Abstract
Giovannetti, Lloyd, and Maccone (2008) proposed a quantum random access memory (QRAM) architecture to retrieve arbitrary superpositions of N (quantum) memory cells via quantum switches and \(O(\log (N))\) address qubits. Toward physical QRAM implementations, Chen et al. (2021) recently showed that QRAM maps natively onto optically connected quantum networks with \(O(\log (N))\) overhead and builtin error detection. However, modeling QRAM on large networks has been stymied by exponentially rising classical compute requirements. Here, we address this bottleneck by: (1) introducing a resourceefficient method for simulating largescale noisy entanglement, allowing us to evaluate hundreds and even thousands of qubits under various noise channels; and (2) analyzing Chen et al.’s networkbased QRAM as an application at the scale of quantum data centers or nearterm quantum internet; and (3) introducing a modified networkbased QRAM architecture to improve quantum fidelity and access rate. We conclude that networkbased QRAM could be built with existing or nearterm technologies leveraging photonic integrated circuits and atomic or atomlike quantum memories.
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Introduction
A quantum random access memory (QRAM) is an essential computational primitive for many quantum algorithms. The ability to perform a QRAM query in \(\log (N)\) time steps, where N = 2^{n} is the number of memory cells, implies polynomial speedups for applications such as quantum machine learning^{1}, matrix inversion^{2}, quantum imaging^{3}, and quantum searching^{4}. Despite its clear importance to quantum information processing, a QRAM has yet to be realized experimentally. Hence, finding a suitable architecture that can be realized in the near future remains an active research subject in the theoretical and experimental domains.
In this article, we present a method to simulate largescale entanglement accounting for various sources of noise. We are able to efficiently simulate circuits with thousands of qubits under dephasing, amplitude damping, and CNOT errors. Based on our simulation model, we present a QRAM architecture for photonic networkbased QRAM based on ref. ^{5}. The feasibility assessment is based on realistic parameters extracted from recent experiments, which we will refer to throughout the article.
A classical RAM^{6} consists of a binary tree leading to a final layer of memory cells, each corresponding to a unique address. The address is represented as a series of bits, with each bit corresponding to a layer of the binary tree. Each bit of an address describes how the bus signal propagates in the layer: to the right or to the left child node. Hence, the nodes of the binary tree act as switches for the address. When provided with a nbit address, the RAM returns a bit string f_{k} associated with the memory cell labeled k. This is called the fanout scheme^{7}.
A QRAM is the quantum analog of the RAM, similarly consisting of addresses, quantum switches, and memory cells in the form of qubits. In particular, with a quantum address state, over the set of address qubits a, given by \(\left\vert {\psi }_{{{{\rm{in}}}}}^{{\prime} }\right\rangle =\mathop{\sum }\nolimits_{j = 1}^{{{{\rm{n}}}}}{\alpha }_{j}{\left\vert j\right\rangle }_{a}\), one can retrieve data from a superposition of memory cells. A QRAM query is defined via the following transformation,
where \(\left\vert {{\emptyset}}\right\rangle\) represents an ancillary state over the bus qubit b, which transforms into the retrieved data state after querying. In this article, we will restrict our investigations to classical data, i.e., \(\left\vert {D}_{j}\right\rangle\) are separable bits. A direct conversion of classical fanout protocol to the quantum realm is inefficient since it requires maintaining quantum coherence over an exponential number of connections^{7}.
Three main schemes have been investigated to date: the fanout scheme that was already described, the bucket brigade model, and the teleportationbased scheme. Important figures of merit for the QRAM are the fidelity of the above transformation and the query time. For a detailed study and comparison of the first two schemes, please refer to ref. ^{8}.
In the bucket brigade (BB) model^{7,9}, the number of qubits of the device scales as O(2^{n}), as does the number of gates. Moreover, the original protocol^{7} includes an additional third state in each node, called the “wait” state, in order to prevent the exponential scaling of the amount of decoherence with respect to the memory size. However, Hann et al.^{8} have shown that the origin of the noise resilience of the BB model is the amount of entanglement among the memory’s components and not the presence of the “wait” state, as one can devise a BB model without the “wait” state that still achieves a polynomial scaling of the decoherence with respect to the number of memory addresses n.
More recently, Chen et al. presented a photonic networkbased QRAM scheme^{5} that makes use of quantum teleportation of addresses from a quantum computer to the QRAM binary tree. Such a scheme greatly increases the protocol’s efficiency by teleporting the registers to the layers (initially prepared in GHZ states) in parallel as opposed to in series, thereby circumventing the event of a single qubit loss collapsing the entire tree state. Additionally, the proposed QRAM maps onto quantum networks, leading to potential applications in distributed quantum computing and sensing.
However, Chen et al. left as an open challenge the simulation of the scheme on largescale networks since the computational complexity scales exponentially with the number of qubits. In this work, we bypass this problem resorting to more efficient ways of modeling the noise in stabilizer states. Moreover, this method generalizes to other quantum networking tasks with similar constructions, such as protocols for distributed quantum computation.
This comes in line with the fact that distributing entanglement is central in quantum information processing schemes ranging from quantum computing to sensing to communications^{10,11,12}. Simulation of distributed entanglement in a network setting, be it a longdistance network such as a possible future quantum internet^{13} or smalldistance quantum local area network (QLAN)^{14}, is important to assess the limitations imposed by nearterm quantum technologies. The architecture of the QRAM considered in this paper, building on the photonic networkbased QRAM proposed in ref. ^{5}, involves a series of exponentially growing GHZ states, with the largest having as many qubits as there are memory cells. Each GHZ state spans across a physical layer in the QRAM architecture, and the number of nodes per layer grows exponentially with the number of memory cells 2^{n} ≡ N to be addressed, as shown in Fig. 1.
Computer simulations of noisy quantum processes in such a system quickly become computationally intensive^{15,16,17} due to the density matrices growing exponentially in size with the number of qubits. Even though the entire QRAM protocol definition, i.e., the retrieval of data given an input address (see Eq. (1)), requires more than just Clifford operations, creating the routing state over the QRAM architecture only uses Clifford gates. These operations are the ones used to create the GHZ states and to teleport the address state onto the QRAM access layers. Moreover, the operations required to access the QRAM after the routing state is distributed over the routing nodes only grow with the logarithm of the number of qubits of the QRAM, in comparison to the linear amount of operations required to create the routing state. This is the reason why noise in the system mostly comes from the GHZ states before access. In particular, this set of operations to create the routing state can be classically simulated efficiently^{16}. This approach enables an explicit and efficient description of all the intermediary states, up to local unitary corrections. In this article, we develop efficient methods to simulate largescale noisy entanglement by characterizing the impact of noise at all intermediate steps and apply these tools to simulate a noisy QRAM.
There are several architectures for a QRAM. Here, we focus on the optically mediated quantum networkbased QRAM architecture introduced in ref. ^{5}, as it offers several key benefits: implementation in quantum networks compatible with envisioned quantum internet architecture and quantum data centers, faster query times and the possibility of executing in a nonlocal manner by means of teleportation. Hence, this scheme works under any networklike architecture, be it locally (e.g., on a chip) or across large distances (e.g., over a quantum internet). Without loss of generality, we characterize each node of the architecture as one of a spinphoton network that could be implemented in photonic integrated circuits (PIC).
The architecture of the QRAM is similar to previous models, such as BB and the fanout models. The main difference concerns the execution of the protocol and the resources available at each node. In this architecture, one considers two agents: the quantum computer, which prepares the addresses, and the QRAM or quantum access tree (see Fig. 1). The quantum computer must provide an address state with \(n={\log }_{2}N\) qubits, where N is the total number of memories (for simplicity assume \(n\in {\mathbb{N}}\)). The QRAM has a binary tree architecture, with n physical layers, where the kth layer (k ∈ {1, . . . , n − 1}) has 2^{k−1} quantum nodes. As we describe next, in each physical layer, all the nodes share a GHZ state, which is used to teleport the address state onto the QRAM itself, allowing for an ancilla qubit to access the memories in the correct superposition.
As for the type of physical implementation chosen, and without loss of generality, we focus on a QRAM implementation involving solidstate spin qubits integrated into PICs, an approach that is promising in terms of scalability. In particular, we consider diamond nanophotonic cavities coupled with siliconvacancy centers^{18,19} as each QRAM tree node. Each emitter contains an electronic spin that directly interacts with the photonic address register qubits and an accompanying nuclear spin acting as a longlived memory. By entangling the electronic spin with the photon via cavity reflection, consecutive reflection of a photon off two neighboring nodes and subsequent heralding achieves spinspin entanglement. This remote entangling strategy is repeatedly used to generate a GHZ state across each layer. Such operations are probabilistic (see Fig. 2): the photon has a nonzero probability of being lost to the environment before reflecting off two cavities and arriving at the detector. On the other hand, it is possible to perform close to deterministic twoqubit gates between the electronic and nuclear spin qubits, albeit with a larger error^{20,21}. For this reason, we term this architecture teleportationbased deterministic QRAM or TDQRAM.
In these types of systems, the main contributors to errors are (1) spin phase errors (at rate 1/T_{2}), (2) spinflip errors (at rate 1/T_{1}), and (3) errors in hyperfine gates between electron and nuclear spins (see Supplementary Table I). We leave out photonelectron interactions, as one could conceive trading off the efficiency η for arbitrarily high fidelity in the cavityreflectionbased scheme proposed in ref. ^{22} in the highcooperativity and overcoupling regime.
Hence, we explore different values for T_{1}, T_{2} of both electronic and nuclear spin qubits and p_{e} and p_{n} for the probabilities of error in electronic and nuclear spin CNOTs. For the remainder of this article, we set \({T}_{1}^{{{{\rm{n}}}}}=100\,{T}_{1}^{{{{\rm{e}}}}}\equiv 100\,{T}_{1}\) and \({T}_{2}^{{{{\rm{n}}}}}=100\,{T}_{2}^{{{{\rm{e}}}}}\equiv 100\,{T}_{2}\). Nuclear spins have a higher coherence time as they are much less coupled to the noisy spinbath compared to electronic spins. Reported values of characteristic times go, experimentally, up to \({T}_{1}^{{{{\rm{e}}}}} \sim 1\,{{{\rm{s}}}},{T}_{2}^{{{{\rm{e}}}}} \sim 10\,{{{\rm{ms}}}}\)^{23}, and there are theoretical predictions of being able to reach p_{e}, p_{n} = 10^{−2} ~ 10^{−4}^{24,25}. Moreover, we detail other important physical parameters of this type of system used for the simulations in Supplementary Table I.
Results
Simulating the effects of decoherence for a TDQRAM
To simulate the QRAM initialization protocol, we use NetSquid^{26} under the stabilizer formalism and extract all the parameters of the noise channels before implementing them in simulations, for instance: timing parameters for every qubit used throughout the simulation, all the noisy CNOTs with corresponding error probabilities, and to which qubits and at which step it is applied. From here, we compute the fidelity of the final QRAM state by substituting all these values into the expressions presented in the “Methods”.
We start by presenting the simulation of a 2^{12}qubit QRAM in Fig. 3. Here, we detail individually the fidelities of the GHZ state distributed at each physical layer of the QRAM. The fidelity of the full state of the QRAM is given by:
i.e., the fidelity of the entire tree (or the QRAM) is defined as the product of the fidelities of each physical layer (see Supplementary Methods for more details). We distinguish access fidelity from tree fidelity, where the former refers to the fidelity of the state retrieved after accessing the memory cells (\(\left\vert {\psi }_{{{{\rm{out}}}}}\right\rangle\) in Eq. (1)), and the latter refers to the multipartite state fidelity of the binary tree constituting the QRAM. Only the access fidelity depends on the address and bus qubits.
One observes an exponential decrease of the fidelity with the number of the layer (notice the logarithmic scaling on the yaxis corresponding to the fidelity). This agrees with the GHZ state size increasing exponentially with the number of layers, i.e., scaling 2^{k}. When one qubit in this multipartite state suffers an error, the entire state is affected.
One critical figure of merit that we extract from the NetSquid simulations is the query time. As demonstrated in ref. ^{5}, the query efficiency scales logarithmically with the number of qubits. Extracting from multiple queries of the QRAM, we obtain the query times (apart from a logarithmic factor derived from making the bus qubit traverse the binary tree) in Fig. 3.
Dephasing and damping errors for TDQRAM
Considering only the effects of dephasing and amplitudedamping errors in the spin qubits, we take T_{2} = ∞ ms for amplitudedamping errors only, and then T_{2} = 10 ms and T_{2} = 100 ms with a fixed T_{1} = 2 s^{23}, see Supplementary Table I. We also set the CNOT error rate to 0. We present the simulation results for the TDQRAM scheme under memory dephasing for increasing QRAM size, as shown in Fig. 3.
Looking closely at Fig. 3c, one can observe that the effect of amplitudedamping shows an identical behavior to the one of dephasing and amplitudedamping combined, i.e., with the same type of scaling. However, it is residual compared to the effect of dephasing. This is easily explainable by the timescales of the coherence times of the corresponding noises (T_{1} and T_{2}) in the memory differ by orders of magnitude, with the first, T_{1}, being usually much longer than the latter, T_{2}, i.e., T_{1} ~ 1 s^{23}. For this reason, its impact can be neglected relative to other sources of error.
Dephasing, damping and noisy CNOTs for TDQRAM
The only type of error missing in the analysis is the error derived from the use of noisy CNOTs. Illustrated in Fig. 3, the dephasing and damping errors minimally contribute to infidelity. We now analyze the case for noisy CNOTs on top of fixed T_{1} = 2 s and T_{2} = 100 ms (note we now switch to linear scale in the yaxis for the fidelity due to the set of values present for the different simulations). For simplicity, we consider equal CNOT error probability, ϵ_{CNOT}, for both electronic and nuclear CNOTs, and vary ϵ_{CNOT} from 10^{−5} to 10^{−2}, as shown in Fig. 3:
These simulations show that the CNOT gates dominate the overall error in the QRAM state fidelity in the TDQRAM. For instance, to access a 128qubit QRAM, one needs fidelities of the CNOT gates to be somewhere near 99.9% to obtain an access fidelity exceeding 90%. In this architecture, while the query times do not increase linearly with the size of the memory, the errors do. Expectedly, applying an error to a single qubit of a GHZ state contributes in the same order for the entire state.
The price to pay for performing CNOTs with such large error rates deterministically could be circumvented by nearperfect yet probabilistic CNOTs^{22,27} via cavitybased electron spinphoton interactions, as opposed to deterministic yet errorprone nuclearelectron spin coupling. In light of this, we explore a hybrid teleportationbased QRAM architecture in the following section.
Teleportationbased stochastic QRAM
In the TDQRAM protocol, the entanglement generation and swap (Fig. 8) operation are still probabilistic, given the finite chance of photon loss. Hence, these probabilistic CNOTs are done in parallel throughout each physical layer to improve efficiency. After an EPR pair is created between two electron spins, however, transferring entanglement onto the nuclear spins is a deterministic procedure. Thereby, the query time grows sublinearly. As noted before addressing the TDQRAM scheme, this deterministic CNOT based on nuclearelectron spin interaction mainly dominates the infidelity of the GHZ state, motivating us to contemplate an alternative solution.
Since the decoherence errors from T_{1} and T_{2} contribute much less to the infidelity relative to electronnuclear spin CNOT, replacing some of the noisy deterministic CNOTs with probabilistic CNOTs helps improve fidelity despite reducing efficiency. As we will show, this leads to higher QRAM tree state fidelities, albeit with longer query times. We call this architecture ‘teleportationbased stochastic QRAM’, or TSQRAM.
Relying solely on probabilistic CNOTs in every step of the protocol would be very inefficient since the probability of generating a GHZ state diminishes exponentially with the number of nodes. In other words, if one entanglement attempt fails during the construction of a GHZ state, the entire state collapses. Since each linking process is heralded, there are ways to circumvent this by choosing a specified order to perform the CNOTs, similar to entanglement swapping in a repeater chain^{28,29}. Here, the probabilistic swapping operations are equivalent to the probabilistic CNOTs, and measuring the middle node is analogous to joining smaller GHZ states to form a larger GHZ state. Abstractly, they describe the same problem, which allows us to use the solutions provided by ref. ^{29}. Next, we present an indepth analysis of the tradeoff between fidelity and query rate as a function of error rates and physical implements.
Increasing T _{1} and T _{2}
To decrease the number of employed deterministic CNOTs, and taking into account that these always happen when the electronic spins interact with the nuclear spins, it is natural to consider dropping the nuclear spins altogether. This is motivated by the fact that we can perform CNOTs, albeit probabilistically, between the electron spins. The downside is that electron spins suffer from having shorter coherence times than their nuclear counterparts. Still, it is advantageous to consider such schemes to avoid the use of noisier deterministic CNOTs.
To minimize the consequently increased decoherence, one could conceive schemes for increasing the T_{1} and T_{2} times for the electrons since these are the ones now causing the fidelity bottleneck, together with the required time to query the memory.
Presently, the SiV’s electronic spin’s T_{1} time is shown to be longer than 1 s^{23}, thereby posing no concern over depolarization. On the other hand, its T_{2} coherence time is limited to tens of milliseconds^{23}, even under dynamical decoupling. The main dephasing mechanism is attributed to the surrounding nuclear spin bath, which is weakly coupled to the electronic spin of interest via hyperfine interaction^{30}. A potential avenue to improving the electronic spin’s T_{2} is therefore to “purify” its environment by materials engineering^{31}. By producing SiV in a carbon13 free matrix, for example, the coherence time may be further extended.
Nevertheless, our numerical analyses of the hybrid scheme show fidelities still exceeding 60% for a reasonable CNOT error rate of 10^{−3} and 1024 memory cells, using a T_{2} of 100 ms. For such a result, a probability of success of about 70% for the CNOT is required.
The teleportationbased stochastic QRAM protocol
In the TDQRAM protocol, there are two steps occurring in parallel across each layer in the QRAM: one for generating EPR pairs across every other node and another for linking all the states into a larger GHZ state via sharing EPR pairs inbetween nodes holding the previously shared EPR pairs (see Fig. 8). This could be made in parallel because the linking operations are deterministic.
In the TSQRAM protocol, however, we must now consider an order for the linking step that depends on the node’s position, similar to the quantum repeater chain problem^{26,29,32}. If a linking process fails, the subset of qubits that would have become entangled must be reset. The optimal strategy is then performing the linking process in a binarytreelike approach^{29}. This binarytree order for the linking processes means that now a heralding signal for a successful link must be exchanged within the tree. Each parenting node will have two children, the rightchild and the leftchild. Each node only attempts entanglement if it receives heralding signals from both children nodes that have been successfully entangled themselves. Figure 4 illustrates this procedure and defines the order.
Moreover, as mentioned before, the advantage of the TSQRAM protocol is that probabilistic CNOTs are used to minimize state infidelity. One might consider the optimal placements for the deterministic CNOTs to maximize the GHZ state fidelity across each physical layer. We further introduce having an additional distribution layer. This is the layer of the order binarytree at which a linking step is attempted, as shown in Fig. 4. These abstract layers are only needed to describe the order of the linking steps and help illustrate the optimal placements for the deterministic CNOTs.
For this reason, we present two possible options to solve the placement of deterministic nodes problem: the first is randomly choosing a set of nodes to be deterministic, regardless of their distribution layer. The second option is choosing the nodes that attempt to link entanglement at the higher steps since if those attempts are unsuccessful, they take the biggest toll on the protocol requiring reattempting every preceded step. We illustrate these two possible options in Fig. 4.
As we will verify later, we need a much smaller number of deterministic nodes if we place them in higherlevel distribution layers. We first present simulation results for both cases.
Simulating the effects of decoherence for a TSQRAM
Using the aforementioned results, we compare the TDQRAM protocol with the TSQRAM, which includes both probabilistic EPR pair generation and deterministic linking. For comparison purposes, we start by assuming all probabilistic CNOTs (Prob (node being deterministic) ≡ P_{d} = 0%) have unit efficiency in the latter scheme. If the deterministic and the probabilistic CNOTs are of the same order in gate time, then the binarytree approach is bound to be more timeconsuming, considering its greater number of entanglement attempts. However, the probabilistic CNOT based on cavity reflection is typically several orders of magnitude faster than the deterministic CNOT (10^{1} ns vs. 10^{5} ns). We therefore present both the QRAM’s query time and its fidelity for both schemes in Fig. 5, assuming perfect deterministic CNOTs. We also consider the case where the CNOT efficiency is less than unity for comparison.
Before moving on to the noise simulations, we delve into the query times. It is not obvious that now the query times scale logarithmically (or even polylogarithmically) since the efficiency of the distributed CNOTs can increase the query times depending on the order of the linking steps. In fact, Fig. 5 already shows a nonlogarithmic behavior when considering a completely probabilistic protocol. If one were to choose sequential linking steps, the query times would increase exponentially with the efficiency. By choosing the scheme demonstrated in Fig. 4, we are able to reduce this to polynomial scaling^{26}. However, depending on the noise parameters, this increase in time, compared to the initial twostep scheme, might not be wanted, as we will verify next. In Fig. 5, we present the query times for different efficiencies of the distributed CNOT, under the two possible hybrid schemes, with different numbers of deterministic CNOTs placed strategically (see Fig. 4).
We start by verifying that, for a random placement of the deterministic nodes, there is no clear dependence on the number of deterministic CNOTs. The reason is that, when choosing random placements for the deterministic nodes, the best order for the linking steps immediately changes and is no longer a binary tree. There already exist algorithms^{29} that use linear programming to solve an identical problem of finding the best order to attempt entanglement swapping along a chain, which is virtually identical to our problem. However, the polynomial scaling of these algorithms in terms of the number of nodes of the chain makes it unsuitable for exponentially growing chains. For the intuitive placement of the deterministic nodes, this is not the case, as choosing only the top layers of the linking tree does not change the best order to do the linking. We also consider varying the efficiency of the distribution of the Bell pairs, as shown in Supplementary Fig. 1.
Dephasing and damping errors for a TSQRAM
We start by considering the case where there are no deterministic CNOTs and vary the dephasing and damping parameters, T_{2} and T_{1}, respectively. Note that, as expected, the query times have increased by orders of magnitude (see in greater detail in Supplementary Fig. 2), hence the extent of decoherence in the memories. Moreover, to overcome the necessity of performing noisy deterministic CNOTs, the qubits used are now the electronic spin qubits, whose dephasing and damping times are much smaller than their nuclear counterparts, thereby limiting the fidelity of the QRAM tree state. For this reason, we analyzed a wide range of possible values for \({T}_{1}^{{{{\rm{e}}}}}\) and \({T}_{2}^{{{{\rm{e}}}}}\): {20 ms, 200 ms, 2 s, 20 s} and {10 ms, 100 ms, 1 s, 10 s}, respectively. In this scenario of having only probabilistic distributed CNOTs, we analyze for multiple CNOT efficiencies η and T_{2} values, fixing T_{1} = 2 s, as its contribution to the error is negligible compared to the T_{2}. In Fig. 6, we observe infidelity values scaling exponentially with the number of qubits for a completely probabilistic execution of the hybrid protocol. Only for memory coherence times on the second timescale, i.e., T_{2} = 1 s, does the fidelity reach around 80% under a CNOT efficiency of η = 0.5. For other combinations of parameters, we refer to Supplementary Figs. 3–5.
Dephasing, damping and noisy CNOTs for TSQRAM
Here, we explore adding some noisy deterministic CNOTs to counteract the effect of the decoherence for longer periods of time. As seen previously, the better location for these deterministic CNOTs are the nodes that perform the linking step at higher levels of the linking tree. In our simulations, we evaluate different values of the first deterministic layer \({\log }_{2}(N)D\in \{2,3,4,5,6\}\). The results are presented in Fig. 7.
Depending on the CNOT error, the TSQRAM scheme can surpass the fidelities of access of the TDQRAM scheme under high enough T_{2} times in the order of seconds. For other possible sets of parameters, we refer again to Supplementary Figs. 3–5.
Discussion
In this article, we introduce a method to simulate large quantum networks in an open system model. Specifically, this approach enables us to model networks comprising hundreds of stationary qubits by modeling decoherence processes as noisy channels with spindephasing errors, spinflip errors, and noisy CNOT gates. When applied to the challenging but important problem of networkbased QRAM, we find that the qubit depth of memory calls in the recently proposed TDQRAM architecture becomes limited by CNOT errors. To overcome this bottleneck, we propose a modified networkenabled QRAM in which the noisy deterministic gates of ref. ^{20} are replaced by heralded probabilistic CNOT gates, which can sharply reduce gate errors. This scheme, TSQRAM, trades increased query time for improved memory access fidelity and/or memory depth. The TSQRAM protocol makes use of already demonstrated elements (see Supplementary Table I), suggesting the viability of nearterm demonstrations in platforms of solidstate color centers as well as potentially other atomic memory modalities.
An outstanding problem relates to the compounding loss of photonic qubits with increasing memory depth. Since teleportationbased QRAM^{5} has shown that distributed quantum computers naturally map onto quantum networks, error correction schemes proposed for the former may be applied to address the issue of photon loss for the latter. Approaches include (1) photonic forward error correction using, for example, 2D photonic cluster states^{33,34,35,36,37,38} and (2) errorcorrected cluster states^{39,40,41}. We leave the exploration of error correction schemes in the context of QRAM for future studies.
Methods
Discretetimeeventbased simulations with NetSquid
Given the complexity of a quantum network and its formulations, a tool such as NetSquid^{28} is essential to simulating a QRAM. NetSquid is capable of defining intricate discretetimeeventbased protocols, with a number of steps and operations that are executed conditioned on the signaling and heralding of prior processes. Furthermore, NetSquid can simulate quantum circuits, providing methods for (1) stabilizer circuits, with simpler and faster execution, of complexity \({{{\mathcal{O}}}}\left({m}^{2}\right)\), where m is the number of qubits; (2) graph states formalism, with possibly even faster execution, in \({{{\mathcal{O}}}}\left({d}^{2}\right)\), where \(\log m\, <\, d \,<\, m\) and m is again the number of qubits; (3) density matrix formalism, which is slower in execution, in \({{{\mathcal{O}}}}\left({2}^{3m}\right)\); (4) sparse density matrix formalism that relies on sparse matrix codes to speed up the execution.
Instead of using the density matrix formalism from NetSquid, we begin by retrieving all the noise information in each step of the protocol from a noiseless discretetime event simulation, resorting to the stabilizer formalism in polynomial time. The noise information is constituted by the time qubits spent decohering, together with the information about the channels of decoherence that would have been applied in real noisy simulations, both for waiting times and gate errors. We then incorporate the extracted information to estimate the effects of decoherence at each step a posteriori. With this information at hand, we have access to the timeevolved state of the QRAM tree at all steps of the protocol, which allows us to reconstruct the noise that would have been applied in the system in a noisy simulation. To reconstruct the density matrix, we find the analytical expressions for the density matrices of smaller parts of the system and how they evolve after the required operations under a set of noise channels. We express these as fundamental building blocks in terms of noise parameters, namely the probability of error and time of decoherence. This is what allows us to postpone the noise calculations to the end of the simulation without losing the effects of the natural stochastic behavior of the protocol. In a way, this can be understood as precompiling the error effect on the intermediary states of the protocol to shortcome the exponential complexity of calculating the density matrix at each time step using a quantum simulator.
In the rest of the “Methods” section, we formalize these elementary building blocks for the operations required to create these GHZ states across each physical layer in the QRAM and explain how different types of noise affect each of the intermediate steps, allowing for a reconstruction of the density matrix. We analyzed dephasing, damping and depolarizing channels, and we believe other noise models could be added in a similar manner. The result is an explicit description of the final state of the QRAM access tree state prior to the execution of the teleportation protocol. The error in the state of the access tree encompasses the majority of all the errors of accessing the QRAM, as the number of steps and operations made after creating the GHZ state grows with the total number of memory addresses logarithmically, whereas the process of generating the GHZ state requires a number of operations linear with the number of memories.
Elementary building blocks for TDQRAM
The protocol for generating GHZ states across each layer consists of two steps (see Fig. 8):

1.
Generating entanglement between the oddindexed links. This entails first distributing photonmediated heralded entanglement between the electrons, with a certain efficiency η, followed by electronic CNOTs being applied with the electron qubit acting as the control and the nuclear qubits as the target. Finally, a measurement of the electron spin in the X basis, with posterior corrections sent to the nuclear qubits.

2.
The second step links the entangled pairs, creating a larger GHZ state distributed across each layer. This starts off by generating heralded entanglement, with the same efficiency η, between the evenindexed links, followed by applying nuclear CNOTs, where now the control is the nuclear qubit and the target is the electronic qubit. We then make consequent measurements on the Z basis on both electronic qubits, followed by appropriate Pauli corrections.
After the aforementioned steps, each physical layer hosts a GHZ state shared among all the nodes. Subsequently, each physical layer extends its (2^{k−1})GHZ state into a (2^{k−1} + 1)GHZ state by sharing an additional entangled pair between an outermost node in the QRAM layer and the quantum computer, which holds the address state. After performing a Bell state measurement and corresponding corrections, the address state is teleported to the QRAM. Lastly, the memories can be accessed in superposition to complete the QRAM protocol.
EPR pair creation and transferring for TDQRAM
The first step to creating a GHZ state across each layer is to share entanglement between neighboring nodes. EPR pairs are created by performing a distributed CNOT gate between these nodes’ electronic spin qubits, mediated by a photon. Despite the process being probabilistic with an efficiency dependent on the experimental implementation, it is a heralded entanglement. Hence, the presence (absence) of photon detection informs the success (failure) of the entangling attempt. After this CNOT is applied between the electronic spins, an EPR pair is created and transferred to the nuclear spins in each node via a deterministic electronic CNOT.
Given the different operations and various types of qubits involved, we introduce noise sources in the system to estimate the protocol’s fidelity. We consider amplitude damping, dephasing, and CNOT gate errors for both electronic and nuclear spin qubits.
In this step, illustrated in Fig. 8, the following takes place:

1.
eL and eR decohere for a duration of time t_{eL} and t_{eR}, respectively;

2.
An electronic CNOT is applied between eL and nL, with an error probability of p_{e},

3.
An electronic CNOT is applied between eR and nR with an error probability of p_{e},

4.
nL and nR decohere for a duration of time t_{nL} and t_{nR}, respectively.
Hence, these parameters, plus the parameters associated with the physical systems, namely the T_{1} and T_{2} times, govern the final form of the entangled pairs. Using the notation ϵ(σ) = 1 − e^{−σ} and \(\overline{\epsilon }(\sigma )=1\epsilon (\sigma )={e}^{\sigma }\) for parameters that are functions of other physical parameters, namely the elapsed times and coherence times. We will use ϵ for parameters that go to zero in the absence of noise, as is the case for ϵ(⋅) and p_{n}. We also further assume ϵ ≪ 1. We then apply the following sequence of noise channels (check Supplementary Methods for details on the parameters):

1.
Apply a Dephasing channel with probability \(\epsilon ({t}_{{{{\rm{eL}}}}}/{T}_{2}^{{{{\rm{e}}}}})\equiv {\epsilon }_{{{{\rm{eL}}}}}^{(2)}\) and \(\epsilon ({t}_{{{{\rm{eR}}}}}/{T}_{2}^{{{{\rm{e}}}}})\equiv {\epsilon }_{{{{\rm{eR}}}}}^{(2)}\) to electronic spin qubits eL and eR, respectively;

2.
Apply an Amplitudedamping channel with probability \(\epsilon ({t}_{{{{\rm{eL}}}}}/{T}_{1}^{{{{\rm{e}}}}})\equiv {\epsilon }_{{{{\rm{eL}}}}}^{(1)}\) and \(\epsilon ({t}_{{{{\rm{eR}}}}}/{T}_{1}^{{{{\rm{e}}}}})\equiv {\epsilon }_{{{{\rm{eR}}}}}^{(1)}\) to electronic spin qubits eL and eR, respectively;

3.
Apply Depolarizing channels with probability p_{e} to all qubits after applying CNOTs (modeling a noisy CNOT);

4.
Apply a Dephasing channel with probability \(\epsilon ({t}_{{{{\rm{nL}}}}}/{T}_{2}^{{{{\rm{n}}}}})\equiv {\epsilon }_{{{{\rm{nL}}}}}^{(2)}\) and \(\epsilon ({t}_{{{{\rm{nR}}}}}/{T}_{2}^{{{{\rm{n}}}}})\equiv {\epsilon }_{{{{\rm{nR}}}}}^{(2)}\) to nuclear spin qubits nL and nR, respectively;

5.
Apply an Amplitudedamping channel with probability \(\epsilon ({t}_{{{{\rm{nL}}}}}/{T}_{1}^{{{{\rm{n}}}}})\equiv {\epsilon }_{{{{\rm{nL}}}}}^{(1)}\) and \(\epsilon ({t}_{{{{\rm{nR}}}}}/{T}_{1}^{{{{\rm{n}}}}})\equiv {\epsilon }_{{{{\rm{nR}}}}}^{(1)}\) to nuclear spin qubits nL and nR, respectively;
The final state for each entangled pair becomes:
where
and
For intuition regarding the ϵ function, consider the following two limits: (1) σ → 0 in the noiseless regime where the memory coherence time goes to infinity (no decoherence) and (2) σ → ∞ where there only exists noise and all the information is scrambled. In these limits, we retrieve: \(\mathop{\lim }\nolimits_{\sigma \to 0}\epsilon (\sigma )=0,\mathop{\lim }\nolimits_{\sigma \to \infty }\epsilon (\sigma )=1,\mathop{\lim }\nolimits_{\sigma \to 0}\overline{\epsilon }(\sigma )=1\) and \(\mathop{\lim }\nolimits_{\sigma \to \infty }\overline{\epsilon }(\sigma )=0\).
Linking of Bell pairs for TDQRAM
The following step is crucial to extending entanglement from bipartite to GHZ states across the entire physical layer of the QRAM. It relies on using an entangled pair to combine two GHZ states of smaller sizes into a larger GHZ state, whose number of qubits equals the sum of each of the elementary GHZ states (i.e., n_{1}GHZ linked with a n_{2}GHZ becomes a (n_{1} + n_{2})GHZ state).
In this step, we account for decoherence before applying CNOTs, therefore entering the previous expressions for the form of each pair. The decoherence to be analyzed in this step stems from:

1.
A nuclear CNOT gate on eL and nL with probability of error p_{n};

2.
A nuclear CNOT gate on eR and nR with probability of error p_{n};

3.
Nuclear qubits nL and nR decohere after a CNOT for \({t}^{{\prime} }\).
Additionally, for each block, we analyze the impact of decoherence by applying the following noise channels:

1.
Apply depolarizing channels with probability p_{n} to all qubits (eL, eR, nL, nR) after applying CNOTs (modeling a noisy CNOT);

2.
Apply a dephasing channel with probability \(\epsilon ({t}_{{{{\rm{nL}}}}}^{{\prime} }/{T}_{2}^{{{{\rm{n}}}}})\equiv {\epsilon }_{{{{\rm{nR}}}}}^{{\prime} (2)}\) and \(\epsilon ({t}_{{{{\rm{nR}}}}}^{{\prime} }/{T}_{2}^{{{{\rm{n}}}}})\equiv {\epsilon }_{{{{\rm{nL}}}}}^{{\prime} (2)}\) to nuclear spin qubits nL and nR, respectively;

3.
Apply an amplitudedamping channel with probability \(\epsilon ({t}_{{{{\rm{nL}}}}}^{{\prime} }/{T}_{1}^{{{{\rm{n}}}}})\equiv {\epsilon }_{{{{\rm{nR}}}}}^{{\prime} (1)}\) and \(\epsilon ({t}_{{{{\rm{nR}}}}}^{{\prime} }/{T}_{1}^{{{{\rm{n}}}}})\equiv {\epsilon }_{{{{\rm{nL}}}}}^{{\prime} (1)}\) to nuclear spin qubits nL and nR, respectively;
Note that all the following calculations are now lower bounds for the fidelity, as the calculation of the full analytical expressions grows exponentially with the number of qubits. Because of this, we keep only the terms up to \({{{\mathcal{O}}}}\left(\epsilon \right)\). In Supplementary Methods, we detail and test the validity of our approximations.
The final GHZ state in each layer is described by a matrix with the following form:
where all ϵ terms are of at least order \({{{\mathcal{O}}}}(\epsilon )\) and do not contribute to infidelity, as they are orthogonal to the GHZ state.
The diagonal elements that we consider are only the first and the last, as the remaining ones have at least \({{{\mathcal{O}}}}\left(\epsilon \right)\) and, when expanding to a larger GHZ state, contribute in \({{{\mathcal{O}}}}\left({\epsilon }^{2}\right)\) or higher orders, hence negligibly affecting the fidelity.
Let us first consider the form of the state after executing the linking protocol in a noiseless manner, with previously noisy states, as the ones that result from the entangling step given by Eq. (3). Starting with the simple case of a 4qubit GHZ state built from three states of the form of Eq. (3), with parameters (μ_{j}, ν_{j}), j = 1, 2, 3 respectively, the final matrix is:
where we, again, denote a bar over a variable as 1 minus itself, \({\overline{\mu }}_{i}\equiv 1{\mu }_{i}\). Note that each of the μ_{i} comes from one of the pairs used to create the GHZ state, as these pairs are solely described by two numbers (μ_{i}, ν_{i}) (see Eq. (3)). There exists a rule for each entry in the diagonal, which we detail in Supplementary Methods, and the same rule holds for any number of qubits of the final state. The GHZ diagonal entries then become:
Now, adding the effect of the noisy CNOTs on the state, we calculate the diagonal terms that are shown to be identical, given by:
where we recall that every term with p_{n}, μ_{i} ≪ 1 converges to zero in the noiseless limit. For the other diagonal entries, we multiply them by h(p_{n}).
Finally, incorporating memory decoherence after CNOTs, we perform another approximation. For the diagonal terms, only the damping channel plays a role. The first and last entries of the diagonal become:
In this approximation, the extra terms that appear for the first entry are already of order \({{{\mathcal{O}}}}\left({\epsilon }^{3}\right)\) and could be neglected.
Lastly, we compute the offdiagonal terms by multiplying every contribution from each noise channel applied in the correct manner. The expression is given by:
When extending the linking protocol to a larger number of qubits, the expressions maintain their form. We only need to add all the terms in a similar manner as in the case of the 4GHZ state. The complete analysis is detailed in Supplementary Methods.
Thus far, we show how three types of noise (one for the twoqubit operations and two for individual memories) influence the final state of the GHZ states generated across each physical layer of the QRAM access tree. Note that we always present the full expressions accounting for all the noise channels. In fact, if we include a specific noise channel or a subset of what we have considered, we may simply set the parameters corresponding to other noises to zero. For example, it is straightforward to verify that setting p_{e} and p_{n} to zero and T_{1} to infinity recovers the case for only having dephasing, thus affecting only the offdiagonal terms. The same is valid for all the other noises.
Modified protocol and building blocks for TSQRAM
We now consider an alternative architecture that enables different subsets of each layer to create GHZ states independently. As illustrated in Fig. 9, this architecture assumes two electron spins and one nuclear spin (instead of each node of the QRAM having an electronic spin and a nuclear spin assumed for the TDQRAM). As we will show, this architecture still retains similar building blocks as the aforementioned TDQRAM protocol.
EPR creation for TSQRAM
As in the nonhybrid version of the protocol, the first step to creating an EPR pair between two physically separated electronic spins is sending a photon that interacts with them sequentially. A subsequent measurement heralds the successful production of a spinspin EPR pair. Notably, there are no deterministic CNOTs applied to transfer the qubit states onto the nuclear spins, as we only work with the electron spins at this stage.
The final state shared between the electronic spins is the one of Eq. (3) in the limit of the absence of electronic CNOT error (p_{e} → 0) and altering the memory decoherence noise from nuclear to electronic (\({\epsilon }_{{{{\rm{nL}}}}}\to {\epsilon}_{\rm{eL}} ^{\prime}\) and \({\epsilon }_{{{{\rm{nR}}}}}\to {\epsilon}_{{\rm{eR}}} ^{\prime}\)):
where
and again,
where we use the same abbreviation \(\overline{\epsilon }(\sigma )=1\epsilon (\sigma )={e}^{\sigma }\).
Linking pairs in the probabilistic scenario for a TSQRAM
TSQRAM differs from TDQRAM in that the operation of linking pairs has a nonunity probability of succeeding—let us call this probability p_{CNOT}. Moreover, it is executed in a similar way as that of creating an EPR pair:

1.
Interact photon γ with the left electronic spin qubit eL, executing a local CNOT,

2.
Send the single photon γ to the right cavity,

3.
Interact the photon γ with the right electronic spin qubit eR, executing a local CNOT,

4.
Measure the photon γ,

5.
Measure the right (or left) electronic spin in X.
Importantly, both cavities belong to the same node in this step. This results in a controlled gate applied between the right and left electronic spin qubits. Unlike before, it is still necessary to measure one of the nodes’ electronic spin qubits, as the state has twice the number of qubits as the final state (we chose to measure the right electron, but one could choose to keep the right and measure the left instead; the choice is arbitrary and translates to the same practical outcome). This measurement should be on the X basis in order to not destroy the entanglement shared among all the qubits and rendering the state useless. Moreover, a correction must be made depending on the outcome of the measurement of the electronic spin qubit and the photonic qubit.
Afterward, the GHZ states shared between the left and right nodes are linked into a larger GHZ state via an intermediary node. Inside this intermediary node, its left electronic spin merges into the larger GHZ state. We again take into account the previous calculations for detailing the density matrix of the final state. The decoherence sources are now only provenient from the memories of where each qubit is being held (which we chose to be the left cavity of the node). As we used nearperfect probabilistic CNOTs mediated by a photon, only its memory affects the state fidelity. Thus, for the remainder of the protocol, we:

1.
Apply a dephasing channel with probability \(\epsilon ({t}_{{{{\rm{eL}}}}}^{{\prime\prime} }/{T}_{2}^{{{{\rm{e}}}}})\equiv {\epsilon }_{{{{\rm{eL}}}}}^{{\prime\prime} (2)}\) to electronic spin qubit eL;

2.
Apply an amplitudedamping channel with probability \(\epsilon ({t}_{{{{\rm{eL}}}}}^{{\prime\prime} }/{T}_{1}^{{{{\rm{e}}}}})\equiv {\epsilon }_{{{{\rm{eL}}}}}^{{\prime\prime} (1)}\) to electronic spin qubit eL.
Importantly, the following calculations are again lowerbound approximations for fidelity. Performing the calculations for a simple link of two entangled pairs described by Eq. (13), with parameters (μ_{j}, ν_{j}), j = 1, 2, the final density matrix of the 3GHZ state, prior to any memory decoherence, is:
Adding the memory decoherence accounting for both dephasing and amplitudedamping leads to a matrix similar in form to one shown in Eq. (7), except with entries changing to:
Linking pairs in the deterministic scenario for TSQRAM
We next describe a scheme for deterministic CNOT gates. Since the composition of each node is now different, the operations one needs to execute to deterministically link smaller GHZ states into larger GHZ states changes as well:

1.
Apply a deterministic electronic CNOT controlled by the left electronic spin qubit eL and targeted at the nuclear spin qubit n,

2.
Measure the left electronic spin qubit eL in X,

3.
Apply a deterministic nuclear CNOT controlled by the nuclear spin qubit n and targeted at the right electronic spin qubit eR,

4.
Measure the right electronic spin qubit eR in Z.
Measurementconditioned corrections result in a GHZ state consisting of the nuclear spin and the remaining electronic spin qubits. Notice that by not involving the photonmediated CNOT, this has been done in a deterministic fashion. In this case, we must consider additional errors, namely those that arise from using deterministic electronic and nuclear CNOTs. The sequence of noise channels becomes:

1.
Apply depolarizing channels with probability p_{e} to the electronic spin qubit eL and to the nuclear spin qubit n,

2.
Apply depolarizing channels with probability p_{n} to the nuclear spin qubit n and to the electronic spin qubit eR,

3.
Apply a dephasing channel with probability \(\epsilon ({t}_{n}^{{\prime} }/{T}_{2}^{{{{\rm{n}}}}})\equiv {\epsilon }_{n}^{{\prime} (2)}\) to nuclear spin qubit n,

4.
Apply an amplitudedamping channel with probability \(\epsilon ({t}_{n}^{{\prime} }/{T}_{1}^{{{{\rm{n}}}}})\equiv {\epsilon }_{n}^{{\prime} (1)}\) to nuclear spin qubit n.
From here, we calculate the final state’s density matrix. Performing the calculations for the same simple link of two entangled pairs described by Eq. (13), with parameters (μ_{j}, ν_{j}), j = 1, 2, the final density matrix of the 3GHZ state, prior to any memory decoherence and without any CNOT errors is the same as Eq. (17). Adding the effect of the CNOTs leads to:
where we set p_{e} = p_{n} ≡ p. In fact, all diagonal entries can be decomposed into terms of the form \({(1p)}^{2}\,{{\mbox{diag}}}\,(\rho )+{\mathbb{1}}p/2(1p/2)\). Using this fact, we incorporate the posterior amplitudedamping noise channels:
where \(\tilde{h}(p)={(1p)}^{2}+p/2(1p/2)\). In the Supplementary Methods, we present the derivation for a chain of an arbitrary number of channels, as well as proof of the validity of our approximations.
Validity of the results
In this section, we discuss our methods and provide empirical proof for our noise analysis robustness. In our work, we have considered an analysis of a discrete protocol, where each gate takes a fixed amount of time, and qubits stay in memory waiting for instructions from the protocol. Given the discreteness of the problem, solving the master equations in each of the specific time periods where the noise actually happens is equivalent to applying the corresponding noise channels.
To support our claims and demonstrate the equivalence of using the full density matrix and our methods of postponing the noise analysis to the end of the protocol simulation, we present one additional figure. This figure shows the comparison of the fidelity of the GHZ states distributed at each of the layers of the QRAM for a small QRAM (5 layers, ~36 qubits) obtained from our simulation methods and those derived using the density matrix formalism. We present the comparison for simulations for the TDQRAM, as for TSQRAM, we use the same methods, just under different assumptions over which noise channels are applied. Let F_{DM} be the fidelity calculated using the full density matrix and F_{S} be the one from our simulation methods. The figure showcases how the difference of the fidelities calculated using the density matrix and our methods evolve for different layers (with increasing amounts of qubits), weighted by the overall error (or infidelity of the state), i.e., ∣F_{DM} − F_{S}∣/(1 − F_{DM}).
One can observe in Fig. 10 that the error is always less than 1%, showing a trend of either maintaining or decreasing as the size of the system increases. This provides empirical proof of the power of our methods across the different scenarios.
Data availability
The data supporting the results in this work are available from L.B. at https://github.com/luisbugalho/HeraldedQRAM.
Code availability
The code supporting the results in this work is available from L.B. at https://github.com/luisbugalho/HeraldedQRAM.
References
Biamonte, J. et al. Quantum machine learning. Nature 549, 195–202 (2017).
Harrow, A. W., Hassidim, A. & Lloyd, S. Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 150502 (2009).
Kiani, B. T., Villanyi, A. & Lloyd, S. Quantum medical imaging algorithms. Preprint at https://arxiv.org/abs/2004.02036 (2020).
Grover, L. K. A fast quantum mechanical algorithm for database search. In Proc. TwentyEighth Annual ACM Symposium on Theory of Computing, STOC ’96, 212–219 (Association for Computing Machinery, 1996).
Chen, K. C., Dai, W., ErrandoHerranz, C., Lloyd, S. & Englund, D. Scalable and highfidelity quantum random access memory in spinphoton networks. PRX Quantum 2, 030319 (2021).
Jaeger, R. & Blalock, T. Microelectronic Circuit Design 4th edn (McGrawHill Education, 1997).
Giovannetti, V., Lloyd, S. & MacCone, L. Architectures for a quantum random access memory. Phys. Rev. A 78, 052310 (2008).
Hann, C. T., Lee, G., Girvin, S. & Jiang, L. Resilience of quantum random access memory to generic noise. PRX Quantum 2, 020311 (2021).
Giovannetti, V., Lloyd, S. & MacCone, L. Quantum random access memory. Phys. Rev. Lett. 100, 160501 (2008).
Tóth, G. Multipartite entanglement and highprecision metrology. Phys. Rev. A 85, 022322 (2012).
Sidhu, J. S. & Kok, P. A geometric perspective on quantum parameter estimation. AVS Quantum Sci. 2, 014701 (2019).
Murta, G., Grasselli, F., Kampermann, H. & Bruß, D. Quantum conference key agreement: a review. Adv. Quantum Technol. 3, 2000025 (2020).
Wehner, S., Elkouss, D. & Hanson, R. Quantum internet: a vision for the road ahead. Science 362, eaam9288 (2018).
Alshowkan, M. et al. Reconfigurable quantum local area network over deployed fiber. PRX Quantum 2, 040304 (2021).
Van den Nest, M. Simulating quantum computers with probabilistic methods. Quantum Inf. Comput. 11, 784–812 (2011).
Jozsa, R. & van den Nest, M. Classical simulation complexity of extended Clifford circuits. Quantum Inf. Comput. 14, 633–648 (2014).
Takahashi, Y., Takeuchi, Y. & Tani, S. Classically simulating quantum circuits with local depolarizing noise. Theor. Comput. Sci. 893, 117–132 (2021).
Bhaskar, M. K. et al. Experimental demonstration of memoryenhanced quantum communication. Nature 580, 60–64 (2020).
Wan, N. H. et al. Largescale integration of artificial atoms in hybrid photonic circuits. Nature 583, 226–231 (2020).
Nguyen, C. T. et al. An integrated nanophotonic quantum register based on siliconvacancy spins in diamond. Phys. Rev. B 100, 165428 (2019).
Bradley, C. E. et al. Robust quantumnetwork memory based on spin qubits in isotopically engineered diamond. NPJ Quantum Inf. 8, 122 (2022).
Chen, K. C., Bersin, E. & Englund, D. A polarization encoded photontospin interface. NPJ Quantum Inf. 7, 1–6 (2021).
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).
Duan, L.M. & Kimble, H. J. A scheme for preparation of multiatom entanglement by detecting the cavity decay and analysis of its implementation. In Proc. SPIE Quantum Communications and Quantum Imaging, Vol. 5161, 40–47 (SPIE, 2004).
CalderonVargas, F. A. et al. Fast highfidelity entangling gates for spin qubits in Si double quantum dots. Phys. Rev. B 100, 035304 (2019).
Coopmans, T., Brand, S. & Elkouss, D. Improved analytical bounds on delivery times of longdistance entanglement. Phys. Rev. A 105, 012608 (2022).
Duan, L.M. & Kimble, H. J. Scalable photonic quantum computation through cavityassisted interactions. Phys. Rev. Lett. 92, 127902 (2004).
Coopmans, T. et al. NetSquid, a NETwork Simulator for QUantum Information using Discrete events. Commun. Phys. 4, 164 (2021).
Dai, W., Peng, T. & Win, M. Z. Optimal remote entanglement distribution. IEEE J. Sel. Areas Commun. 38, 540–556 (2020).
Childress, L. et al. Coherent dynamics of coupled electron and nuclear spin qubits in diamond. Science 314, 281–285 (2006).
Findler, C., Lang, J., Osterkamp, C., Nesládek, M. & Jelezko, F. Indirect overgrowth as a synthesis route for superior diamond nano sensors. Sci. Rep. 10, 22404 (2020).
Brand, S., Coopmans, T. & Elkouss, D. Efficient computation of the waiting time and fidelity in quantum repeater chains. IEEE J. Sel. Areas Commun. 38, 619–639 (2020).
Pichler, H., Choi, S., Zoller, P. & Lukin, M. D. Universal photonic quantum computation via timedelayed feedback. Proc. Natl Acad. Sci. USA 114, 11362–11367 (2017).
Larsen, M. V., Guo, X., Breum, C. R., NeergaardNielsen, J. S. & Andersen, U. L. Deterministic generation of a twodimensional cluster state. Science 366, 369–372 (2019).
Russo, A., Barnes, E. & Economou, S. E. Generation of arbitrary allphotonic graph states from quantum emitters. New J. Phys. 21, 055002 (2019).
Pant, M., Towsley, D., Englund, D. & Guha, S. Percolation thresholds for photonic quantum computing. Nat. Commun. 10, 1070 (2019).
Uppu, R. et al. Scalable integrated singlephoton source. Sci. Adv. 6, eabc8268 (2020).
Michaels, C. P. et al. Multidimensional cluster states using a single spinphoton interface coupled strongly to an intrinsic nuclear register. Quantum 5, 565 (2021).
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).
Nemoto, K. et al. Photonic architecture for scalable quantum information processing in diamond. Phys. Rev. X 4, 031022 (2014).
Choi, H., Pant, M., Guha, S. & Englund, D. Percolationbased architecture for cluster state creation using photonmediated entanglement between atomic memories. NPJ Quantum Inf. 5, 104 (2019).
Acknowledgements
L.B., E.Z.C., and Y.O. thank the support from Fundação para a Ciência e a Tecnologia (FCT, Portugal), namely through projects UIDB/04540/2020 and UIDB/50008/2020. L.B. acknowledges the support of FCT through scholarship BD/05268/2021 and of the PEPR integrated project EPiQ ANR22PETQ0007 part of Plan France 2030. E.Z.C. also acknowledges funding by FCT through project 2021.03707.CEECIND/CP1653/CT0002. W.D. is supported by the National Science Foundation to the Computing Research Association for the CIFellows 2020 Program. K.C.C. and D.E. acknowledge funding support by the National Science Foundation (NSF) Engineering Research Center for Quantum Networks (CQN), awarded under cooperative agreement number 1941583, and the ARO MURI on ‘Theory and Engineering of LargeScale Distributed Entanglement’ (W911NF2110325). D.E. further acknowledges support from the NSF CAccel program, grant number 2040695.
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L.B., E.Z.C., W.D. and K.C. performed the simulation of the protocol and consequent analysis. All authors discussed the analysis of the data and contributed to writing or proofreading the manuscript. Y.O. and E.D. supervised the project.
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Bugalho, L., Cruzeiro, E.Z., Chen, K.C. et al. Resourceefficient simulation of noisy quantum circuits and application to networkenabled QRAM optimization. npj Quantum Inf 9, 105 (2023). https://doi.org/10.1038/s4153402300773x
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DOI: https://doi.org/10.1038/s4153402300773x