Abstract
Despite linearoptical fusion (Bell measurement) being probabilistic, photonic cluster states for universal quantum computation can be prepared without feedforward by fusing small nphoton entangled clusters, if the success probability of each fusion attempt is above a threshold, \({\mathrm{\lambda }}_{\mathrm{c}}^{(n)}\). We prove a general bound \({\mathrm{\lambda }}_{\mathrm{c}}^{(n)} \ge 1/(n  1)\), and develop a conceptual method to construct longrangeconnected clusters where \({\mathrm{\lambda }}_{\mathrm{c}}^{(n)}\) becomes the bond percolation threshold of a logical graph. This mapping lets us find constructions that require lower fusion success probabilities than currently known, and settle a heretofore open question by showing that a universal cluster state can be created by fusing 3photon clusters over a 2D lattice with a fusion success probability that is achievable with linear optics and single photons, making this attractive for integratedphotonic realizations.
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Introduction
Optical qubits are a promising candidate for scalable universal quantum computing because of the relative ease of generating photons, fabricating photonic circuits^{1,2,3}, and low decoherence rates. Assembling a universal photonic cluster state, e.g., by fusing small entangled clusters using linear optical interferometers into progressively larger ones, is limited by probabilistic operations because of the limitations of linear optics. For example, a linearāoptical circuit for a twoqubit Bellstate measurement (BSM)āthe smallest primitive to āfuseā two cluster statesācan succeed with at most probability 1/2ā=ā0.5^{4,5}. However, since photons are the base resource to encode qubits anyway, it is reasonable to allow photons to be used as an ancilla resource to boost the success probability of linearāoptic BSMs^{6,7,8}. The maximum known BSM success probability attainable using linear optics, boosted by injecting eight ancilla single photons, is 25/32ā=ā0.78125^{8}. It is not known if a higher BSM success probability is possible with linear optics boosted with more single photons. Very little is understood about similar successprobability limits associated with linearāoptical realizations of larger (three or more qubits) projective measurements, and those for creating small photonic entangled clusters^{9}. The lack of our understanding of the maximum efficiency in creating universal photonic cluster states goes back to the mathematical structure of manipulations of multiphoton entangled states using linear optics and photon detection being complex, as evidenced by the hardness of boson sampling^{10}.
Linear optical quantum computing (LOQC) in its widely known circuitmodel formācoined by the seminal KLM paper^{11}āemploys singlephoton dualrail qubits (0ćāā”ā10ć, 1ćāā”ā01ć), probabilistic linearāoptical gates, singlephoton detection, and measurementinduced feedforward. Even though KLMās scheme in principle can do universal quantum computing, the overheads are astronomically high. Many variants of KLM were proposed^{12,13,14} that use entangled ancillas to reduce the overheads, but the hardness was pushed into creating those ancillas. Schemes for efficient generation of these ancillas from simpler resources such as single photons remain unknown. Furthermore, the need for detectioninduced feedback creates additional experimental issues. This resulted in LOQC being largely disfavored in lieu of other qubit technologies, such as superconducting and trappedion qubits.
Kieling, Rudolph, and Eisert proposed a clustermodel version of LOQC^{15}, which mitigates these issues. Their scheme places d + 1 photon āstarā clusters at nodes of a regular lattice G with degreed nodes and stitches them together using twophoton fusions (BSMs) along the lattice edges (see Fig. 1). If the success probability of each fusion attempt Ī» is above the bond percolation threshold p_{c}(G) of the underlying lattice G, the resulting cluster state will be a longrangeconnected (random) subgraph of G, which can then be renormalized into a universal cluster state by identifying a subarray of nodes in the percolated latticeāthe intersection points of vertical and horizontal āpercolation highwaysā^{16,17}āto serve as the nodes of a logical universal lattice and identifying paths connecting them through other nodes of the percolated subgraph as edges of this logical lattice (see Fig. 2a). The inspiration behind this model is that if small d + 1 photon clusters can be created, and a lattice can be identified whose bond percolation threshold is below a practically attainable fusion success probability, then the clusters can be linearāoptically fused into a large universal cluster state in a oneshot (or āballisticā) fashion, with no detectioninduced feedforward. In order to produce conservative estimates for the fusion success probability required to produce a renormalizable cluster state, this work disregarded the āfailure modesā of the fusion attempts, i.e., the leftover cluster states when fusion attempts fail.
Significant progress in this model of LOQC came recently from Rudolph and collaborators, who by paying closer attention to the failure modes, and a clever design of 3D lattices with a lowpercolation threshold, e.g., the diamond^{18} and the pyrochlore lattice^{19}, were able to show that ballistic creation of a universal cluster state is possible starting with a large supply of threephotonentangled (GHZ state) clusters, with twophoton fusions that succeed with a probability ā0.625. Since 0.625ā<ā25/32ā=ā0.78125, the bestknown success probability attainable with linear optics boosted with unentangled singlephoton ancillas^{8}, this gave the first realizable ballistic construction of a universal cluster if threephoton GHZ clusters were available as a starting point.
The results of GimenoSegovia et al. and Zaidi et al.^{18,19} on ballistic (feedbackfree) creation of universal clusters, in conjunction with a host of progress in experimental proposals to directly create small photonic entangled clusters using quantum emitters^{20,21,22} (rather than starting with single photons and fusing them probabilistically using heralded methods into larger clusters), have resurrected the optimism in scalable optical quantum computing. However, the fundamental limits of efficiency and resource overhead associated with this philosophyāthat of direct generation of a universal cluster state using small photon clusters and linear opticsāremain unanswered. Refs.ā^{18,19} present two (clever, yet ad hoc) examples of constructions that use threephoton clusters as the starting point.
The goal of this paper is to explore this fundamental limit. We address the following general question:
Given an unlimited supply of nphotonentangled dualrailbasis clusters, and repeated use of a linearāoptical circuit that realizes twoqubit fusion with probability of success Ī», but with no restrictions on how the clusters to be fused are aligned in space and time, what is the minimum value \({\lambda }_{\mathrm{c}}^{(n)}\) of each fusionās success probability that will allow for ballistic creation of a universal cluster state?
Before we proceed, it should be clear that the two examples we discussed aboveāone in Fig. 1^{15} and that in ref.ā^{18}āestablish the following bounds:
and
where we use ā² sign instead of ā¤ in Eq. (2), since the 0.625 threshold was an approximate numerical estimation of the threshold using a smallsized diamond lattice^{18}.
We focus on Bellmeasurementbased fusion operations. Although Browne and Rudolphās original paper that coined the term āfusionā^{13} also proposed an alternative linear optical fusion circuit (called fusionI), subsequent work has focussed on Bellmeasurementbased fusion (fusionII of ref. ^{13}) because of its natural loss tolerance and recent progress in boosting its success probability by injecting unentangled singlephoton ancillas^{8}.
In this paper, we prove that starting with nphoton clusters, a fusion success probability of at least 1/(nā1) is required to create a resource state for universal quantum computation without feedforward. We map a planned set of fusion operations on photon pairs among a regular spatial array of nphoton clustersāthe clusterfusion lattice, G_{1}āto an instance of bond percolation on a logical lattice G_{2}. We use this map to find new constructions to produce universal photonic cluster states starting with threephoton cluster states as the initial resource, without feedforward, that require a lower fusion success probability than was previously known. Finally, we settle a heretofore open question by showing that a universal cluster state can be created by fusing threephoton clusters over a 2D lattice with a fusion success probability that is achievable with linear optics and single photons, making this attractive for integratedphotonic realizations.
Results
Summary of the main results
The main results in this paper are as follows:

1.
We prove a general lower bound \({\lambda }_{\mathrm{c}}^{(n)} > 1/(n  1)\), and provide evidence in favor of our conjecture that it is tight (achievable as closely as we wish).

2.
We develop a systematic graphical construction of a universal cluster, wherein \({\lambda }_{\mathrm{c}}^{(n)}\) becomes the usual bondpercolation threshold of a logical graph that we specify, while accounting for the fusion failure modes. This is a key realization that was missing in both Kieling et al.ās construction^{15} as well as in refs. ^{18,19}. This not only enabled us to readily generate achievable thresholds (upper bounds on \({\lambda }_{\mathrm{c}}^{(n)}\)) using existing literature on bond percolation, but also helped us to prove the aforesaid lower bound. We believe that this logical construction also naturally generalizes to future extensions that may employ multiqubit fusions (projective measurements). We describe this construction systematically in the section āReconciling prior results in a new frameworkā, and present several improved thresholds using this new formalism. However, we develop a fully axiomatic construction of this logical graph for the most general case if left open for future work.

3.
Using our logical construction, we prove \({\lambda }_{\mathrm{c}}^{(3)} \le 0.5898\) as an achievable threshold, which improves upon the best existing result in ref. ^{18}. As a byproduct of our logical construction allowing a standard bondpercolation interpretation, we were able to reinterpret the construction of ref. ^{18} in our formalism and used Newman and Ziffās efficient Monte Carlo algorithm to refine their threshold (2) to \({\lambda }_{\mathrm{c}}^{(3)} \le 0.627\).

4.
We settle a heretofore open question posed by Rudolph and collaborators^{18,19,23} by showing that a universal cluster can be created by linearāoptically fusing threephoton GHZ clusters over a 2D (brickwork) lattice with fusion success probability 0.746, which is below 25/32, the maximum known fusion success probability achievable with a singlephotonboosted linearāoptic scheme^{8}, making this attractive for integratedphotonic realizations.
In the next section, we will begin with a discussion of the highlevel problem setup and the constraints therein, following which we will present a logical technical progression leading to our main results and the key intuitions. Detailed derivations and technical graphical constructions not central to following the main arguments in the paper are presented in the Supplementary Note 2.
Generality and constraints of our problem setup
Figure 3 and its caption explains our highlevel problem setup. It follows from our discussion above that we are limiting ourselves to only those multimode linearāoptical unitaries U that can be pieced together using a sequence of twoqubit linearāoptic BSMs. Our problem formulation, however, subsumes the special cases of the constructions in refs. ^{15,18,19}. Furthermore, our formulation allows for yetundiscovered fusion circuits that may be boosted with up to nphoton entangled clusters (which may attain a success probability greater than the currently best known, 25/32). It also admits generalizations to multiqubit fusion, i.e., mqubit GHZstate projections, the theory behind optimal linearāoptical realizations of which it remains not well understood.
The detectors shown in Fig. 3 are those corresponding to the linearāoptical fusion gates pushed (deferred) all the way to the right with no loss of generality. Similarly, any photons or photon clusters used to boost any fusion gate are pushed to the left of U. This way, one can envision the cluster preparation as a multimode linear optical circuit presented with individual photons and photon clusters as input, and producing (deterministically) a renormalizable universal cluster state as its output. The individual detector outputs are random, but if we design U in a way that the percolation condition is met, the undetected outputs of U would, with high probability, carry a random instance of a universal longrangeconnected (percolated) cluster and the vector of detector outputs would simply tell us what cluster state got created on the undetected outputs, and thus give us the necessary data to compute crisscrossing edgedisjoint paths through the percolated lattice to renormalize it into a universal logical cluster state^{24}.
General lower bounds
In this section, we present an intuitive explanation of our lower bound on \({\lambda }_{\mathrm{c}}^{(n)}\). A detailed formal proof is provided in Supplementary Note 1.
We have the following general lower bound:
Proof sketchāstarting with nphoton clusters, and any sequence of fusion attempts, the resulting instance of the logical graph is a bondpercolation instance, with bond success probability Ī», on some graph of maximum degree n. Of all infinite graphs of maximum degree n, the minimum bondpercolation threshold is that of the degreen Bethe lattice, and equals 1/(nā1). This implies that if Ī»ā<ā1/(nā1), the percolation condition cannot be met no matter what. Hence, \({\lambda }_{\mathrm{c}}^{(n)} \ge 1/(n  1)\).
Since the Bethe lattice is a tree and since trees are not a universal resource, this does not prove 1/(nā1) to be an achievable threshold for universal cluster creation (see Supplementary Note 3 for more details). However, we conjecture (and provide evidence in its favor) that Ī»ā=ā[1/(nā1)] + Īµ with any Īµā>ā0 that may suffice to create a universal cluster state. This conjecture is supported by the fact that starting with N copies of photon clusters of size n and probabilistic operations that succeed with any probability greater than 1/(nā1), it is possible to obtain an O(N)sized tree graph (detailed in Supplementary Note 3). We also present a random graph construction, which provides intuition for why probabilistic operations on photon clusters of size n may provide enough connectivity to obtain a large connected cluster when Ī»ā>ā1/(nā1) (Supplementary Note 3).
We also show that if mqubit fusions are used to fuse nqubit clusters for māā„ā2, the optimum threshold on the fusion success probability required to generate a universal cluster ballistically must satisfy
Very little is known about linearāoptical circuits for mā>ā2 qubit fusion^{9} (e.g., projecting three qubits to one of the eight orthogonal threequbit GHZ states), their associated success probabilities, and various failure outcomes. Thus, it remains unclear if the above bound on \({\lambda }_{\mathrm{c}}^{(n,m)}\) is tight.
Reconciling prior results in a new framework
Our next contribution is a reinterpretation of the constructions in refs. ^{15,18,19} as a standard bondpercolation problem on a logical lattice, which lets us identify (and sharpen) the 0.627 threshold found in ref. ^{18} as the standard bondpercolation threshold of a 3Dmodified (10,3)b lattice^{25}.
We first describe the mapping of a spatially regular arrangement of fusion attempts among photons in nphoton clustersāthe clusterfusion lattice G_{1}āto a logical lattice G_{2}, each of whose nodes correspond to an nphoton cluster. We then show that the standard bondpercolation threshold of G_{2} equals the threshold on the success probability of fusion attempts in G_{1} exceeding, which generates, with high probability, a longrangeconnected (random) photonic lattice \({\mathrm{G}}_1^\prime\) among the unmeasured photons in G_{1}, which is renormalizable for universal quantum computing. The mapping is illustrated via an example shown in Fig. 4. We will restrict, for simplicity, to nā=ā3 photonline clusters as the initial resource to explain the mapping. But the G_{1}toG_{2} mapping we describe below, holds for clusters of any size n and shape.
G_{1} represents the pattern of pairwise fusion attempts we intend to perform on a subset of photons across many threephoton clusters. To construct G_{1}, we begin with (1) a (two or higherdimensional) spatial arrangement of threephoton line clusters, (2) identifying pairs of photons (belonging to distinct clusters), upon which fusions will be attempted, and (3) one of two types for each of those fusions: blue or green, linear optical circuits to realize, which are shown in Figs. 2 and 3 of ref. ^{18}, respectively. Success and failure outcomes of an example green and blue fusion are shown in Fig. 4a, b.
We thus form the clusterfusion lattice G_{1}:

The nodes of G_{1} are photons, colored white or black: referring to whether the photons will be left unmeasured vs. will be measured (and hence destroyed) in fusion attempts.

The edges of G_{1} are of three kinds: black corresponding to the preexisting entanglement between photons within a threephoton cluster, and, blue and green (dashed) edges connecting black nodes corresponding to two types of (planned) fusion attempts.
Now, we construct the logical graph G_{2}, each of whose nodes correspond to one threephoton cluster of G_{1}. The color of a node of G_{2}āwhite, red, blue, and black, is determined based on the number of black (measured) photons in the corresponding photonic cluster in G_{1}: (i) zero, (ii) one, (iii) two, or (iv) three, respectively (see Fig. 4c). There are two kinds of edges in G_{2}: blue or green, corresponding to the two types of fusion attempts.
We impose four conditions on the graphs we consider, which are not required for the above G_{1}āāāG_{2} mapping, but needed for our percolationthreshold equivalence to hold. Each condition can be stated either for G_{1} or G_{2}.

Condition 1: Any loop in G_{2} must contain at least three nonblack nodes.

Condition 2: One endpoint of a bluedashed edge in G_{1} must be the degree2 (middle) node of a threephoton cluster. We will call this node A. The other end node is a degree1 (leaf) node of a threephoton cluster. We call this node B.

Condition 3: Both endpoints of a greendashed edge in G_{1} must be degree1 (leaf) nodes of two threephoton clusters.

Condition 4: For each cluster in G_{1}, all whose photons are measured in fusion attempts, there is at least one cluster with one unmeasured photon at a constant distance (number of hops) away. Translated to G_{2}, this implies that every black node has a nonblack node at a constant distance from it (e.g., one bond away for the example shown in Fig. 4e).
We now describe the action of the blue and green fusion operations on photonic cluster states. In the following discussion, we will use ācluster edgeā and ācluster nodeā to refer to edges and nodes in a photonic cluster state, to avoid any confusion with the logical graph. A ācluster edgeā refers to entanglement between neighboring photons (e.g., black edges in G_{1}) and not fusion attempts (e.g., blue and greendashed edges of G_{1}). A ācluster nodeā is a photon.
After the fusion attempts in G_{1} have been made, all the black cluster nodes in G_{1} are destroyed. If a fusion attempt is successful, then additional cluster edges (depicting a newly created entanglement) appear among cluster nodes that were former neighbors of the measured (black) cluster nodes. Each fusion attempt succeeds or fails with probability Ī» or 1āĪ», respectively. Depending upon the successāfailure outcomes of all the fusion attempts, a new photonic lattice is createdāa random graph state \({\mathrm{G}}_1^\prime\)āwhose cluster nodes are the white cluster nodes (unmeasured photons) of G_{1}.
In order to describe the \({\mathrm{G}}_1 \to {\mathrm{G}}_1^\prime\) mapping, we need to describe the success and failure outcomes of both kinds of fusion. To do so, we need a few additional definitions:

Cluster edge parity (of a pair of cluster nodes): The parity of a cluster edge is 1 if the cluster edge exists, and 0 otherwise.

Neighborhood inversion (on a cluster node): A unitary action on a cluster node in a photonic cluster state, which flips the cluster edge parity between every pair of neighbors of the cluster node. So, some cluster edges can be deleted, and new cluster edges can be created.
Let us now describe the action of the two fusion operations. Either type of fusion attempt destroys the two photons fused, regardless of whether the fusion succeeds.

Blue fusion: Fig. 4a shows an example of a blue fusion on a pair of photons in two threephoton clusters. The blue fusion is asymmetric across the two cluster nodes it acts upon. Let us label the measured cluster nodes A and B. If the fusion succeeds, it flips the cluster edge parity between every neighboring pair of A and B. In other words, if one neighboring cluster node of A and one neighboring cluster node of B had no direct cluster edge between them prior to the fusion attempt, after successful fusion, they get a direct cluster edge between them. Also, if a neighbor of A and a neighbor of B had a direct cluster edge between them before fusion, this cluster edge is removed after successful fusion^{18}. Condition 1 above prevents the latter to ever happen, i.e., we never attempt to fuse two cluster nodes whose neighbors had a direct cluster edge prior to the fusion attempt. If a blue fusion fails, we first perform a neighborhood inversion on cluster node A, i.e., we flip the cluster edge parity between every pair of neighbors of cluster node A and then remove both cluster nodes A and B from the graph.
If the blue fusion shown in Fig. 4a succeeds, we obtain a fourphoton cluster state. If it fails, we perform a neighborhood inversion on cluster node A and remove both measured cluster nodes, leaving the two cluster fragments with two photons in each. Condition 2 above ensures that one of the two cluster nodes for blue fusions is a leaf cluster node of a cluster. This is the B cluster node, which does not undergo neighborhood inversion after a fusion attempt failure.

Green fusion: The green fusion, shown in an example in Fig. 4b, has the same behavior as the blue fusion when it succeeds. As in the case of the blue fusion, we never fuse two cluster nodes whose neighbors already have a direct cluster edge, due to Condition 1 we imposed above. In the case of failure, we simply remove the two measured cluster nodes and all their cluster edges. The green fusions are symmetric, and as stated above in Condition 3, always used between a pair of leaf cluster nodes of two clusters.
We introduced above the (deterministic) lattice G_{1} denoting clusterfusion attempts, and mapped it to a (deterministic) logical lattice G_{2}, whose edges denote fusion attempts. The probabilistic successāfailure fusion outcomes map G_{1} to a (random) photonic lattice \({\mathrm{G}}_1^\prime\) among the unmeasured photons, and map G_{2} to a (random) lattice \({\mathrm{G}}_2^\prime\), which is a standard bondpercolation subgraph of G_{2} where each edge is missing (independent of the other edges) with probability 1āĪ».
Under a given successāfailure fusion outcome pattern, we will now argue that if \({\mathrm{G}}_2^\prime\) has a giantconnected component (GCC), which will happen if Ī»ā>āp_{c}(G_{2}), the bondpercolation threshold of G_{2}, then \({\mathrm{G}}_1^\prime\)āthe photonic lattice involving the unmeasured nodesāwill also have a GCC, which is a lattice renormalizable for universal clustermodel quantum computing.
The black nodes in G_{2} disappear after fusion attempts (they have no photons left in them), but help provide connections between (the unmeasured photons within) the nonblack nodes.
If N is the number of nodes in G_{2}, bond percolation on G_{2}, i.e., Ī»ā>āp_{c}(G_{2}) ensures that \({\mathrm{G}}_2^\prime\)āthe random bondpercolation instance (subgraph) of G_{2}ācontains a unique GCC, i.e., a single subgraph of \({\mathrm{G}}_2^\prime\) has O(N) nodes (black and nonblack nodes combined). However, since every black node has a nonblack node at a constant distance from it (Condition 4 above), the probability of a nonblack node being connected to the nearest black node is a constant. Therefore, the presence of O(N) black nodes in the GCC implies the presence of O(N) nonblack nodes in the GCC, and hence, there must be O(N) nonblack nodes in the GCC of \({\mathrm{G}}_2^\prime\).
Given any successāfailure pattern of all the fusion attempts, two unmeasured photons, i.e., nodes of \({\mathrm{G}}_1^\prime\), have a connected path in \({\mathrm{G}}_1^\prime\) if and only if the corresponding white nodes in G_{1} have a connected path, via black edges and successful fusion (green or bluedashed) edges. This can be seen through the examples shown in Fig. 4a, b. This, and the fact that nonblack nodes of G_{2} map to unmeasured (white) nodes of G_{1}, a GCC with O(N) nonblack nodes in \({\mathrm{G}}_2^\prime\) implies the presence of a GCC of O(N) (white) nodes in \({\mathrm{G}}_1^\prime\).
Hence, if Ī»ā>āp_{c}(G_{2}), the leftover photonic lattice \({\mathrm{G}}_1^\prime\) will have a GCC, which is a lattice renormalizable for universal clustermodel quantum computing. For the G_{2} in Fig. 4e, we numerically get p_{c}(G_{2})āāā0.672. See plot in Fig. 4f. This establishes that \({\lambda }_{\mathrm{c}}^{(3)} \le 0.672\).
It is instructive to note here that if the logical graph is composed only of black and red nodes, the postfusion random photonic cluster state \({\mathrm{G}}_1^\prime\) is a random subgraph of G_{3}: the effective lattice formed by the red nodes alone (each of which are threephoton clusters with one photon unmeasured). The nodes of G_{3} are the red nodes of G_{2}. In the example shown in Fig. 4e, even though G_{2} is a nonplanar twolayer lattice, \({\mathrm{G}}_1^\prime\) is a random subgraph of G_{3}, a planar (2D) square lattice. All the logical graph examples in this paper will have the aforesaid property, i.e., composed of black and red nodes only.
Further intuition regarding the development of the above construction, and an alternative way to view the generation of \({\mathrm{G}}_1^\prime\) as a modified siteābond percolation on G_{1} directly, are provided in Supplementary Note 2.
Equipped with this new formalism, we construct a 3D logical lattice G_{2}āa modified (10,3)b latticeāshown in Fig. 5. In this case, G_{3}, the lattice connecting the red nodes of G_{2} if all the fusions were to succeed, is the 3D diamond lattice. The bondpercolation threshold of the modified (10,3)b lattice, and hence the threshold on fusion success probability that would ensure \({\mathrm{G}}_1^\prime\) to be longrange connected, is p_{c}(G_{2})āāā0.627. Therefore, \({\lambda }_{\mathrm{c}}^{(3)} \le 0.627\), which gives a better (lower) upper bound than the example we examined above.
Let us see how and why G_{3} for this example is in the diamond lattice. Let us assume that all the fusion attempts succeed, so that the black nodes of G_{2} in Fig. 5 are gone. Let us pick any red node and follow the bluedashed edge up to the neighboring black node. The black node has two neighboring black nodes, each of which is connected to a red node via bluedashed edges. Once the photons within the black nodes have disappeared after the fusions, the red node we started with will inherit these two red nodes as its immediate neighbors. Similarly, if we followed the bluedashed edge down from the original red node we picked, we would find two other rednode neighbors of that red node after the fusions are done. So, if all the fusion attempts were to succeed, we will get a regular 3D lattice of red nodes, where each node has degree 4. This is the diamond lattice.
The authors of ref. ^{18} did exactly the above, attempting to create a percolated instance of the 3D diamond lattice as a longrangeconnected photonic lattice. But our reinterpretation of their (nonstandard) percolation threshold as a standard bondpercolation threshold of the modified (10,3)b lattice not only provides a natural way to account for failure modes of fusion attempts, but allows one to exploit existing literature on efficient Monte Carlo methods (viz., the NewmanāZiff method^{26}) to calculate sharper percolation thresholds. More importantly, this interpretation also opens the door for new constructions and improved thresholds, which we describe next.
Improved thresholds
Leveraging this new insight, we embark upon a progression of improved achievability thresholds for nā=ā3, using higherdimensional generalizations of the modified (10,3)b lattice.
First, we consider a 4D extension of the (10,3)b lattice (see Fig. 6). It consists of a doubly infinite stack of (x, y)plane layersāof parallel 1D line lattices of black (degree3) nodesāstacked along the z and w directions, respectively. Of the three neighboring bonds of a black node, two (green) bondsāconnecting to neighboring black nodes in the line lattice to which it belongsāare in the (x, y) plane, whereas one (blue) bondāconnecting to a red node, which in turns connects via another blue bond to a black node in a neighboring (x, y)plane layerāpoints in either the z direction or in the w direction. Along each line lattice of black nodes, the blue bonds alternate between directions +z, +w, āz, āw, ā¦, and so on. The graph has a period of four in each of the x, y, z, and w dimensions. One period of the lattice is depicted in Fig. 6. The inner axes represent an (x, y) plane at a given value of z and w. This construction results in longer loops compared with the 3D case discussed above, while retaining the 3D graphās coordination number (average node degree), which in turn lowers the bondpercolation threshold. We find, using a NewmanāZiff simulation performed on a 4Dmodified (10,3)b lattice of size Nā~ā10^{7} nodes, that
Next, we consider an infinitedimensional modified (10,3)b lattice, which becomes locally treelike, as shown in Fig. 7. However, note that for any finite dimension of this family of modified (10,3)b lattices, no matter how large the dimension is, it is not a tree, and a percolated subgraph of it will give us a renormalizable universal cluster. Similar to the 3D and 4Dmodified (10,3)b lattices, each black node has two green bonds and one blue bond (which in turn leads to a black node via a red node and another blue bond). We denote the expected number of children of a node when approached via a green bond as E_{1} and the expected number of children of a node when approached via a blue bond as E_{2}. When counting the number of children of a node, we only count red nodes since they are the only nodes with unmeasured qubits. Counting children from the top of Fig. 7, each black node is labeled as 1 or 2 depending on the bond from which it is approached. Counting children at the points labeled E_{1} and E_{2} yields the equations E_{1}ā=āĪ»E_{1} + Ī» + Ī»^{2}E_{2} and E_{2}ā=ā2Ī»E_{1}, where Ī» is the bond probability. For percolation, E_{1}āāāā and solving the equations with this condition, we find that \({\lambda }_{\mathrm{c}} + 2{\lambda }_{\mathrm{c}}^3 = 1\), which leads to Ī»_{c}āāā0.5898. Therefore,
thereby bringing the bestknown achievable threshold with threephoton clusters as the initial resource closer to our general lower bound applied to nā=ā3, \({\lambda }_{\mathrm{c}}^{(3)} \ge 0.5\).
Universal cluster creation on a 2D planar lattice
We found a 2D logical lattice, a modified brickwork (see Fig. 8), for which Ī»_{c}ā=ā0.746. This gives a bound \({\lambda }_c^{(3)} \le 0.746\), which is looser, compared with the ones we obtained from the constructions in the previous section. However, since 0.746ā<ā25/32, this result shows that singlephotonboosted linear optical fusion and threephoton GHZstate clusters are sufficient to generate a universal lattice ballistically by fusing the clusters on a twodimensional lattice. Fusion on a 2D lattice is significantly simpler, compared with 3D (or higherdimensional) lattices from an experimental standpoint, since 2D programmable linear optics is a fastmaturing technology. Whether or not the above is possible was posed as a challenging problem by Rudolph and colleagues in refs. ^{18,19}.
Effects of photon loss on the thresholds
We prove an extension of our lower bound on \({\lambda }_{\mathrm{c}}^{(n)}\). We use a loss model inspired by proposals to produce photonic microclusters using quantum emitters^{21,27,28}, which have been recently realized experimentally^{22}. In these proposals, since each node of the microcluster is added sequentially, we assume that the total transmissivity of the photon in its lifetime is of the form \(\eta = \eta _0^n\) (the loss is 1āĪ·) where n is the size of the microcluster and Ī·_{0} represents the transmissivity in one step. Hence, the total loss seen by the photon increases with the size of the microcluster. The import fidelity of the clusters generated due to unheralded photon losses and other forms of errors (e.g., mode mismatch, detector dark clicks) have not been considered. To obtain a lower bound, we only look at loss in the photons undergoing fusion and observe that a fusion gate only succeeds if both input photons are detected. Furthermore, for our bound, we treat loss as the removal of the corresponding node from the cluster state or a measurement in the Z basis. In general, loss of a qubit is not equivalent to a Zbasis measurement. Loss results in tracing out the qubit, but treating it as a Zbasis measurement leads to a lower value of Ī»_{c} (Zbasis measurement gives us more information than tracing out over the Z basis) and is sufficient for a lower bound on \({\lambda }_{\mathrm{c}}^{(n)}\). Hence, if \({\lambda }\eta _0^{2n} < 1/(n  1)\), there cannot be a giantconnected component in the cluster, and we obtain the lossdependent lower bound of \({\lambda }_{\mathrm{c}} \ge 1/[(n  1)\eta _0^{2n}] \equiv {\lambda }_{\mathrm{c}}^{({\mathrm{LB}})}\). Figure 9 plots \({\lambda }_{\mathrm{c}}^{({\mathrm{LB}})}\) as a function of n for different values of Ī·_{0} and we find that there is an optimum value of n for any Ī·_{0}ā<ā1, which gives the lowest \({\lambda }_{\mathrm{c}}^{({\mathrm{LB}})}\). In other words, there is an optimum size of the starting microcluster, for which the required fusion probability is minimized.
Conversely, for a given fusion success probability Ī», there exists a threshold Ī·_{0c}, s.t., if Ī·_{0}ā<āĪ·_{0c}, the postfusion cluster cannot be percolated. We thus have a lower bound \(\eta _{0{\mathrm{c}}} \ge \eta _{0{\mathrm{c}}}^{{\mathrm{(LB)}}}\), where \(\eta _{0{\mathrm{c}}}^{{\mathrm{(LB)}}} = \left[ {1/{\lambda }(n  1)} \right]^{1/(2n)}\). In Fig. 9b, we plot \(\eta _{0{\mathrm{c}}}^{({\mathrm{LB}})}\) for different values of n, for Ī»ā=ā0.75 and Ī»ā=ā1. There is an optimum value of n which gives the best lower bound on loss tolerance, e.g., for Ī»ā=ā0.75, sixphoton microclusters give the lowest bound of \(\eta _{0{\mathrm{c}}}^{{\mathrm{(LB)}}} = 0.8957\) which corresponds to a loss (1āāāĪ·) of 48.36% seen by each photon. Furthermore, we find that going from Ī»ā=ā0.75, which is attainable using four single ancilla photons and (lossless) linear optics^{8} to deterministic fusion (Ī»ā=ā1), \(\eta _{0{\mathrm{c}}}^{({\mathrm{LB}})}\) only decreases slightly, i.e., the equivalent perphoton loss threshold increases from 89.6 to 87.1%. Hence, when losses are accounted for in ballistic cluster state creation, the advantage in having a fully deterministic fusion may be relatively small. It is important to note, however, that the numbers presented here are only lower bounds on Ī·_{0c} and Ī»_{0c}, and may not be tight. For example, our results show that any ballistic cluster creation process that starts with sixphoton microclusters and uses destructive fusion with Ī»ā=ā0.75 cannot tolerate more than 48.36% loss. However, these results do not prove that 48.36% loss is sufficient for ballistic cluster creation.
Discussion
Despite recent progress in LOQC making it resurface as a strong contender to scalable quantum computing, many questions remain, whose answers will be crucial to its eventual success. A major challenge, as expounded in ref. ^{23} as well, is an experimental scheme to build an ondemand source of highfidelity small entangled cluster states, e.g., threephoton GHZ states. The other big experimental challenge is to realize highquality quantum interference of the photon clusters using a linear optical interferometer that minimizes inline losses and mode mismatch.
A few immediate problems we leave open in this paper are as follows: (a) proving our conjecture on the achievability of \({\lambda }_{\mathrm{c}}^{(n)} = 1/(n  1) + \varepsilon\); and (b) for a given n, finding the construction that gives the smallest overhead (i.e., edge length of an optimally chosen renormalized universal lattice), while not limiting only to those linearāoptical unitaries U that can be assembled with twoqubit fusion gates (i.e., allow for higherdimensional linearāoptical projective operations to potentially fuse more than two clusters at once).
On the theoretical front, some of the most important open problems are as follows: (1) extending this work to include mode mismatch within the linear optic fusion circuits; (2) finding the minimum size clusters and the most resourceefficient schemes for ballistic cluster creation that are immune to unheralded losses.
In this paper, we only considered losses in photons undergoing fusion. This was sufficient for obtaining bounds on \({\lambda }_{\mathrm{c}}^{(n)}\) and Ī·_{0c}. However, any losses in photons that do not undergo fusion are unheralded, i.e., we do not know which photons were lost. The strategy of renormalizing by finding crisscrossing highways and choosing the intersection of these highways to be logical qubits, which gives us a perfect universal cluster state in the lossless case, fails in the presence of such unheralded loss. This is because even if we have a percolated cluster, we need to know the obtained graph state exactly in order to compute the highways. In order to correct for these unheralded photon losses (and other forms of qubit errors), an additional layer of error correction is needed^{29,30,31,32,33}. Let us consider fivephoton GHZ states as a starting point, which correspond to five node clusters in the āstarā configuration, and place them at the nodes of a Raussendorf lattice (which has degree4 nodes). The central photon of the star is placed on a node of the Raussendorf lattice and the four leaf nodes of the star are placed along the four neighboring bonds. We now attempt twoqubit fusion on each bond, attempting to construct a bondpercolated instance of the Raussendorf lattice following the recipe in ref. ^{15}. If each fusion attempt succeeds with probability Ī», we obtain a subgraph of the Raussendorf lattice where each bond is present independently with probability Ī». Recent work has shown that a subgraph of the Raussendorf lattice with 0.145 or a smaller fraction of missing bonds^{33} (or with 0.249 or a smaller fraction of missing qubit sites^{30}) can be used for universal quantum computation while correcting for unheralded qubit errors. The amount of unheralded error on each physical qubit that can be corrected using such a thinned Rausendorf lattice depends upon how far below 0.145 the actual fraction of missing bonds is. While unheralded photon losses can be accounted for with such a construction starting with fivephoton clusters as described above, it requires a fusion probability exceeding 0.85, which is greater than the best known linear optic Bell measurements boosted with single photons or squeezing. Future work should look at ways of reducing the required fusion probability to within 0.78āwhich is achievable with singlephoton boostingāby using more efficient constructions and with smaller clusters as the starting point, and one should investigate experimentally realizable methods that improve upon the achievable fusion success probability beyond 0.78. It will also be important to extend the errorcorrection method to account for modemismatch errors in the linear optic mixing in the fusion gates, as well as detector excess noise, such as dark clicks.
Code availability
All the numerical data presented in this paper are the results of C simulations conducted by M.P. The code used to generate thiese data will be made available to the interested reader upon reasonable request.
Data availability
The data sets generated during and or analyzed during the current study are available from the corresponding author on reasonable request.
References
Reck, M. & Zeilinger, A. Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73, 58ā61 (1994).
Carolan, J. et al. Universal linear optics. Science 349, 711ā716 (2015).
Harris, N. C. et al. Quantum transport simulations in a programmable nanophotonic processor. Nat. Photonics 11, 447ā452 (2017).
Calsamiglia, J. & LĆ¼tkenhaus, N. Maximum efficiency of a linearoptical Bellstate analyzer. Appl. Phys. B 72, 67ā71 (2001).
Michler, M., Mattle, K., Weinfurter, H. & Zeilinger, A. Interferometric Bellstate analysis. Phys. Rev. A. 53, R1209āR1212 (1996).
Grice, W. P. Arbitrarily complete Bellstate measurement using only linear optical elements. Phys. Rev. A. 84, 042331 (2011).
Zaidi, H. A. & van Loock, P. Beating the onehalf limit of ancillafree linear optics Bell measurements. Phys. Rev. Lett. 110, 260501 (2013).
Ewert, F. & van Loock, P. 3/4efficient Bell measurement with passive linear optics and unentangled ancillae. Phys. Rev. Lett. 113, 140403 (2014).
Pan, J.w & Zeilinger, A. GreenbergerHorneZeilingerstate analyzer. Phys. Rev. A. 57, 2208ā2211 (1998).
Aaronson, S. & Arkhipov, A. The computational complexity of linear optics. Proceedings of the 43rd annual ACM symposium on Theory of computingāSTOC ā11. 333ā342 (ACM Press: New York, San Jose, California, USA, June 06ā08, 2011. https://doi.org/10.1145/1993636.1993682.
Knill, E., Laflamme, R. & Milburn, G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 46ā52 (2001).
Raussendorf, R. & Briegel, H. J. A oneway quantum computer. Phys. Rev. Lett. 86, 5188ā5191 (2001).
Browne, D. E. & Rudolph, T. Resourceefficient linear optical quantum computation. Phys. Rev. Lett. 95, 010501 (2005).
Kok, P., Nemoto, K., Ralph, T. C., Dowling, J. P. & Milburn, G. J. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys. 79, 135ā174 (2007).
Kieling, K., Rudolph, T. & Eisert, J. Percolation, renormalization, and quantum computing with nondeterministic gates. Phys. Rev. Lett. 99, 130501 (2007).
Browne, D. E. et al. Phase transition of computational power in the resource states for oneway quantum computation. New J. Phys. 10, 023010 (2008).
Grimmett, G. Percolation, vol. 321 of Grundlehren der mathematischen Wissenschaften. (Springer Berlin Heidelberg, Berlin, Heidelberg, 1999).
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  At., Mol., Opt. Phys. 91, 042301 (2015).
Kieling, K. & Eisert, J. Percolation in quantum computation and communication. Lect. Notes Phys. 762, 287ā319 (2009).
Rao, D. D. B., Yang, S. & Wrachtrup, J. Generation of entangled photon strings using NV centers in diamond. Phys. Rev. B  Condens. Matter Mater. Phys. 92, 081301 (2015).
Schwartz, I. et al. Deterministic generation of a cluster state of entangled photons. Science 49, 1804ā1807 (2016).
Rudolph, T. Why I am optimistic about the siliconphotonic route to quantum computing. APL Photonics 2, 030901 (2017).
MorleyShort, S. et al. Physicaldepth architectural requirements for generating universal photonic cluster states. Quantum Sci. Technol. 3, 015005 (2018).
Tran, J., Yoo, T., Stahlheber, S. & Small, A. Percolation thresholds on threedimensional lattices with three nearest neighbors. J. Stat. Mech.: Theory Exp. 2013, P05014 (2013).
Newman, M. E. J. & Ziff, R. M. Fast Monte Carlo algorithm for site or bond percolation. Phys. Rev. E  Stat., Nonlinear, Soft Matter Phys. 64, 1ā16 (2001).
Lindner, N. H. & Rudolph, T. Proposal for pulsed Ondemand sources of photonic cluster state strings. Phys. Rev. Lett. 103, 113602 (2009).
Economou, S. E., Lindner, N. & Rudolph, T. Optically generated 2dimensional photonic cluster state from coupled quantum dots. Phys. Rev. Lett. 105, 093601 (2010).
Varnava, M., Browne, D. & Rudolph, T. Loss tolerance in oneway quantum computation via counterfactual error correction. Phys. Rev. Lett. 97, 120501 (2006).
Barrett, S. D. & Stace, T. M. Fault tolerant quantum computation with very high threshold for loss errors. Phys. Rev. Lett. 105, 200502 (2010).
Michael, M. H. et al. New class of quantum errorcorrecting codes for a bosonic mode. Phys. Rev. X 6, 031006 (2016).
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).
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).
Acknowledgements
MP, DE, and SG acknowledge support from the DARPA seedling project Scalable Engineering of Quantum Optical Information Processing Architectures (SEQUOIA), under US Army contract number W31P4Q15C0045. MP and DE acknowledge support from the Air Force Office of Scientific Research MURI (FA95501410052). SG acknowledges support from the Office of Naval Research MURI on Optical Computing under US Navy contract number N0001416C2069. MP, DE, and SG acknowledge extremely valuable discussions with Terry Rudolph, Pete Shadbolt, and Mercedes GimenoSegovia during their visit to Boston in April 2016, which was partially supported by the MITImperial College London Seed Fund.
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M.P., S.G., and D.T. conceived the idea of mapping fusions between microclusters to an instance of bond percolation on a logical graph. M.P. and S.G. performed the percolation simulations and found the new lattices with lower percolation thresholds and lower dimensionality. D.E. provided a physically meaningful model for photon loss. All authors contributed to the writing of the paper.
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Pant, M., Towsley, D., Englund, D. et al. Percolation thresholds for photonic quantum computing. Nat Commun 10, 1070 (2019). https://doi.org/10.1038/s4146701908948x
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DOI: https://doi.org/10.1038/s4146701908948x
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