires ‘feedforward’, operations conditioned on the previous measurement results. In the absence of errors, the missing bonds can be compensated with constant overhead.^{8,9,10,11} In the presence of errors, the scheme can be adapted to be faulttolerant.^{12,13} Our approach also provides tolerance for site imperfections, and we show tradeoffs between reaching percolation threshold and sitelossresilience. Combined with transparent nodes implemented on nanophotonic platform, longrange connections are possible, reducing the percolation threshold. We derive a theoretical lower bound on the minimum time required to percolate in any lattice, and found that the proposed lattices are only a factor of \(1.6 \sim 3\) above this limit.
We focus on nitrogen vacancy (NV) centers in diamond^{14} and propose the control sequence to map the physical properties to cluster state quantum computation. NV centers have favorable properties as quantum memories. The NV\({}^{}\) charge state has a robust optical transition for heralded entanglement between remote NV centers^{15,16} and a long electronic spin (\(S\)=1) coherence time approaching 1 second.^{17} Recently, singlequbit gates with fidelities up to 99\(\%\) were achieved with optimal control techniques.^{18} Electronic spins of NV centers can be coupled with nearby nuclear spins, which have coherence times exceeding 1 minute.^{19} In addition, they can be coupled with integrated nanophotonic devices.^{20}
Our approach possibly applies to a number of physical systems, including atomic gases,^{21} ion traps,^{22} semiconductor quantum dots,^{23} and rare earth ions.^{24}
Results
Protocol
Figure 1 illustrates the percolation approach to generate cluster states with NV centers. We work in the framework of cluster states where nodes represent qubits in the superposition state \((\left0\right\rangle +\left1\right\rangle )/\sqrt{2}\) and bonds represent the controlledZ gate. Consider a square lattice where edges exist with probability \(p\) (Fig. 1(a–c)). The computational power of the cluster state corresponding to the graph is related to the size of the largest connected component (LCC) (shown in red). When \(p\;<\;0.5\), the graph produces small disconnected islands. For a lattice with \(N\) nodes, the size of the LCC is \(O(\mathrm{log}(N))\).^{25} Singlequbit measurements on such a cluster state can be efficiently simulated on classical computers. When the bond probability \(p\) exceeds the critical value \({p}_{c}=0.5\), called the percolation threshold of the lattice, there is a sudden transition in the size of the LCC. On crossing this boundary, the size of the LCC jumps to \(\Theta (N)\) and the lattice is ‘percolated’. A large percolated lattice can be ‘renormalized’ with singlequbit measurements to obtain a perfect lattice.^{8} Renormalization consumes a constant fraction of qubits^{8} and requires classical computation (pathfinding).^{26,27} Resulting perfect lattice is a resource for universal quantum computation. Thus, the percolation transition is accompanied by a sudden transition in computational power; adaptive singlequbit measurements on the cluster state enable universal quantum computing.^{11}
Figure 1d shows the physical implementation of bond creation (entanglement) with NVs. The nuclear spins (red spheres) are ‘client qubits’ that store entanglement. They are coupled to NV electronic spins (‘broker qubits’), that can be entangled remotely by photonmediated Bell measurements. In each time step, we attempt to create one bond (entanglement) at each node. We consider the Barrett–Kok protocol^{28} on the broker qubits of neighboring nodes. If the probabilistic Bell measurement succeeds, the electron spins in each node are entangled. This entanglement is transferred to the nuclear spins with the entanglement swapping procedure, as illustrated in Fig. 1(d)^{1}. The whole cycle from initialization to entanglement swapping is assumed to be approximately t_{0} = 5 µs based on experimental demonstrations^{16} (See Supplemental Material, which includes a discussion of emitters coupled with nanophotonic circuits,^{29} ultrasmall mode volume cavities,^{30,31} waveguideintegrated singlephoton detectors,^{32} low jitter detectors,^{33} effects of frequency mismatch between two emitters,^{34,35} spectral diffusion,^{36,37} nitrogen implantation,^{38,39} and charge state initialization^{40,41}).
If the Bell measurement fails, one needs to restore the nuclear spins as before the trial. In addition, the electronic spins should be initialized to spin ground state without disrupting the nuclear spin. For the first problem, we just wait for the nuclear spin and electronic spin to be decoupled, which happens after a time period of the hyperfine interaction. This time period serves as a global clock and can be synchronized across the whole graph only using the \({}^{15}\)N nuclear spin, which is the host atom of NV centers. For the electron spin initialization, optical pumping technique cannot be used as in many experiments,^{15,16} because hyperfine coupling persists during a random amount of time when the electronic spin is in m_{s} = 1 states. Instead, we readout the spinstate via m_{s} = 0 optical transition where the electronic spin decouples from the nuclear spin. If the spin is in the other state, one can use a fast microwave pulse to put it in the spin ground state (See Supplemental Material).
We assume the Barrett–Kok protocol because it does not require ancilla single photons or high cooperativity cavities, unlike reflectionbased protocols.^{1} Furthermore, unlike singlephoton detection protocols,^{42} photon loss does not degrade the entanglement fidelity \(\left\langle \Psi \right\hat{\rho }\left\Psi \right\rangle\) where \(\Psi\) is the Bell state, and \(\hat{\rho }\) is density matrix operator of the system after successful heralding. High entanglement fidelity is important for reducing the error correction overhead in a faulttolerant architecture. High fidelity comes at the price of low bond success probability,^{28} that can be overcome in the percolation based architecture.
Physical unit
The physical unit cell could be very small, on the order of tens of microns, so that the entire lattice may be integrated on a chip, as illustrated in Fig. 2. Each node in the architecture requires an atomic memory and a \(1\times d\) switch, where \(d\) is the number of nearest neighbors in the underlying lattice, i.e., the degree of the lattice. Each bond in the lattice requires waveguides, a beamsplitter, and two detectors for the Bell measurement.
At each time step, a stateselective optical \(\pi\)pulse entangles a photonic mode with the electronic spin. Photonic modes coupled to neighboring electronic spins undergo probabilistic linear optic Bell measurement. Successful Bell measurement corresponds to the creation of a bond in the lattice.
Let us define \({n}_{{\rm{a}}tt}\) as the total number of entanglement trials per node. Then each bond is attempted \({n}_{{\rm{a}}tt}/d\) times. If a bond is created before \({n}_{{\rm{a}}tt}/d\) allocated trials, the node idles on the rest of the attempts for that bond. Each switch only needs to be flipped \(d1\) times during cluster generation, and hence the switching time is negligible compared with the whole process. For example, electrooptic modulators can switch at subnanosecond time scales, but the time spent on each bond is on the order of milliseconds as we will see.
The probability of successfully heralding entanglement of two NV centers is \({p}_{0}={\eta }^{2}/2\),^{28} where \(\eta\) is the efficiency of emitting, transmitting, and detecting the photon entangled with the electronic spin (zero phonon line) from the NVexcited state. Table 1 summarizes \({p}_{0}\) for three representative types of emitterphoton interfaces: lowefficiency interfaces with \({p}_{0}=5\times 1{0}^{5}\) representative of today’s stateof the art circular gratings or solid immersion lenses (SILs),^{15,16,43} mediumefficiency interfaces with \({p}_{0}=2\times 1{0}^{4}\) for NV centers coupled to diamond waveguides,^{20} and highefficiency \({p}_{0}=5\times 1{0}^{2}\) for nanocavitycoupled NV centers.^{44} For all three coupling mechanisms, we assumed coupling efficiencies that are realistic today (See Supplemental Material). After \({n}_{{\rm{a}}tt}/d\) entanglement attempts with a nearest neighbor, the probability of having generated a bond is \(p=1{(1{p}_{0})}^{{n}_{{\rm{a}}tt}/d}\).
Percolation thresholds
We evaluate the growth of clusters using the NewmanZiff algorithm with nine million nodes. Figure 3a plots the fraction of nodes that are in the largest cluster component (\({f}_{{\rm{L}}CC}\)), as a function of time from the start of the protocol for the three values of \({p}_{0}\), assuming t_{0} = 5 µs, with an underlying square lattice. Initially, \({f}_{{\rm{L}}CC}\) is O(\(\mathrm{log}(N)/N\))^{25} where \(N\) is the total number of nodes in the lattice. As the bond success probability passes the bond percolation threshold (\({p}_{c}\)), \({f}_{{\rm{L}}CC}\) rapidly rises approaching \(\Theta (1)\). For a degree \(d\) lattice, the bond probability after time \(t\) is \(p=1{(1{p}_{0})}^{t/{t}_{0}d}\). The time required to obtain a resource for universal quantum computation is \({t}_{c}={t}_{0}d\mathrm{ln}(1{p}_{c})/\mathrm{ln}(1{p}_{0})\), which is shown with vertical dashed lines in the figure. The transition becomes sharper as the number of nodes in the lattice (\(N\)) increases.
In all collection schemes, the bond success probability exceeds the percolation threshold within 1 second, which is a conservative estimate of the nuclear spin coherence time in nanostructured, integrated systems based on the experimentally demonstrated coherence time, 1 minute.^{19} These simulations reveal that an arbitrarily large cluster can be generated even with free space couplings.
Degree of the lattice and imperfection
It is known that higher degree lattices have a lower percolation threshold. However, \({t}_{c}\) is nearly the same for the honeycomb (\(d=3\)), the square (\(d=4\)) and the triangular (\(d=6\)) lattices (Fig. 3b). This is because increasing \(d\) lowers the bond percolation threshold, but it also decreases the number of entanglement attempts per bond, \({n}_{{\rm{a}}tt}/d\); a single broker qubit per NV requires entanglement attempts to proceed serially. Increasing \(d\) would in fact substantially lower \({t}_{c}\) if each site contained multiple broker qubits that could be entangled simultaneously. We do not explore the possibility of multiNV nodes owing to a lack of studies.^{45} Ion trap systems are presently more mature for this purpose.^{2}
Let’s consider the most general case where we can attempt Bell measurements on any pair of NVs at any time step. What is the minimum time, \({t}_{c}^{(LB)}\), required to obtain a resource for universal quantum computation over all lattice geometries? The bond probability after time \(t\) is \(p=1{(1{p}_{0})}^{t/{t}_{0}d}\). For percolation, \(p\ge {p}_{c}\), i.e., \(t\ge {t}_{0}d\cdot \mathrm{ln}(1{p}_{c})/\mathrm{ln}(1{p}_{0})\). For a degree \(d\) lattice, \({p}_{c}\ge 1/(d1)\),^{46} with equality for a degree\(d\) Bethe lattice (infinite tree with fixed degree at each node). This leads to \({t}_{c}\ge {t}_{0}d\cdot \mathrm{ln}(11/(d1))/\mathrm{ln}(1{p}_{0})\). \({t}_{c}\) is minimized as \(d\to \infty\) in which case we obtain \({t}_{c}^{(LB)}={t}_{0}/\mathrm{ln}(1{p}_{0})\). \({t}_{c}^{(LB)}\) is plotted as a black dashed line in Fig. 3b. The lattice corresponding to the lower bound is the infinitedegree Bethe lattice, and such lattices are not a resource for universal quantum computing.^{47} Meanwhile, we find that the simple 2D lattices with nearest neighbor connectivity are only a factor \(\,3\) above this limit and are resources for universal quantum computing.
Practically, a scalable architecture should be able to tolerate nonfunctional sites. For example, trapping in a metastable state, a fardetuned transition, failed charge initialization greatly reduced the system performance in the recent stateofthe art experiment.^{48} Even if all faulty nodes and the bonds connected to it are removed, the lattice can retain enough bonds for a percolated lattice (Fig. 4a, inset). The problem maps to sitebond percolation. We define the site yield \(q\) as the fraction of functional nodes. Figure 4a plots the minimum time required to obtain a percolated lattice as a function of \(q\), assuming NVs coupled to diamond waveguides (\({p}_{0}=0.02 \%\)). In general, a reduced site yield can be compensated with a larger bond probability, which would require a longer time (more attempts). The site percolation threshold, \({q}_{c}\), corresponds to the minimum possible site yield for percolation with all bonds having succeeded (\(p=1\)). The triangular (\({q}_{c}=0.5\)) performs better than the square lattice (\({q}_{c}\approx 0.593\)) that outperforms the honeycomb lattice (\({q}_{c}\approx 0.697\)).
The architecture that we have discussed thus far only allows for nearest neighbor interaction. Adding longrange connections shown in the inset of Fig. 4b can decrease the threshold time and increase tolerance to imperfect site yield. Such an architecture can be implemented by replacing the \(1\times 4\) switch in Fig. 2 with a \(5\times 5\) optical switch, which is depicted in Supplementary Fig. S1. Seven MachZehnder interferometers (MZIs) with two phase shifters can implement switching between the set of input and output modes (Fig. S1c). The MZI arrays allow a node to be ‘transparent’. If a node is transparent, photons pass through the node to the next adjacent node for interference (Fig. 4b inset). By turning multiple adjacent nodes transparent, this bypass enables longrange entanglement in a planar architecture.
Interestingly, this transparent node architecture can be used to reduce percolation threshold by randomly turning a fraction \(1\epsilon\) of the nodes transparent. Figure 4b plots \({f}_{{\rm{L}}CC}\) vs. time. As \(\epsilon\) decreases, the maximum possible value of \({f}_{{\rm{L}}CC}\) is reduced from one to \(\epsilon\) because only a fraction \(\epsilon\) nodes are active. However, reduced \(\epsilon\) decreases \({t}_{c}\) because the transparent nodes increase the effective dimensionality of the lattice. Therefore, there is an optimum value of \(\epsilon\) that maximizes LCC for a given time. We numerically found a minimum \({p}_{c}\) of \(0.33\) with transparent nodes, which is achieved when \(1/N\ll \epsilon \ll 1\), i.e., \(\epsilon \to 0\) but the number of nontransparent nodes in the lattice is still \(\Theta (N)\). Faulty sites can be incorporated into the fraction of transparent nodes as long as the yield exceeds \(1/N\).
Small size system
We used nine million qubits (nodes) in the simulations above in order to show the limiting behavior of the cluster creation. However, the same qualitative behavior is also observed in smaller systems. In Fig. 5a, we plot the mean (line) and the standard deviation (shade) of \({f}_{LCC}\) as a function of time for a \(5\times 5\) (green) and \(10\times 10\) (red) square lattices (\({p}_{0}=0.02 \%\)). In the simulation, we generated 300 lattice instances at each time with a periodic boundary condition and calculated the statistical mean and standard deviation. The transition from small disconnected islands to a large cluster on the order of the lattice size becomes sharper as the size of the system increases, but even for a \(5\times 5\) qubit system, there is a clearly visible transition near the percolation threshold (dashed line). Because of the small system size, there is statistical variation in \({f}_{LCC}\) and the shaded regions represent one standard deviation. As expected, the relative variation become smaller in larger systems.
Fault tolerance
It is possible to obtain a regular lattice from the percolated lattices in Fig. 3 with a constant overhead by finding crossing paths and using singlequbit measurements (renormalization).^{8,9,10,11} The singlequbit measurements used to obtain the regular lattice may bring additional errors, but we can adapt our architecture to the Raussendorf lattice.^{12,13} The Raussendorf lattice is a 3D lattice with degree4 (Fig. S3a) and a means of translating surface code error correction into the cluster state model of quantum computation. The Raussendorf lattice can be constructed in a 2+1D architecture^{49} where qubits are arranged in 2D, and an additional dimension is constructed in time (Fig. S3 b)^{1}. Then, one needs only two layers of 2D lattices to implement a 2+1D lattice because memories are reused after measurement (Fig. S3c). This architecture requires a small number of waveguidecrossings, which can be implemented with low loss.^{50} Alternatively, they can be absorbed in the optical switches.
Recent results have evaluated the fault tolerance in the Raussendorf lattice with nondeterministic entangling gates.^{26,51} Following,^{51} fault tolerance requires the bond probability (\(p\)) and measurement error to be in the two shaded regions in Fig. 5b, which correspond to adaptive and nonadaptive measurement schemes; in the nonadaptive scheme, qubits are measured in the \(X\)basis regardless of the failure of bonds; in the adaptive scheme, one of the qubits connected to the missing bond is measured in the \(Z\)basis transforming bondloss to siteloss. The error threshold of each scheme vanishes at \(p \sim 0.935\) and \(p \sim 0.855\), respectively. These correspond to the site percolation threshold of the Raussendorf lattice (\({q}_{c} \sim 0.75\)) when lost bonds are transformed to missing sites (see more details in ref. ^{51}).
We assume that the singlequbit measurement error probability increases with time as \(1{e}^{t/{t}_{coh}}\), where we use \({t}_{coh}=1\) sec. At \(t=0\), the measurement error probability and bond probability (\(p\)) are both zero. Both of these probabilities increase as we spend more time attempting entanglement generation, resulting in the curves shown in Fig. 5b depending on the entanglement success probability per attempt (\({p}_{0}\)). Assuming a bond trial time \({t}_{0}=5\)\(\mu\)s (Table 1), we find that a value of \({p}_{0}\approx 4\times 1{0}^{3}\) is required to meet the fault tolerance threshold, which corresponds to an NV to detector coupling efficiency of \(\sim 9\)\(\%\).
As an example, we estimated the resource overhead for faulttolerant factorization of 2000bit numbers using Shor’s algorithm (Fig. 6). Various methods from refs. ^{52,53,54,55,56,57,58,59,60,61,62,63,64} are used for the calculation. Reference ^{54} in particular, has the least resource overhead even with the additional overhead from spacetime layout and routing. With the assumptions on the error model described in the Supplementary Discussion, 2000bit Shor’s algorithm requires 2.9 billion (280 million) qubits and 2.2 hours (43 mins) of computation given a physical error rate \(p=1{0}^{3}\,(1{0}^{5})\). Alternatively, 57 million (3.8 million) physical qubits can run the algorithm in a day (half a day) as a result of spacetime tradeoff. Essentially, a large number of high purity \(T\)gates for modular exponentiation in Shor’s algorithm require a formidable amount of physical qubits. On the other hand, a lower bound is set at 12.2 million (1.46 million) physical qubits for 6000 logical qubits (2028 (243) physical qubits per a logical qubit) for distance25 (8) surface code.
Discussion
We assumed here that both bit and phaseflip probabilities are the same. However, our results may be significantly improved using recent work on tailoring the surface code^{65} if the noise is biased. We consider other important properties of NVs in the Supplementary Notes. For example, NVs can be ionized under strong optical pulses, but their charge state can be recovered by optical repumping. We designed the microwave and optical pulse sequences so that the nuclear spin is not disturbed by failed trials and subsequent spinstate initialization. However, precise control of the microwave and optical transitions presents technical challenges and mark an area of active research.^{18,66} Future work should also refine the error model to explicitly include contributions from the higher order internal dynamics of NVs.
Though there have been huge progress in reducing the resource overhead in faulttolerant quantum computing, it still requires a large number of physical qubits and long computation time owing to slow clock cycles. In this aspect, noisy intermediate scale quantum (NISQ) technology^{67} can be a more near to medium term path for our proposal. Two observations are favorable to the NISQ direction; the proposed system well behaves with only 25 qubits as shown in Fig. 5a; the LCC size scales linearly with the total number of qubits in the supercritical regime. Thus, the quantum simulation on this architecture quickly reaches thermodynamic behavior.^{68}
We emphasize that the architecture is compatible with any quantum memories with enough coherence time. Especially, silicon vacancy centers and other group IV emitters such as GeV, SnV, or PbV showed promising emission properties.^{69,70} For example, silicon vacancy centers have higher Debye–Waller factor of \(\sim 0.7\), with narrow inhomogeneous distribution of \(\sim 51\) GHz.^{71} Emission wavelength of emitters can be matched by strain tuning with tuning range upto \(\sim 440\) GHz.^{72} The spin coherence time of \(10\) ms has been shown at dilution fridge temperature.^{73}
Data availability
All the numerical data presented in this paper is the results of C simulations. The code used to generate this data will be made available to the interested reader upon request.
References
Nemoto, K. et al. Photonic architecture for scalable quantum information processing in diamond. Phys. Rev. X 4, 031022 (2014).
Monroe, C. et al. Largescale modular quantumcomputer architecture with atomic memory and photonic interconnects. Phys. Rev. A 89, 022317 (2014).
Nickerson, N. H., Fitzsimons, J. F. & Benjamin, S. C. Freely scalable quantum technologies using cells of 5to50 qubits with very lossy and noisy photonic links. Phys. Rev. X 4, 041041 (2014).
Cai, J., Retzker, A., Jelezko, F. & Plenio, M. B. Towards a largescale quantum simulator on diamond surface at room temperature. Nat. Phys. 9, 168–173 (2013).
Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nat. Photonics 5, 222–229 (2011).
Raussendorf, R. & Briegel, H. J. A oneway quantum computer. Phys. Rev. Lett. 86, 5188–5191 (2001).
Duan, L.M. & Raussendorf, R. Efficient quantum computation with probabilistic quantum gates. Phys. Rev. Lett. 95, 080503 (2005).
Kieling, K., Rudolph, T. & Eisert, J. Percolation, renormalization, and quantum computing with nondeterministic gates. Phys. Rev. Lett. 99, 130501 (2007).
GimenoSegovia, M., Shadbolt, P., Browne, D. E. & Rudolph, T. From threephoton greenbergerhornezeilinger states to ballistic universal quantum computation. Phys. Rev. Lett. 115, 020502 (2015).
Zaidi, H. A., Dawson, C., Van Loock, P. & Rudolph, T. Neardeterministic creation of universal cluster states with probabilistic Bell measurements and threequbit resource states. Phys. Rev. A 91, 042301 (2015).
Browne, D. E. et al. Phase transition of computational power in the resource states for oneway quantum computation. New J. Phys. 10, 023010 (2008).
Raussendorf, R., Harrington, J. & Goyal, K. A faulttolerant oneway quantum computer. Ann. Phys. 321, 2242–2270 (2006).
Raussendorf, R., Harrington, J. & Goyal, K. Topological faulttolerance in cluster state quantum computation. New J. Phys. 9, 199–199 (2007).
Doherty M. W. et al. The nitrogenvacancy colour centre in diamond. Phys. Rep. 528, 1 (2013).
Bernien, H. et al. Heralded entanglement between solidstate qubits separated by three metres. Nature 497, 86–90 (2013).
Hensen, B. et al. Loopholefree Bell inequality violation using electron spins separated by 1.3 kilometres. Nature 526, 682–686 (2015).
Abobeih, M. H. et al. Onesecond coherence for a single electron spin coupled to a multiqubit nuclearspin environment. Nat. Commun. 9, 2552 (2018).
Dolde, F. et al. Highfidelity spin entanglement using optimal control. Nat. Commun. 5, 345–380 (2014).
Bradley, C. et al. A 10qubit solidstate spin register with quantum memory up to one minute. arXiv:1905.02094 (2019).
Mouradian, S. L. et al. Scalable integration of longlived quantum memories into a photonic circuit. Phys. Rev. X 5, 031009 (2015).
Chou, C. W. et al. Measurementinduced entanglement for excitation stored in remote atomic ensembles. Nature 438, 828–832 (2005).
Moehring, D. L. et al. Entanglement of singleatom quantum bits at a distance. Nature 449, 68–71 (2007).
Delteil, A. et al. Generation of heralded entanglement between distant hole spins. Nat. Phys. 12, 218–223 (2015).
Kolesov, R. et al. Optical detection of a single rareearth ion in a crystal. Nat. Commun. 3, 1029 (2012).
Bazant, M. Z. Largest cluster in subcritical percolation. Phys. Rev. E 62, 1660 (2000).
Herr, D., Paler, A., Devitt, S. J. & Nori, F. A local and scalable lattice renormalization method for ballistic quantum computation. npj Quantum Inf. 4, 27 (2018).
MorleyShort, S. et al. Physicaldepth architectural requirements for generating universal photonic cluster states. Quantum Sci. Technol. 3, 015005 (2017).
Barrett, S. D. & Kok, P. Efficient highfidelity quantum computation using matter qubits and linear optics. Phys. Rev. A 71, 060310 (2005).
Sipahigil, A. et al. An integrated diamond nanophotonics platform for quantumoptical networks. Science 354, 847–850 (2016).
Choi, H., Heuck, M. & Englund, D. Selfsimilar nanocavity design with ultrasmall mode volume for singlephoton nonlinearities. Phys. Rev. Lett. 118, 223605 (2017).
Choi, H., Zhu, D., Yoon, Y. & Englund, D. Indistinguishable singlephoton sources with dissipative emitter coupled to cascaded cavities. arXiv:1809.01645 (2018).
Marsili, F. et al. Siliconvacancy spin qubit in diamond: a quantum memory exceeding 10 ms with singleshot state readout. Nat. Photonics 7, 210–214 (2013).
Calandri, N., Zhao, Q. Y., Zhu, D., Dane, A. & Berggren, K. K. Superconducting nanowire detector jitter limited by detector geometry. Appl. Phys. Lett. 109, 152601 (2016).
Choi, H., Zhu, D., Yoon, Y. & Englund, D. Cascaded cavities boost the indistinguishability of imperfect quantum emitters. Phys. Rev. Lett. 122, 183602 (2019).
Metz, J. & Barrett, S. D. Effect of frequencymismatched photons in quantuminformation processing. Phys. Rev. A 77, 042323 (2008).
Ruf, M. et al. Optically coherent nitrogenvacancy centers in μmthin etched diamond membranes. Nano Lett 19, 3987 (2019).
Chu, Y. et al. Coherent optical transitions in implanted nitrogen vacancy centers. Nano Lett. 14, 1982–1986 (2014).
Rabeau, J. R. et al. Implantation of labelled single nitrogen vacancy centers in diamond using N_{15}. Appl. Phys. Lett. 88, 023113 (2006).
Bayn, I. et al. Generation of ensembles of individually resolvable nitrogen vacancies using nanometerscale apertures in ultrahighaspect ratio planar implantation masks. Nano Lett. 15, 1751–1758 (2015).
Siyushev, P. et al. Optically controlled switching of the charge state of a single nitrogenvacancy center in diamond at cryogenic temperatures. Phys. Rev. Lett. 110, 167402 (2013).
Ji, P. & Dutt, M. V. G. Charge state dynamics of the nitrogen vacancy center in diamond under 1064nm laser excitation. Phys. Rev. B 94, 024101 (2016).
Cabrillo, C., Cirac, J. I., GarciaFernandez, P. & Zoller, P. Creation of entangled states of distant atoms by interference. Phys. Rev. A 59, 1025 (1999).
Li, L. et al. Efficient photon collection from a nitrogen vacancy center in a circular bullseye grating. Nano Lett. 15, 1493–1497 (2015a).
Li, L. et al. Coherent spin control of a nanocavityenhanced qubit in diamond. Nat. Commun. 6, 6173 (2015b).
Dolde, F. et al. Roomtemperature entanglement between single defect spins in diamond. Nat. Phys. 9, 139 (2013).
Broadbent, S. R. & Hammersley, J. M. Percolation processes. Math. Proc. Camb. Philos. Soc. 53, 629–641 (1957).
Van den Nest, M., Miyake, A., Dür, W. & Briegel, H. J. Universal resources for measurementbased quantum computation. Phys. Rev. Lett. 97, 150504 (2006).
Humphreys, P. C. et al. Deterministic delivery of remote entanglement on a quantum network. Nature 558, 268 (2018).
Rudolph, T. Why I am optimistic about the siliconphotonic route to quantum computing. APL Photonics 2, 030901 (2017).
Han, H.L. et al. High performance ultracompact soi waveguide crossing. Opt. Express 26, 25602 (2018).
Auger, J. M., Anwar, H., GimenoSegovia, M., Stace, T. M. & Browne, D. E. Faulttolerant quantum computation with nondeterministic entangling gates. Phys. Rev. A 97, 030301 (2018).
Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: towards practical largescale quantum computation. Phys. Rev. A 86, 032324 (2012).
Gidney, C. & Fowler, A. G. Efficient magic state factories with a catalyzed ccz to 2t transformation. arXiv:1812.01238 (2018).
Gidney, C. & Ekera, M. How to factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits. arXiv:1905.09749 (2019).
Vedral, V., Barenco, A. & Ekert, A. Quantum networks for elementary arithmetic operations. Phys. Rev. A 54, 147 (1996).
Beauregard, S. Circuit for shor’s algorithm using 2n+ 3 qubits. Quantum Inf. Comput. 3, 175 (2002).
Cuccaro, S. A., Draper, T. G., Kutin, S. A., Moulton, D. P. A new quantum ripplecarry addition circuit. arXiv:quantph/0410184 (2004).
Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, 2010).
Jones, C. Lowoverhead constructions for the faulttolerant toffoli gate. Phys. Rev. A 87, 022328 (2013).
Gidney, C. Halving the cost of quantum addition. Quantum 2, 10–22331 (2018).
Bravyi, S. & Kitaev, A. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A 71, 022316 (2005).
Horsman, C., Fowler, A. G., Devitt, S. & Van Meter, R. Surface code quantum computing by lattice surgery. New J. Phys. 14, 123011 (2012).
Fowler, A. G. & Gidney, C. Low overhead quantum computation using lattice surgery. arXiv:1808.06709 (2018).
Gidney, C. & Fowler, A. G. Flexible layout of surface code computations using autoCCZ states. arXiv:1905.08916 (2019).
Tuckett, D. K., Bartlett, S. D. & Flammia, S. T. Ultrahigh error threshold for surface codes with biased noise. Phys. Rev. Lett. 120, 050505 (2018).
Reiserer, A. et al. Robust quantumnetwork memory using decoherenceprotected subspaces of nuclear spins. Phys. Rev. X 6, 021040 (2016).
Preskill, J. Quantum computing in the nisq era and beyond. Quantum 2, 79 (2018).
Mohammady, M. H. et al. Lowcontrol and robust quantum refrigerator and applications with electronic spins in diamond. Phys. Rev. A 97, 042124 (2018).
Trusheim, M. E. et al. Leadrelated quantum emitters in diamond. Phys. Rev. B 99, 075430 (2019).
Trusheim, M. E. et al. Transformlimited photons from a tinvacancy spin in diamond. arXiv:1811.07777 (2018).
Schröder, T. et al. Scalable focused ion beam creation of nearly lifetimelimited single quantum emitters in diamond nanostructures. Nat. Commun. 8, 15376 (2017).
Sohn, Y.I. et al. Controlling the coherence of a diamond spin qubit through its strain environment. Nat. Commun. 9, 2012 (2018).
Sukachev, D. D. et al. Siliconvacancy spin qubit in diamond: a quantum memory exceeding 10 ms with singleshot state readout. Phys. Rev. Lett. 119, 223602 (2017).
Acknowledgements
We thank Simon Devitt for helpful comments on the manuscript. H.C., M.P., and D.E. acknowledge support from the Air Force Office of Scientific Research MURI (FA95501410052) and the Army Research Laboratory (ARL) Center for Distributed Quantum Information (CDQI). H.C. and D.E. acknowledge support from the Defense Advanced Research Projects Agency (DARPA) DRINQS (HR001118S0024) and the National Science Foundation (NSF) RAISE TAQS (CHE1839155) and EFRI ACQUIRE (EFMA1838911). H.C. was also supported in part by a Samsung Scholarship. S.G. acknowledges support from the Office of Naval Research MURI (N0001416C2069).
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H.C. and M.P. contributed equally to this work. H.C., M.P. and D.E. conceived the idea. M.P. derived the analytical limits. H.C. and D.E. analyzed the properties and entanglement protocols of NVs as a physical layer of the architecture. H.C. estimated resource overheads. H.C, M.P., and S.G. developed the simulation codes. D.E. supervised the project.
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Choi, H., Pant, M., Guha, S. et al. Percolationbased architecture for cluster state creation using photonmediated entanglement between atomic memories. npj Quantum Inf 5, 104 (2019). https://doi.org/10.1038/s4153401902152
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DOI: https://doi.org/10.1038/s4153401902152
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