Design rules for fault-tolerant multi-gate teleportation

Design rules for fault-tolerant multi-gate teleportation Mathys Rennela

arXiv:2607.01342v1 [quant-ph] 1 Jul 2026

Unitary Foundation, France

Abstract— Multi-gate teleportation (MGT) packages n remote gates into a single ebit via a 1-ebit fan-out quantum circuit, saving n−1 entangled pairs relative to sequential gate teleportation. The cost is a correlated failure mode: a single network fault propagates through the fan-out tree, injecting a weight-n Pauli error. We derive a design rule for fault-tolerant packet sizes, ncorr max (d) = ⌈d/2⌉ for rotated surface codes of distance d with a correlationaware decoder (nnaive max = ⌊d/2⌋ without), bounding how many gates can be packaged whilst preserving fault tolerance. Simulation with PyMatching shows that the standard MWPM decoder built from the packet circuit’s noise model naturally corrects the correlated error: at network-to-local noise ratios γ = pnet /pgate up to 100, the packet matches or surpasses the per-link sequential LER at moderate-to-high γ, with the advantage growing with both γ and d, whilst reducing the entanglement cost from n ebits to 1. Packetisation wins when the network is the bottleneck (γ ≫ 1); at γ ≈ 1 the n−1 extra local fan-out gates offset the network savings. No custom decoder is required: the circuit-level noise model already encodes the correlation. These results enable noiseaware distributed circuit compilers to favour fan-out packetisation without sacrificing fault tolerance.

I. I NTRODUCTION Distributed quantum computing (DQC) promises to scale the computational power of quantum processors by interconnecting them. A basic paradigm of DQC is to partition a monolithic circuit into modules which communicate via remote gates, typically implemented by teleportation over an ebit (most often, a shared Bell pair) [1], [2]. For a circuit requiring n nonlocal CNOTs between the same pair of modules, sequential teleportation consumes n ebits. Multi-gate teleportation (MGT) collapses this to a single ebit by packaging the n gates into a packet: one non-local Bell pair drives a local CNOT fan-out tree that distributes the control to all n targets [3]. The resource savings are clear (n−1 ebits saved), but the noise profile is not. A single failure on the Bell pair propagates through the fan-out tree, injecting a weight-n correlated Pauli error rather than n independent weight-1 errors. This correlated failure raises a basic question: when does the ebit savings survive fault-tolerant quantum error correction? The decisive parameter is the network-to-gate noise ratio γ = pnet /pgate . When γ ≳ 1, the network error dominates and the correlated weight-n error becomes the bottleneck. Whether an MGT packet remains fault-tolerant, and whether a decoder can exploit the known correlation structure, depends on the interplay between γ, the code distance d, and the packet size n. Related work Recent works establish noise thresholds for distributed surface and qLDPC codes [4], [5], [6], [7], [8], [9], [10],

including Shalby et al. [11] who use the same γ parameter for surface-code teleportation interfaces. All address singlegate teleportation under independent per-link noise; none models a single gate failure injecting a weight-n correlated error. On the resource side, Loke [12] showed that GHZbased fan-out reduces O(n2 ) non-local resources to O(n), and compiler frameworks [13] minimise non-local gate counts as a communication-cost proxy, but neither accounts for the correlated noise that fan-out introduces. This paper connects the two threads, accounting for noise and communication cost simultaneously. Contributions 1) Error model. A single network failure in a 1-ebit fanout MGT packet injects a weight-n contiguous Pauli error (section II). The correlated structure is forced by the 1-ebit fan-out bottleneck and is independent of the teleportation interface. 2) Design rule. The well-known MWPM failure threshold at weight ⌈d/2⌉ [14], [15] implies a maximum safe packet size ncorr max (d) = ⌈d/2⌉ with a correlation-aware decoder, or nnaive max (d) = ⌊d/2⌋ without (section III). The correlation-aware decoder extends the bound by one because the correlated error appears as a single lowweight edge. 3) Numerical results. Detector-error-model (DEM)-level simulation with PyMatching [16] under a simplified circuit-level depolarising model (section IV) shows that a naive decoder cannot exploit the correlation: the LER gap between the naive and correlation-aware decoders grows with code distance at moderate-to-high γ. However, the standard MWPM decoder built from the packet circuit’s DEM closes the gap and extends the correctable packet size by one: the packet matches or surpasses the per-link sequential baseline (each link at rate pnet ) at d ∈ {5, 7, 9} at moderate-to-high γ, with the advantage growing with both γ and d. No custom decoder construction is required: the gap closes because the natural DEM already encodes the packet structure. Assumptions: Simplified circuit-level depolarising noise (gate noise only); the Eisert–Jacobs–Papadopoulos–Plenio (EJPP) protocol [2] for quantitative claims; rotated surface codes. Qualitative conclusions are interface-independent by structural argument. II. E RROR M ODEL Consider a sequence of consecutive CNOT gates, sharing the same control qubit. A same-control multi-gate teleportation

q t1 t2 t3 Fig. 1. Fan-out decomposition of a same-control MGT packet of 3 CNOTs. A single non-local CNOT(q → t1 ) (red) is the only operation that crosses the network; all other gates are local fan-out CNOTs (blue). The total entanglement cost is one ebit, independent of n.

(MGT) packet is formed by a single non-local CNOT(q → t1 ), preceded and followed by local fan-out CNOTs for i = 2, . . . , n, as shown in fig. 1. The non-local CNOT can for example be executed via the EJPP protocol [2], which consumes one ebit of entanglement and two rounds of classical communication. The total entanglement cost of an MGT packet is therefore one ebit, independent of the number of targets n. An X error on q before the non-local CNOT(q → t1 ) propagates as Xq → Xq Xt1 , and each subsequent local fan-out CNOT(t1 → tk ) copies Xt1 onto tk . The result is a weight(n+1) error Xq Xt1 · · · Xtn ; restricting to the target register, this is a contiguous weight-n X string. An X error on t1 (between the non-local step and the fan-out) produces the same weight-n target string without q. Faults on downstream targets tk (k > 1) remain localised. Thus a single network failure injects a weight-n correlated X error; that is, an all-ornothing event that mimics a logical error chain on the logical X̄ operator. Note that under depolarising network noise with per-Pauli rate pnet , the dangerous errors on control q include both Xq and Yq = iXq Zq : the X component of Y propagates identically through CNOT(q → t1 ) to produce the same weightn target string, whilst the Z component commutes forward and remains on q alone (see below). Therefore, the effective rate for producing the weight-n X string is 2pnet /3 rather than pnet /3. Z errors behave asymmetrically and remain secondary: a Z error on q commutes through the non-local CNOT and stays localised, while a Z error on a downstream target tk propagates backward as a weight-2 localised error Zt1 Ztk . No Z error produces a weight-n string. The same-target MGT packet (dual configuration) has the opposite behaviour: a Z error on the target produces a weight-n Z string on the controls, while X errors remain benign. Without loss of generality, the remainder of this paper treats same-control MGT. The weight-n correlated error mechanism extends beyond CNOT to all controlled-Clifford gates. The Clifford property ensures that Pauli propagation is closed under conjugation, so a

Pauli error on the control before the non-local step produces a weight-n Pauli string on the targets (the specific Pauli type may differ per gate, but the weight and contiguity are identical). Note that the weight-n correlated error is forced by the 1-ebit fan-out bottleneck, not by the non-local gate implementation protocol specifically: any 1-ebit scheme has a single classical-orquantum crossing, and an error there is broadcast to all n targets via the local fan-out tree. The present analysis is therefore interface-independent. However, whilst the quantitative claims in this paper are computed for the EJPP protocol [2], other quantum interconnect approaches may carry additional resource overhead [11]. III. A NALYTIC D ESIGN RULE We established that a single quantum link failure in a MGT packet produces a weight-n correlated error, but how large can a packet be before this correlated error breaks fault tolerance? We derive the answer from the correctability condition [17] and the MWPM degenerate syndrome failure mechanism, leading to a design rule that depends on whether the decoder is built from the packet circuit’s DEM. Decoder-independent correctability bound A quantum code of distance d can correct any error of weight at most ⌊(d − 1)/2⌋ = ⌊d/2⌋ for odd d (as used throughout) [17]. For two weight-n errors Ea , Eb with n ≤ ⌊d/2⌋, the product Ea† Eb has weight at most 2⌊d/2⌋ = d − 1 < d and therefore cannot be a nontrivial logical operator. This bound is decoder-independent and geometry-independent: a packet of size n ≤ ⌊d/2⌋ is correctable regardless of where the target qubits are located. Naive decoder: failure at n = ⌈d/2⌉ Now consider the failure Q mechanism at n = ⌈d/2⌉. Place a weight-n X error E = j∈S Xqj on a contiguous subset S of the minimum-weight logical X̄ operator L. Define its compleQ ment E ′ = j∈L\S Xqj on the remaining d − ⌈d/2⌉ = ⌊d/2⌋ qubits. Since E · E ′ = X̄ commutes with all stabilizers, E and E ′ produce a degenerate pair of identical syndromes. Because |E| = ⌈d/2⌉ exceeds the correctable weight ⌊d/2⌋ while |E ′ | = ⌊d/2⌋ does not, a naive MWPM decoder (built from the base DEM, without the correlated-error edge) selects E ′ as the minimum-weight correction. The residual E · E ′ = X̄ is a logical failure: an error of weight ⌈d/2⌉ is uncorrectable whenever its complement along a logical operator has weight ⌊d/2⌋. This mechanism is purely algebraic: it depends on the stabilizer structure, not on the matching-graph geometry. Correlation-aware decoder: extension to n = ⌈d/2⌉ The failure mechanism above assumes the decoder treats the weight-n correlated error as n independent gate-error edges. A correlation-aware decoder (which includes the correlated error as a single CORRELATED_ERROR edge) assigns it a weight of − ln pnet rather than n · (− ln pgate ). At n = ⌈d/2⌉, the complement E ′ has weight ⌊d/2⌋ · (− ln pgate ) in the matching graph. Whenever γ ≥ 1, the single packet edge

is cheaper: − ln pnet ≤ − ln pgate < ⌊d/2⌋ · (− ln pgate ), so the correlation-aware decoder prefers the correct correction E, leaving no residual. Computational confirmation at d=5 shows the correlation-aware LER at n=3 remains at the gate-noise floor (∼ 10−4 ) for all γ ∈ {1, 10, 100}, whilst the naive decoder fails (LER → 0.01–0.1). However, this extension is limited to n = ⌈d/2⌉: at n > ⌈d/2⌉, no lower-weight complement exists, and both decoders fail (LER ∼ 10−3 –10−1 , growing with γ).

TABLE I PACKET LER WITH THE STANDARD MWPM DECODER , USING THE CORRELATION - AGNOSTIC DEM (LERbase ) AND THE CORRELATION - AWARE DEM (LERpkt ), COMPARED WITH THE PER - LINK SEQUENTIAL BASELINE (LERseq , n INDEPENDENT FAULTS AT RATE pnet EACH ). T HE RATIO LERpkt /LERseq DROPS WELL BELOW 1 AT HIGH γ, WHERE THE SEQUENTIAL BASELINE ACCUMULATES n INDEPENDENT NETWORK FAULTS WHILE THE CORRELATION - AWARE DECODER ABSORBS THE SINGLE PACKET FAULT AT THE GATE - NOISE FLOOR . A LL AT pgate = 10−3 ; LERbase AND LERpkt USE 108 SHOTS (d=5, d=7 AT γ≤10, AND ALL d=9), WHILE LERseq AND THE REMAINING POINTS USE 106 SHOTS .

Design rule

d

Combining the bounds gives two design rules for MGT packet size:     d d naive corr nmax (d) = , nmax (d) = (1) 2 2 for odd d, where nnaive max applies to a decoder built from the base DEM and ncorr max to a correlation-aware decoder. For d = 5, 7, 9, 11, this gives ncorr max = 3, 4, 5, 6. The decoder-

n

γ

LERbase

LERpkt

LERseq

pkt/seq

7.1 × 10−5

6.9 × 10−5

5 5 5 5

2 2 2 2

1 10 30 100

9.2 × 10−5 1.9 × 10−4 4.1 × 10−4 1.3 × 10−3

7.4 × 10−5 7.7 × 10−5 7.9 × 10−5

7.0 × 10−5 8.2 × 10−5 1.9 × 10−4

1.03 1.06 0.94 0.41

7 7 7 7

3 3 3 3

1 10 30 100

2.05 × 10−5 1.7 × 10−4 4.8 × 10−4 1.6 × 10−3

5.9 × 10−6 5.3 × 10−6 5.2 × 10−6 5.9 × 10−6

5.2 × 10−6 5.5 × 10−6 8.0 × 10−6 3.5 × 10−5

1.13 0.97 0.65 0.17

9

4

1

2.04 × 10−5

4.4 × 10−7

2.0 × 10−6

0.22

independent correctability floor is ⌊d/2⌋ in both cases; exceed9 4 10 1.9 × 10−4 4.2 × 10−7 1.0 × 10−6 0.42 9 4 30 5.8 × 10−4 4.0 × 10−7 < 10−5 — ing ncorr forces a degenerate syndrome with no lower-weight max 9 4 100 1.9 × 10−3 3.5 × 10−7 3.0 × 10−6 0.12 complement. The upper bound requires worst-case placement: all n targets lie on a minimum-weight logical operator. Off-chain placements produce shorter complements and are strictly easier to decode; to the gate-noise floor, γ-independent, because the decoder we take the central on-chain placement as the canonical worst absorbs the single packet fault. Against the per-link sequential case throughout. Only one Pauli type per configuration is baseline (n independent faults at rate pnet each), the packet dangerous (weight-n X on targets for same-control, weight-n matches or surpasses sequential at moderate-to-high γ, with Z on controls for same-target); the benign type’s worst case is the advantage growing with both γ and d. At low γ (γ ≈ 1), the packet is slightly worse (ratio ≤ 1.13): the network noise a weight-2 localised error, correctable for all d ≥ 5. corr For packets exceeding nmax , a hybrid k-ary fan-out tree is comparable to gate noise, so the packet’s n−1 additional restores fault tolerance at the cost of k ebits: each of k bridge local fan-out CNOTs contribute gate noise without the network qubits fans out to a subgroup of ≈ n/k targets, reducing savings yet being decisive, while the sequential baseline has no the worst-case error weight from n to ≈ n/k, with kmin = extra local gates. At high γ, the sequential baseline accumulates ⌈n/⌈d/2⌉⌉ interpolating between the 1-ebit fan-out (k=1) and n independent network faults whose contribution grows with γ, while the correlation-aware decoder absorbs the single packet fully sequential teleportation (k=n). fault at the gate-noise floor. IV. PACKET FAULT-T OLERANCE P ERFORMANCE All sweeps use pinned random seeds for reproducibility. The packet architecture saves n−1 ebits per logical CNOT V. C ONCLUSION (section II), but a single network failure injects a correlated weight-n error. We compare the packet model (single Multi-gate teleportation [3] saves n−1 ebits by packaging n CORRELATED_ERROR(pnet ), weight n) against the per-link remote CNOTs behind a single non-local crossing, at the cost sequential baseline (n independent X_ERROR(pnet ) per link), of a correlated failure mode: a single network fault produces a which is the physically honest comparison: pnet is the per- weight-n Pauli error (section II). Two design rules govern faultlink rate (section II), and each ebit is an independent channel tolerant MGT for rotated surface codes. First, the packet size use. The ratio γ = pnet /pgate parametrises the network-to-gate must satisfy n ≤ ncorr max (d) = ⌈d/2⌉ when using a correlationnoise ratio; pgate = 10−3 throughout. aware decoder, or the stricter n ≤ nnaive max (d) = ⌊d/2⌋ with a When the decoder does not account for the correlated error, naive decoder (section III). The correctability floor n ≤ ⌊d/2⌋ the relative cost grows with code distance at moderate-to-high is decoder-independent; the failure threshold is MWPM-specific. γ (see LERbase in table I). Beyond ncorr max , a degenerate syndrome with no lower-weight However, this gap is an artefact of the decoder’s DEM, not complement forces both decoders to fail. Second, the decoder of the packet architecture. The packet circuit’s DEM naturally must be built from the packet circuit’s DEM, which encodes includes the correlated error as a single edge (the correlated the correlated error as a single edge (section IV). This not only error appears automatically in the DEM’s construction). Build- closes the LER gap but extends the correctable regime by one: ing the MWPM decoder from this DEM closes the gap: table I the correlation-aware decoder corrects n = ⌈d/2⌉ because the shows that the correlation-aware packet LER (LERpkt ) drops single correlated-error edge (weight − ln pnet ) is cheaper than

the complement path (⌊d/2⌋ independent gate-error edges) at any γ ≥ 1. Without this correlation awareness, the packet’s LER penalty grows with code distance at moderate-to-high γ; with it, the packet matches or surpasses the per-link sequential baseline at moderate-to-high γ—at worst slightly worse for low d and low γ (where the n−1 extra local fan-out gates dominate), decisively lower at high γ or high d, where the sequential baseline accumulates n independent network faults while the decoder absorbs the single packet fault. The correlated error originates from the 1-ebit bottleneck, not the teleportation interface: any scheme that packages n gates behind a single non-local crossing (EJPP, measurement-based, or CAT-state) shares the same failure mode, so the conclusions are interface-independent. For packets exceeding ncorr max , a k-ary fan-out tree restores fault tolerance at the cost of k ebits: each of k bridge qubits fans out to a subgroup of ≈ n/k targets, reducing the worst-case error weight from n to ≈ n/k, with kmin = ⌈n/⌈d/2⌉⌉ interpolating between the 1-ebit fan-out (k=1) and fully sequential teleportation (k=n). Limitations and future work: The simplified depolarising model (gate noise only) underestimates absolute LER by 3– 5× relative to the full four-channel model, though relative comparisons are unaffected. The packet’s LER advantage over per-link sequential appears at γ ≳ 10 (table I); at γ ≈ 1 the packet incurs a small penalty (≤ 13%) from the extra local fan-out gates. Recent photonic experiments report teleported CNOT gate fidelities of ∼ 87% (error rate ∼ 10−1 ) [18], and distributed architectures routinely consider entanglementgeneration error rates of ∼ 5% [19]; with pgate = 10−3 , these correspond to γ ∈ [50, 100], squarely in the regime where packetisation outperforms sequential teleportation. Injecting the full pnet rather than 2pnet /3 makes the reported packet LER conservative. The simulation places the packet error at round 0, giving the decoder maximal syndrome information; a final-round injection would represent the hardest decoding scenario. The quantitative claims are computed for the EJPP protocol [2] under rotated surface codes with MWPM decoding; the qualitative conclusions extend by structural argument to any surface code and any minimum-weight decoder. The nmax bound applies to any CSS code, but the simulation evidence is limited to rotated surface codes; extending to qLDPC codes [4], [6], biased noise, and full circuit-level noise is deferred to future work. Integrating the design rules into distributed circuit compilers [20], [21], [22], [23], [24] would test whether they translate into measurable resource savings. ACKNOWLEDGEMENTS This work has been supported by the Mozilla Foundation. AI assistance (GLM-5.2, Mimo-V2.5) was used for copy-editing, proofreading, and formatting, and helped with numerical data generation and analysis. All scientific content and conclusions are the sole responsibility of the author. DATA AVAILABILITY The detector error models (DEMs) underlying all simulations in this work are publicly available on Zenodo at 10.5281/zen-

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