Encodings and Node Architecture: Lessons from the Fault-Tolerant Distributed Transversal Fanout

Encodings and Node Architecture: Lessons from the Fault-Tolerant Distributed Transversal Fanout Seng W. Loke

arXiv:2610.22966v1 [quant-ph] 16 Oct 2026

School of Information Technology, Deakin University, Burwood, VIC 3125, Australia. seng.loke@deakin.edu.au

Abstract—An encoding choice specifies the quantum errorcorrecting code used to encode logical qubits at a given node. Different encoding choices may differ in the number of physical qubits, the number of encoded logical qubits, the code distance, and the available fault-tolerant operations. We study different encodings for a distributed transversal fanout operation and look at its implications on node architecture. Our experiments suggest that an encoding with higher distance, even if requiring more ebits, can yield a lower logical error rate for distributed operations, and that “upgrading” an encoding prior to distributed operations and having a cache, as part of node architecture, for qubits can be useful for distributed operations. Index Terms—distributed quantum computing, tansversal gate gadgets, fault-tolerant quantum computing, multipartite entanglement, distributed fanout gates

I. I NTRODUCTION Fanout operations, with one control qubit for multiple target qubits, are amenable to efficient implementation in some types of quantum hardware such as trapped-ion quantum computers, enabling reduced depth quantum computations [9], [12]. It has also been considered in other platforms, e.g., [16], [28]. In the context of distributed or modular quantum computing, often touted as a means of scaling up quantum computation, beyond monolithic platforms [2], [5], [8], [14],1 distributed fanout gadgets using distributed GHZ states have been proposed in [30], and investigated in [17], [19]. Towards faulttolerant distributed quantum computing, different encodings for quantum error correction have been proposed and are being implemented, including surface codes (e.g., [6], [22], [23]), Bivariate-Bicycle (BB) codes (e.g., transversal nonlocal CNOTs [25] and inter-module operations [31]), and floquet codes [27]. A fault-tolerant distributed transversal fanout was discussed in [20]. Previous work looked at encoding choices for transversal non-local CNOTs and corresponding LER estimates for distributed global gates based on different encodings [18]. This paper studies transversal implementations of faulttolerant distributed fanout with qubits in different BivariateBicycle (BB) encodings. Our results suggest (i) that with good enough encodings and low enough physical error rates, distributed quantum operations can achieve comparable Logical 1 For example, see [13], https://spectrum.ieee.org/quantum-computers, https://newsroom.ibm.com/2025-11-20-ibm-and-cisco-announce-plans-to-b uild-a-network-of-large-scale,-fault-tolerant-quantum-computers. See also https://quantnet.lbl.gov and https://www.ox.ac.uk/news/2025-02-06-first-distr ibuted-quantum-algorithm-brings-quantum-supercomputers-closer.

Dc-U c

≡ t

U

Fig. 1: Distributed controlled-U (i.e., dCNOT is when U=X (⊕) as shown) between nodes QC and QC ′ (the computation qubits are c, the control qubit, and t, the target qubit), and the wavy line illustrates a Bell pair involving the communication qubits from the two nodes both initially |0⟩. The right hand side shows the notation we will use for a distributed controlledU operation.

Error Rates (LERs) with equivalent (entirely) local computations, and (ii) that even if an encoding uses more physical qubits causing non-local transversal operations to consume more ebits, this can be compensated for, in terms of LER, by the higher code distance. We also explore the implications of the results for designing nodes participating in distributed quantum computations. In the rest of this paper, we first briefly review distributed fanout operations in §2, and then look at distributed or nonlocal transversal fanout operations in §3. In §4, we provide a simulation analysis of distributed transversal fanouts with different encodings, and then explore implications for node architecture design in §5. We conclude with avenues for future work in §6. II. B RIEF R EVIEW OF D ISTRIBUTED FANOUT O PERATIONS We first consider the distributed CNOT gate (or dCNOT, in short, a.k.a. non-local CNOT), which is needed as a resource for each dCNOT gate, as shown in Figure 1. By a distributed fanout operation, we refer to an operation where the same control qubit is used for multiple target qubits, on different nodes. This can arise from the structure of a quantum circuit itself, or from a control-U operation where the unitary U spans multiple qubits and is decomposed into operations executed on qubits distributed on different nodes [21], [29], [30]. For example, the quantum circuit such as:

fanout c t1 t2 t3

c U1

t1

U1

≡

U2 U3

t2

U2

t3

U3

i.e., a multitarget control-U , where U1 acts on qubit t1, U2 acts on qubit t2, and U3 acts on qubit t3, each qubit on a different node, will result in the distributed operation as shown in Figure 2; note that the wavy line in the figure connecting the four (communication) qubits a0 , a1 , a2 , a3 represents a distributed 4-qubit GHZ state of the form √12 (|0000⟩ + |1111⟩)a0 a1 a2 a3 , over four nodes A′ , A, B and C. In the rest of the paper, we particularly focus on the case where Ui = X, unless otherwise noted.

III. T RANSVERSAL N ON -L OCAL FANOUT A transversal operation can be defined as follows. Let there be m code blocks, each consisting of n physical qubits. A quantum Nn operation U is transversal if it can be expressed as U = i=1 Ui , where each Ui acts only on the i-th physical qubits of the code blocks and does not couple two qubits within the same code block. From [25], we see that a logical CNOT between two logical qubits is transversal, when the logical qubits are encoded using the Bivariate-Bicycle (BB) qLDPC code; multiple logical CNOT operations are performed simultaneously. Following [20], because the logical fanout decomposes into CNOTs between the same code blocks, and because each of those logical CNOTs is transversal, their product can be reordered into a set of independent physical fanout operations, one per physical coordinate; a logical fanout gadget is transversal if it can be composed using transversal logical CNOT gadgets. This comes from [10], where “a product of transversal gate gadgets is a transversal gate gadget”, and since, a logical fanout is a product of (transversal) CNOT operations, by definition. Using the BB encoding [[n, k, d]] of logical qubits, with operations on a block of n qubits, we effectively perform k independent logical fanout operations simultaneously. Using the [[n, k, d]] BB encoding, where the logical CNOT is transversal, i.e., between two blocks C with qubits ci and T with qubits ti , denoted by CN OT (C; T ) = Πi∈{1,...,n} CN OT (ci ; ti ), consider one control block C = {c1 , ..., cn } with n qubits, and m target blocks Ti , i ∈ {1, ..., m}, each with qubits Ti = {ti1 , ..., tin }, where block C encodes k logical qubits {c̄(l) | l ∈ {1, ..., k}} and each of the (l) m blocks, Ti , encodes k logical qubits {t̄i | l ∈ {1, ..., k}}.

Suppose we wanted to perform the following operation: F AN OU T (C; T1 , ..., Tm ) O (l) := F AN OU T L (c̄(l) ; t̄1 , ..., t̄(l) m) l∈{1,...,k}

where F AN OU T L denotes the logical fanout acting on the corresponding l-th encoded logical qubit, 1 ≤ l ≤ k. We do this by performing n independent physical fanout operations, one for each coordinate (corresponding to a physical qubit) of the code block: O F AN OU T (ci ; t1i , ..., tmi ) i∈{1,...,n}

where F AN OU T denotes a physical fanout operation, each physical qubit ci fans out as control to the corresponding physical qubits in every target block [20]. Note that the above applies to non-local CNOTs and non-local FANOUTs, e.g., with one block per node (or k logical qubits per node). IV. S IMULATION ANALYSIS WITH D IFFERENT E NCODINGS We consider a 4-node configuration with one code block per node in most of our analysis below; for comparison, we have a 4 blocks on one node configuration. With all four blocks, we have a total of 16 logical qubits involved in 4 concurrent distributed transversal fanout operations (each involving 4 logical qubits, one from each node). Figure 3 illustrates the configurations used in our experiments. We used the Transversal Multiple Code Block Simulator (TMCBS) library code from [25] (developed for simulating non-local transversal CNOTs), which uses stim,2 and we use sinter3 for Monte Carlo sampling. Figure 4 shows the graph of LERs vs PERs, using pebit = pghz = 10p, for PER p, using three different BB encodings [[18, 4, 4]], [[36, 4, 6]], and [[54, 4, 8]], for two simulated realizations of a distributed (or non-local) fanout, one using a one shot approach to create distributed GHZ states (1shotghzfanout), and one using a sequence of 3 (non-local) CNOTS (since there are four blocks), equivalent to the fanout (3cnotseq). We construct (blocks of) GHZ states for 1shotghz-fanout via a transversal protocol: among corresponding qubits in the communication qubit blocks on each node, perform the Hadamard H on one “root” (control) qubit, and then perform non-local CNOT between the “root” and each of the other qubits; we count the entire process as taking one simulation step, to simulate the one shot approach to creating GHZ states (one shot GHZ approaches are possible, e.g. [1], [24]). The respective noise models and syndrome extraction are similar to [20] and is summarised here in Table I. For each data point, we also show error bars corresponding to 95% confidence intervals. Table II in the Appendix shows the figures plotted in Figure 4. It can be seen that even when larger blocks are used with more physical qubits and higher distance encodings, even 2 https://github.com/quantumlib/stim 3 https://pypi.org/project/sinter/

Z m1 ⊕m2 ⊕m3

c

c

A’ a0

m0

a1

X m0

Dfanout c

m1

H

A U1

t1

t1

U1

t2

U2

t3

U3

≡ a2

X

m0

m2

H

B U2

t2 a3

X m0

m3

H

C U3

t3

Fig. 2: Distributed fanout operation with single control qubit (on A′ ) for multiple target qubits (one on A, one on B and one on C) - all target qubits on different nodes from the control qubit. The right hand side shows the notation we will use for distributed fanout throughout the paper. Note that mi ∈ {0, 1}. Here, the qubits a0 , a1 , a2 and a3 form a distributed GHZ state. TABLE I: Syndrome-extraction schedule for the different distributed fanout implementations, and the distributed CNOTs (dCNOT)s they are constituted by. For each, initial extraction consists of one first-pass round followed by three repeated rounds. For 3cnot-seq, an additional extraction round is performed between successive non-local CNOT operations; this isolates successive logical operations and provide the same fault-tolerance boundary used by the TMCBS non-local-CNOT protocol. No such intermediate extraction is required in the GHZ-fanout circuit because the fanout is treated as one logical operation. DEPOLARIZE2 in Stim is a two-qubit depolarizing noise channel, with identity with probability 1−p Q ebit , and pebit , otherwise, i.e. P r(II) = 1 − pebit , P r(P ) = pebit , for P ∈ {IX, IY, IZ, XI, . . . , ZZ}, and E (p ) = GHZ ghz q D1 (q, pghz /2), where the 15 implementation applies independent D1 (DEPOLARIZE1) channels to the (distributed) GHZ qubits (where DEPOLARIZE1 is Stim’s single-qubit depolarizing channel) giving identity with probability 1 − pghz /2 and (pghz /2)/3 for X, Y and Z error, in this case. Tesseract is the decoder used. Circuit 3cnot-seq 1shotghz-fanout

Before non-local operation

Between operations

After operation

Non-local noise model

4 rounds 4 rounds

1 round after each non-local CNOT —

3 rounds 3 rounds

DEPOLARIZE2 per dCNOT EGHZ per dGHZ

though more ebits are required for the same number of logical qubits, the higher distances can compensate for and in fact enables a lower LER for the fanout operation. In particular, [[54, 4, 8]] achieves a lower LER than [[36, 4, 6]] even with 54/36 = 1.5 times more ebits (or more distributed GHZ states), and in contrast to [[18, 4, 4]], achieves LER < P ER with more than double the number of ebits used for each dCN OT . Figure 5 shows a similar trend with fixed pebit where the distributed fanouts are implemented using sequences of nonlocal transversal CNOTs (3 dCN OT s for each fanout since there are 4 nodes). We introduced the two cases of pebit = 0 in the simulation with the [[18, 4, 4]] and [[36, 4, 6]] encodings for comparison - effectively modelling all qubits (or all blocks) and so, all operations on the same node (note that there are extra operations involving local ebits which would not be needed compared to a totally local fanout using all local

CNOTs - hence, this can be viewed as an upper bound or higher estimate of a completely local fanout). One observation is that the [[54, 4, 8]] encoding, when pebit = 102 (quite a high error rate), just managed to keep the LER below the P ER for low enough p, but when with a lower pebit = 103 , the [[54, 4, 8]] encoding comfortably pushes LER well below the P ER. This means that high network error rates (modelled by the pebit ) can reduce the fault tolerance of the encoding substantially. Also, at pebit = 103 , the [[54, 4, 8]] encoding achieves a similar LER as the entirely local operation with [[36, 4, 6]] blocks, that is, the distributed fanout involving 16 logical qubits over 4 nodes, each node with 4 logical qubits in a [[54, 4, 8]] block (in Figure 3(b)), has equivalent LER as the local fanout involving 16 logical qubits encoded in 4 blocks of [[36, 4, 6]] on one node (in Figure 3(a)), at low enough pebit ; in terms of LER, the additional 4x54-4x36=216-144=72 physical computation/data qubits to increase the distance from

Fig. 3: Configurations used in our experiments - four nodes: (a) a configuration with four blocks on one node (all operations local) and (b) a configuration with four nodes, one block per node, but with a higher distance code.

Fig. 5: Four node 3-CNOT sequential fanout: logical error rate (LER) versus physical error rate (PER), with shaded exact twosided 95% Clopper–Pearson confidence intervals. For varying PER=p, we used a fixed pebit . Table III in the Appendix shows the data for this graph.

setting pghz = 0). (A deeper comparison between distributed fanouts using dCNOT sequences and dGHZ states is in [20], and is not done in this paper.)

Fig. 4: Two versions of four node fanout for three different encodings: logical error rate (LER) versus physical error rate (PER), with shaded exact two-sided 95% Clopper–Pearson confidence intervals. For varying PER=p, we set pebit = pghz = 10p. Table II in the Appendix shows the figures plotted here.

6 to 8 compensated fully for the non-local operations - in other words, instead of a node with 144 physical data qubits, one can equivalently (in terms of LER) use 4 nodes of 54 physical data qubits, useful if no such 144 physical data qubits machine is available. For a distributed fanout implemented transversally using distributed GHZ states, Figure 6 shows a similar trend with fixed pghz at 102 and 103 . The higher distance compensated for the larger number of distributed GHZ states used for the [[54, 4, 8]] blocks, especially at low enough pghz . A similar equivalence in terms of LER is found between the fanouts with distributed 16 logical qubits and the entirely local operations with four [[36, 4, 6]] blocks on one node (modelled by

Fig. 6: Four node 1-shot GHZ fanout: logical error rate (LER) versus physical error rate (PER), with shaded exact twosided 95% Clopper–Pearson confidence intervals. For varying PER=p, we used a fixed pghz . Table IV in the Appendix shows the data for this graph.

V. I MPLICATIONS ON N ODE A RCHITECTURE D ESIGN The above observations suggest that code teleportation, used to convert from one encoding to another, e.g., [7], also used to bring logical qubit states from memory qubits into processor qubits [3], [4], or code switching (e.g., [11], [15]) could be used to “upgrade” a block of logical qubits, i.e., to convert from a code [[n, k, d]] to [[n′ , k, d′ ]], where n′ > n and d′ > d. This can be done, for example, to: (i) improve code distance and error protection for the logical qubits before they get involve in distributed computations; and/or (ii) cache logical quantum qubit states, e.g., if there are network delays, by

moving the qubits into higher distance blocks, as illustrated in Figure 7.

Fig. 7: Two uses of “upgrading” logical qubit encodings: (a) to increase “protection” before distributed operations, and (b) to temporarily cache qubits while waiting for distributed operations, e.g., delays in obtaining ebits, etc; how much (higher) the distance encoding required may depend on the required wait time. The use of such caching encoding blocks of qubits might call for an architecture of nodes involved in distributed quantum computations to have, beyond blocks of memory, ancilla, and processor qubits (as mentioned in [3], [4]), blocks of qubits for communication (holding ebits), and cache qubits for the two purposes above, as illustrated in Figure 8. It could be that the blocks of communication qubits and cache qubits (additional to memory, ancilla and processor qubit blocks) are hosted on a separate node (call these comm-nodes), to be usable by different nodes (the comp-nodes), as illustrated in Figure 8(b); for this architecture, logical qubits from a node can be teleported into the remote cache on a comm-node - the cache may have a higher distance encoding then the compnodes for the logical qubits (but assuming a high quality classical and quantum link between the comp-node and the comm-node, e.g., within the same locality), depending on the required wait time. Such code teleportation between encoding blocks on different nodes can utilise logical Bell pairs which can be efficiently formed (e.g., the work in [26], which needs to be adapted for BB codes); for example, imagining three blocks of logical qubits, one on the comp-node (holding the logical data qubits), and two hosting logical Bell pairs hosted across the comp-node and comm-node, the remote code teleportation then can be done with local joint logical qubit measurements on the comp-node with corrections on the comm-node (as in a typical quantum teleportation protocol) but on logical qubits and using logical operations - for blocks with [[n, k, d]] encoding, concurrent teleportation of k logical qubits can be carried out. Several possibilities exist - e.g., one is direct (remote) teleportation from a [[n, k, d]] block on the comp-node to one with [[n′ , k, d′ ]] on the comm-node, with n′ > n and d′ > d, or teleportation first among equal encoding blocks and then locally “upgrade” (as needed). Assuming a

high quality link between the comm-node and comp-node (e.g., pebit = p), teleportation can have a relatively low LER, indicated by Figure 9, among equal sized blocks, e.g., at least one or several orders of magnitude lower than the LER of the distributed transversal fanout itself.

Fig. 8: Communication and cache blocks of qubits as (a) part of a node, or (b) on their own node (usable by comp-nodes).

Fig. 9: LER for fault-tolerant teleportation with [[36, 4, 6]] blocks across two nodes with pebit = 10p and pebit = p. Table V in the Appendix shows the data for this graph. Local operations could work with a “weaker protection” encoding for the logical qubits, requiring fewer physical qubits per logical qubit, while distributed operations may require a higher encoding to increase chances of success. However, additional resources such as fault-tolerant operations are required for the “upgrading” (or for “downgrading” later if required), so that a further cost-benefit analysis is required. VI. C ONCLUSION AND F UTURE W ORK We investigated distributed fanout gadgets using different BB encodings of logical qubits. We observed that higher distance encodings with more physical qubits (even though using more ebits or more distributed GHZ states, due to distributed transversal operations), can help deal with logical errors to achieve LER < P ER, but also comparable with entirely local operations (with lower distances), at low enough

pebit and pghz . This suggests a node architecture to consider “upgrading” encoding to maintain quantum information before they are involved in more error-prone distributed operations, or to cache quantum information in case of network delays. Further work is needed to estimate the trade-off in faulttolerant code switching versus the benefits. Also, similar studies can be done with other distributed quantum circuits, e.g., to see when such “upgrading” is useful, to study under what network conditions caching is useful, and to understand what actual encodings to use for a given circuit and a given network quality, in order to achieve a target LER. While our results have used small sized blocks, much larger encoding blocks and larger circuits can be analyzed with higher code distances, where, hopefully, a similar trend can be observed this is an avenue for future work. ACKNOWLEDGEMENTS The author acknowledges the use of ChatGPT-6 to assist in generating the Python code for plotting the graphs in Figures 4, 5, 6, and 9. The author verified all algorithmic logic and executed the code, and takes full responsibility for the results. R EFERENCES [1] E. M. Ainley, A. Agrawal, D. Main, P. Drmota, D. P. Nadlinger, B. C. Nichol, R. Srinivas, and G. Araneda. Multipartite Entanglement for Multi-node Quantum Networks. Jul 2024. arXiv:2408.00149 [quantph]. [2] David Barral, F. Javier Cardama, Guillermo Dı́az-Camacho, Daniel Faı́lde, Iago F. Llovo, Mariamo Mussa-Juane, Jorge Vázquez-Pérez, Juan Villasuso, César Piñeiro, Natalia Costas, Juan C. Pichel, Tomás F. Pena, and Andrés Gómez. Review of distributed quantum computing: From single qpu to high performance quantum computing. Computer Science Review, 57:100747, 2025. [3] Todd A. Brun, Yi-Cong Zheng, Kung-Chuan Hsu, Joshua Job, and Ching-Yi Lai. Teleportation-based Fault-tolerant Quantum Computation in Multi-qubit Large Block Codes. 4 2015. [4] Madelyn Cain, Qian Xu, Robbie King, Lewis R. B. Picard, Harry Levine, Manuel Endres, John Preskill, Hsin-Yuan Huang, and Dolev Bluvstein. Shor’s algorithm is possible with as few as 10,000 reconfigurable atomic qubits. arXiv preprint arXiv:2603.28627, 2026. [5] Marcello Caleffi, Michele Amoretti, Davide Ferrari, Jessica Illiano, Antonio Manzalini, and Angela Sara Cacciapuoti. Distributed quantum computing: A survey. Computer Networks, 254:110672, 2024. [6] Nitish Kumar Chandra, Reza Nejabati, and Eneet Kaur. Towards the Characterization of Logical Errors in Distributed Lattice Surgery. In 2026 IEEE International Conference on Quantum Computing and Engineering, 7 2026. [7] Byung-Soo Choi. Cost comparison between code teleportation and stabilizer sequence methods for quantum code conversion. In 2013 International Conference on ICT Convergence (ICTC), pages 1083– 1087, 2013. [8] Ang et al. Arquin: Architectures for multinode superconducting quantum computers. ACM Transactions on Quantum Computing, 5(3), September 2024. [9] Stephen A. Fenner and Rabins Wosti. Implementing the quantum fanout operation with simple pairwise interactions. Quantum Inf. Comput., 23(13&14):1081–1090, 2023. [10] D. Gottesman. Surviving as a Quantum Computer in a Classical World. 2026. https://www.cs.umd.edu/ dgottesm/QECCbook-2026.pdf. [11] Sascha Heußen and Janine Hilder. Efficient fault-tolerant code switching via one-way transversal CNOT gates. Quantum, 9:1846, September 2025. [12] Peter Høyer and Robert Spalek. Quantum fan-out is powerful. Theory Comput., 1(1):81–103, 2005. [13] Photonic Inc. Distributed Quantum Computing in Silicon. arXiv e-prints, page arXiv:2406.01704, June 2024.

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A PPENDIX

TABLE II: Data for the 4-node fanout LER-versus-PER graph. The logical error rate is o/s, with o observed logical errors in s shots. Confidence limits are exact two-sided 95% Clopper–Pearson intervals. We set pebit = pghz = 10p, pghz = 10p Method 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential

Code [[18, 4, 4]] [[18, 4, 4]] [[18, 4, 4]] [[18, 4, 4]] [[36, 4, 6]] [[36, 4, 6]] [[36, 4, 6]] [[36, 4, 6]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[18, 4, 4]] [[18, 4, 4]] [[18, 4, 4]] [[18, 4, 4]] [[36, 4, 6]] [[36, 4, 6]] [[36, 4, 6]] [[36, 4, 6]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]]

PER 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3

Shots 796,262 18,978 1,314 137 5,000,000 1,647,206 7,842 118 5,000,000 4,576,102 14,570 110 488,926 18,978 1,178 121 5,000,000 1,092,198 4,834 109 5,000,000 5,000,000 10,370 118

Errors 130 114 167 136 3 120 112 116 1 113 128 108 117 118 207 120 4 113 112 109 1 81 134 118

LER 1.63 × 10−4 6.01 × 10−3 1.27 × 10−1 9.93 × 10−1 6.00 × 10−7 7.29 × 10−5 1.43 × 10−2 9.83 × 10−1 2.00 × 10−7 2.47 × 10−5 8.79 × 10−3 9.82 × 10−1 2.39 × 10−4 6.22 × 10−3 1.76 × 10−1 9.92 × 10−1 8.00 × 10−7 1.03 × 10−4 2.32 × 10−2 1.00 × 100 2.00 × 10−7 1.62 × 10−5 1.29 × 10−2 1.00 × 100

95% CI for LER [1.36 × 10−4 , 1.94 × 10−4 ] [4.96 × 10−3 , 7.21 × 10−3 ] [1.10 × 10−1 , 1.46 × 10−1 ] [9.60 × 10−1 , 1.00 × 100 ] [1.24 × 10−7 , 1.75 × 10−6 ] [6.04 × 10−5 , 8.71 × 10−5 ] [1.18 × 10−2 , 1.72 × 10−2 ] [9.40 × 10−1 , 9.98 × 10−1 ] [5.06 × 10−9 , 1.11 × 10−6 ] [2.04 × 10−5 , 2.97 × 10−5 ] [7.33 × 10−3 , 1.04 × 10−2 ] [9.36 × 10−1 , 9.98 × 10−1 ] [1.98 × 10−4 , 2.87 × 10−4 ] [5.15 × 10−3 , 7.44 × 10−3 ] [1.54 × 10−1 , 1.99 × 10−1 ] [9.55 × 10−1 , 1.00 × 100 ] [2.18 × 10−7 , 2.05 × 10−6 ] [8.53 × 10−5 , 1.24 × 10−4 ] [1.91 × 10−2 , 2.78 × 10−2 ] [9.67 × 10−1 , 1.00 × 100 ] [5.06 × 10−9 , 1.11 × 10−6 ] [1.29 × 10−5 , 2.01 × 10−5 ] [1.08 × 10−2 , 1.53 × 10−2 ] [9.69 × 10−1 , 1.00 × 100 ]

TABLE III: Data for the four-node 3-CNOT sequential fanout LER-versus-PER graph. The logical error rate is k/n, with k observed logical errors in n shots. Confidence limits are exact two-sided 95% Clopper–Pearson intervals. Method 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential 3-CNOT sequential

Code [[18, 4, 4]] [[18, 4, 4]] [[18, 4, 4]] [[18, 4, 4]] [[36, 4, 6]] [[36, 4, 6]] [[36, 4, 6]] [[36, 4, 6]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]]

Parameter pebit = 0 pebit = 0 pebit = 0 pebit = 0 pebit = 0 pebit = 0 pebit = 0 pebit = 0 pebit = 10−2 pebit = 10−2 pebit = 10−2 pebit = 10−2 pebit = 10−3 pebit = 10−3 pebit = 10−3 pebit = 10−3

PER 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3

Shots 772,710 26,146 1,570 151 5,000,000 2,370,150 6,690 138 3,863,142 1,713,203 10,754 105 5,000,000 5,000,000 14,690 110

Errors 121 113 218 148 1 121 112 135 107 111 101 105 1 76 122 110

LER 1.57 × 10−4 4.32 × 10−3 1.39 × 10−1 9.80 × 10−1 2.00 × 10−7 5.11 × 10−5 1.67 × 10−2 9.78 × 10−1 2.77 × 10−5 6.48 × 10−5 9.39 × 10−3 1.00 × 100 2.00 × 10−7 1.52 × 10−5 8.30 × 10−3 1.00 × 100

95% CI for LER [1.30 × 10−4 , 1.87 × 10−4 ] [3.56 × 10−3 , 5.19 × 10−3 ] [1.22 × 10−1 , 1.57 × 10−1 ] [9.43 × 10−1 , 9.96 × 10−1 ] [5.06 × 10−9 , 1.11 × 10−6 ] [4.24 × 10−5 , 6.10 × 10−5 ] [1.38 × 10−2 , 2.01 × 10−2 ] [9.38 × 10−1 , 9.95 × 10−1 ] [2.27 × 10−5 , 3.35 × 10−5 ] [5.33 × 10−5 , 7.80 × 10−5 ] [7.66 × 10−3 , 1.14 × 10−2 ] [9.65 × 10−1 , 1.00 × 100 ] [5.06 × 10−9 , 1.11 × 10−6 ] [1.20 × 10−5 , 1.90 × 10−5 ] [6.90 × 10−3 , 9.91 × 10−3 ] [9.67 × 10−1 , 1.00 × 100 ]

TABLE IV: Data for the four-node 1-shot GHZ fanout LER-versus-PER graph. The logical error rate is k/n, with k observed logical errors in n shots. Confidence limits are exact two-sided 95% Clopper–Pearson intervals. Method 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ 1-shot GHZ

Code [[18, 4, 4]] [[18, 4, 4]] [[18, 4, 4]] [[18, 4, 4]] [[36, 4, 6]] [[36, 4, 6]] [[36, 4, 6]] [[36, 4, 6]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]] [[54, 4, 8]]

Parameter pGHZ = 0 pGHZ = 0 pGHZ = 0 pGHZ = 0 pGHZ = 0 pGHZ = 0 pGHZ = 0 pGHZ = 0 pGHZ = 10−2 pGHZ = 10−2 pGHZ = 10−2 pGHZ = 10−2 pGHZ = 10−3 pGHZ = 10−3 pGHZ = 10−3 pGHZ = 10−3

PER 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3 5.00 × 10−5 2.71 × 10−4 1.47 × 10−3 7.94 × 10−3

Shots 923,238 34,338 2,594 155 5,000,000 2,344,550 11,554 118 4,432,659 955,955 14,642 109 5,000,000 4,754,707 19,106 118

Errors 145 117 282 148 1 124 115 112 119 115 117 107 1 105 122 113

LER 1.57 × 10−4 3.41 × 10−3 1.09 × 10−1 9.55 × 10−1 2.00 × 10−7 5.29 × 10−5 9.95 × 10−3 9.49 × 10−1 2.68 × 10−5 1.20 × 10−4 7.99 × 10−3 9.82 × 10−1 2.00 × 10−7 2.21 × 10−5 6.39 × 10−3 9.58 × 10−1

95% CI for LER [1.33 × 10−4 , 1.85 × 10−4 ] [2.82 × 10−3 , 4.08 × 10−3 ] [9.70 × 10−2 , 1.21 × 10−1 ] [9.09 × 10−1 , 9.82 × 10−1 ] [5.06 × 10−9 , 1.11 × 10−6 ] [4.40 × 10−5 , 6.31 × 10−5 ] [8.22 × 10−3 , 1.19 × 10−2 ] [8.93 × 10−1 , 9.81 × 10−1 ] [2.22 × 10−5 , 3.21 × 10−5 ] [9.93 × 10−5 , 1.44 × 10−4 ] [6.61 × 10−3 , 9.57 × 10−3 ] [9.35 × 10−1 , 9.98 × 10−1 ] [5.06 × 10−9 , 1.11 × 10−6 ] [1.81 × 10−5 , 2.67 × 10−5 ] [5.31 × 10−3 , 7.62 × 10−3 ] [9.04 × 10−1 , 9.86 × 10−1 ]

TABLE V: Logical error-rate data for teleportation using the [[36, 4, 6]] code, for the two datasets with pebit = 10p and pebit = p. The LER is the number of logical errors divided by the number of shots. Confidence intervals are exact two-sided 95% Clopper–Pearson intervals. Dataset 10p 10p 10p 10p 1p 1p 1p 1p

PER

Shots

95% CI for LER Lower Upper 5.00 × 10 5,000,000 2 4.00 × 10 4.84 × 10−8 1.44 × 10−6 2.71 × 10−4 1,841,766 115 6.244 × 10−5 5.155 × 10−5 7.495 × 10−5 1.468 × 10−3 9,762 114 1.168 × 10−2 9.642 × 10−3 1.401 × 10−2 7.943 × 10−3 152 129 8.487 × 10−1 7.817 × 10−1 9.016 × 10−1 5.00 × 10−5 5,000,000 0 < 2.00 × 10−7 ∗ 0 7.378 × 10−7 2.71 × 10−4 4,479,078 112 2.501 × 10−5 2.059 × 10−5 3.009 × 10−5 1.468 × 10−3 15,906 118 7.419 × 10−3 6.144 × 10−3 8.878 × 10−3 7.943 × 10−3 170 123 7.235 × 10−1 6.498 × 10−1 7.893 × 10−1 * This is due to zero errors in 5,000,000 shots though with more shots, errors could emerge. −5

Errors

LER −7

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