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arxiv: 2604.05870 · v1 · submitted 2026-04-07 · 🪐 quant-ph

Recognition: no theorem link

Fault-Tolerant One-Shot Entanglement Generation with Constant-Sized Quantum Devices in the Plane

Dylan Harley , Robert Koenig
Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:31 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum entanglementfault tolerancequantum repeaterslocal stochastic noiseBell pairstabilizer Hamiltonianlocalizable entanglement2D qubit grid
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0 comments X

The pith

A rectangular grid of constant-size quantum devices generates high-fidelity entanglement over arbitrary distances using only local operations in one shot, provided noise stays below a fixed threshold.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to create a Bell pair between qubits that are arbitrarily far apart while using only nearest-neighbor gates on a two-dimensional grid whose one dimension stays polylogarithmic in the distance. The protocol runs in constant time up to a known Pauli correction and keeps the output fidelity constant as long as the local stochastic Pauli noise is weaker than some fixed positive threshold. This matters because existing methods either require the local processor size to grow with distance or need repeated rounds whose number grows with distance. The construction works both for a single device with limited connectivity and for networks of small repeater stations. It also yields the first known short-range entangled 2D state whose long-range localizable entanglement survives the same noise model.

Core claim

Below a constant threshold on local stochastic Pauli noise, a rectangular grid of qubits of size Θ(R) by Θ(poly(log R)) suffices to produce a constant-fidelity Bell pair between qubits distance R apart. The protocol applies single-qubit and nearest-neighbor two-qubit operations in one shot and leaves the output in a Bell state up to a known Pauli correction. The same construction gives a 2D-local stabilizer Hamiltonian whose Gibbs states at constant positive temperature possess long-range localizable entanglement.

What carries the argument

A one-shot local protocol that uses many-body entanglement across the grid to distill long-range entanglement while correcting errors with constant-size blocks.

If this is right

  • Quantum repeater stations can remain constant size while still supporting entanglement over unlimited distances.
  • A 2D-local stabilizer Hamiltonian exists whose thermal states retain long-range localizable entanglement at any fixed positive temperature below some threshold.
  • Entanglement generation between distance-R qubits requires only polylogarithmic growth in one grid dimension rather than linear growth in both.
  • The protocol applies equally to geometrically limited single devices and to planar networks of small repeater units.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the noise threshold is high enough to be reachable in hardware, the construction could simplify the design of large-scale quantum networks by removing the need for distance-dependent processor sizes.
  • The same grid-based many-body state might be adaptable to other noise models or to three dimensions, though the paper does not explore those cases.
  • The existence of a constant-temperature Gibbs state with long-range entanglement suggests that thermal noise alone need not destroy useful quantum correlations in suitably engineered 2D systems.

Load-bearing premise

Local stochastic Pauli noise must remain below some fixed positive strength so that the error-correction and distillation steps keep the output fidelity bounded away from zero.

What would settle it

Implement the protocol on successively larger grids and measure the fidelity of the output Bell pair; if the fidelity falls toward zero as distance R increases while noise stays fixed below the claimed threshold, the scaling claim is false.

Figures

Figures reproduced from arXiv: 2604.05870 by Dylan Harley, Robert Koenig.

Figure 1
Figure 1. Figure 1: Sketch of the setup for Theorem 1. A short-ranged entangled state [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Single-shot entanglement generation: We compare existing one-shot entanglement [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Noise defined by a comb E affecting a subset F ⊂ Loc(C) of (fault) locations of a circuit C 20 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Diagrammatic notation for notions used in the definition fault tolerance [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Diagrammatic notation for fault tolerance [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Conditions for fault tolerance of 1-gadgets for an [[n, 1, d]]-code, see [26]. Here t = ⌊(d − 1)/2⌋. 3.2 Level-L simulation of a circuit Here we describe how the fault-tolerant circuit CFT is constructed from (a description of) the ideal circuit C. We note that – unlike the original circuit C – the circuit CFT generally is an adaptive circuit involving mid-circuit measurements and subsequent operations dep… view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of basic steps in the proof of Lemma 3.3.1. [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The decoder iDec L→0 for L ◦L , for L = 3 concatenation levels associated with an [[n, 1, d]]- code L. For our illustrations, we use n = 3 throughout but the constructions actually require larger codes. Each line represents a physical qubit. We note that by Eqs. (3.5) and (3.6), we have  iDec L→0 ⊗N = (iDec(1)) ⊗N ◦ · · · ◦ (iDec(L) ) ⊗N . (3.8) The following result is an immediate consequence of level r… view at source ↗
Figure 6
Figure 6. Figure 6: Let t := ⌊(d−1)/2⌋. Let L ∈ N and let iDec L→0 be ideal decoding the map defined in Eq. (3.7) decoding L ◦L to a physical qubit. Then there is a threshold error strength p∗ > 0 depending only on the code and gadgets used such that the following holds: Let N ∈ N be arbitrary and let 26 [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 9
Figure 9. Figure 9: An illustration of level reduction (see Lemma 4.4.1) for [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The statement of Corollary 3.3.2 illustrated for [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Diagrammatic notation for decoder and encoding gadgets. As before, thick (thin) [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Properties of the decoder gadget (r ≤ t, s ≤ t and s + P i ri ≤ t) 0 = (a) Encoder gadget = (b) Encoder and ideal decoder gadget [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: Illustration of the modifications used to establish Lemma 4.4.1. This should be [PITH_FULL_IMAGE:figures/full_fig_p033_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Proof of Lemma 4.4.2, illustrated for a circuit [PITH_FULL_IMAGE:figures/full_fig_p034_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Proof of Lemma 4.4.3. The noisy implementation of [PITH_FULL_IMAGE:figures/full_fig_p035_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: A key identity: Conjugating a Pauli F by a classically controlled Pauli ctrlP results in F up to (possibly) a sign (if F and P anticommute), but the sign can be ignored as the controlling register is classical. Lemma 5.1.2 (Pauli error propagation through a Clifford circuit). Let C be a depth-D Clifford circuit consisting only of one- and two-qubit Clifford unitaries. Let F ∼ N P C (p) be local stochastic… view at source ↗
Figure 18
Figure 18. Figure 18: Circuit transformations for propagating adaptive Paulis forward past a subsequent [PITH_FULL_IMAGE:figures/full_fig_p041_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Two non-intersecting valid rectangles R1, R2 associated with unitary subcircuits of a circuit C Proof. In [11, Lemma 11], it is shown that conjugating a strength p local stochastic Pauli by a depth-1 Clifford circuit composed of one- and two-qubit Cliffords results in a local stochastic Pauli of strength g(p). This immediately implies the induction base case k = 1: Since FD−1 is a local stochastic Pauli e… view at source ↗
Figure 20
Figure 20. Figure 20: An adaptive rectangle R Define the “left” or “in”-boundary of an adaptive rectangle R as ∂ℓR := {(t, q) | (t, q) ∈ R} and similarly the “right” or “out”-boundary as ∂rR := {(t + 2, q) | (t + 2, q) ∈ R} . The interior of R consists of the wires Int(R) := {(t + 1, q) | (t + 1, q) ∈ R} . We say that the adaptive rectangle R is “read-once” in the circuit C if the measurement results z ∈ {0, 1} Ω1 are not used… view at source ↗
Figure 21
Figure 21. Figure 21: Illustration of circuit inflation of depth [PITH_FULL_IMAGE:figures/full_fig_p048_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The circuit C˜ obtained by substituting unitary subcircuits in the circuit C defined in [PITH_FULL_IMAGE:figures/full_fig_p050_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Transforming a circuit C into an equivalent circuit C˜ of alternating form. In the substitution moves, a subcircuit of the form id ◦ U (where U is a one- or two-qubit unitary) is replaced by the circuit U ◦ id in every rectangle. 52 [PITH_FULL_IMAGE:figures/full_fig_p052_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: The path graph P2r and the bilinear array graph B2r illustrated for r = 6 6 Robust, geometrically local implementation of circuits In this section we explain how to fault-tolerantly implement circuits under certain structural restrictions. We then consider different geometries: In Section 6.1, we discuss geometrically local circuit implementation in bilinear arrays, before upgrading this to truly 1D-local… view at source ↗
Figure 25
Figure 25. Figure 25: One- and two-qubit gate teleportation circuits implementing a single- respectively [PITH_FULL_IMAGE:figures/full_fig_p060_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: A convenient factorization of the one-qubit teleportation circuit. The qubits after [PITH_FULL_IMAGE:figures/full_fig_p061_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Construction of the circuit C 1D post starting from a fault-tolerant 1D-local implemen￾tation of a circuit C. The key building block are two- respectively one-qubit teleportation subcircuits, see [PITH_FULL_IMAGE:figures/full_fig_p063_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: The two-qubit gate teleportation circuit from Fig. 25b with the Pauli correction [PITH_FULL_IMAGE:figures/full_fig_p063_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Illustration of the “unfolding” of the 1D-local circuit C 1D post in space, resulting a constant-depth state preparation circuit C 2D. For concreteness, we illustrate the construction for an ideal circuit C with Nin = 0 and Nout = N, i.e., the circuit starts with the state [PITH_FULL_IMAGE:figures/full_fig_p066_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: The two-qubit circuit C prep, which prepares a maximally entangled state on two qubits. This circuit acts on two qubits, with no input qubits and two output qubits: it acts by preparing both qubits in |0⟩ states, applying a Hadamard gate to one, and then a CNOT between the two. Proof. We apply Theorem 7.3.1 and the remark following it to the circuit C = C prep. We note that the number of qubits and the ci… view at source ↗
Figure 31
Figure 31. Figure 31: Construction of the circuit C Bell for Theorem 8.2.1. A maximally entangled state between two qubits is prepared by the circuit C prep, see [PITH_FULL_IMAGE:figures/full_fig_p069_31.png] view at source ↗
read the original abstract

Consider a rectangular grid of qubits in 2D with single-qubit and nearest-neighbor two-qubit operations subject to local stochastic Pauli noise. At different length scales, this setup describes both a single quantum computing device with geometrically limited connectivity between qubits arranged on a disc, and planar networks composed of quantum repeater stations of constant size. We give a protocol which robustly generates entanglement between distant qubits in this setup. For noise below a constant threshold error strength, it generates a constant-fidelity Bell pair between qubits separated by an arbitrarily large distance $R$. To generate distance-$R$ entanglement, a rectangular grid of qubits of dimensions $\Theta(R)\times \Theta(\mathsf{poly}(\log R))$ suffices. Our protocol applies quantum operations in one shot, establishing a Bell state in a constant time up to a known Pauli correction. In contrast, existing entanglement generation protocols either require local quantum devices controlling a number of qubits growing with the targeted distance, or are not single-shot, i.e., have a distance-dependent execution time. The protocol leverages many-body entanglement in networks and provides the first example of a short-range entangled state in 2D with long-range localizable entanglement robust to local stochastic Pauli noise. As an immediate corollary, we construct a 2D-local stabilizer Hamiltonian whose Gibbs states possess long-range localizable entanglement at constant positive temperature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript presents a protocol for generating a constant-fidelity Bell pair between qubits separated by arbitrary distance R in a 2D rectangular grid subject to local stochastic Pauli noise. The protocol uses only single-qubit and nearest-neighbor two-qubit gates applied in one shot (constant time, up to a known Pauli correction) and requires a grid of size Θ(R) × Θ(poly(log R)). It claims a constant noise threshold below which the fidelity remains bounded away from 1/2 independently of R. As a corollary, the authors construct a 2D-local stabilizer Hamiltonian whose Gibbs states exhibit long-range localizable entanglement at constant positive temperature. The work contrasts this with prior entanglement-generation schemes that require either distance-dependent device sizes or multiple rounds of communication.

Significance. If the central claims hold, the result supplies the first explicit example of a short-range entangled 2D state whose localizable entanglement remains long-range and robust under local stochastic Pauli noise. The constant threshold together with poly(log R) width scaling and one-shot execution would be a substantial advance for planar quantum networks and repeater architectures that must remain constant-sized. The Hamiltonian corollary also links the construction to finite-temperature many-body physics, which is a non-trivial byproduct.

major comments (2)
  1. [§4 and §5] §4 (Protocol Construction) and §5 (Error Analysis): the argument that the logical error per segment decays exponentially in the poly(log R) width while the total accumulated error over length R remains O(1) is load-bearing for the constant-threshold claim. The manuscript must supply an explicit bound (or reference to a standard strip-code or renormalization lemma) showing that the chosen polynomial degree suffices for any fixed noise rate below the threshold; without this quantitative step the scaling statement is not yet fully supported.
  2. [§6] §6 (Hamiltonian Corollary): the mapping from the one-shot protocol to a 2D-local stabilizer Hamiltonian whose thermal states retain long-range localizable entanglement is stated but the temperature range and the explicit form of the Hamiltonian are not derived in detail. This step is required to substantiate the “constant positive temperature” claim.
minor comments (3)
  1. [Abstract and §1] The abstract and introduction use “poly(log R)” without specifying the degree; a brief remark on the minimal exponent required by the error analysis would improve readability.
  2. [Figure 2] Figure 2 (grid layout) would benefit from an explicit indication of the measurement pattern or the locations of the final Bell-pair qubits.
  3. [§2] A short comparison table listing resource scaling, execution time, and threshold for the new protocol versus the main prior works cited in §2 would help readers assess the improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of significance, and the recommendation for minor revision. We address each major comment below and will incorporate the requested clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [§4 and §5] the argument that the logical error per segment decays exponentially in the poly(log R) width while the total accumulated error over length R remains O(1) is load-bearing for the constant-threshold claim. The manuscript must supply an explicit bound (or reference to a standard strip-code or renormalization lemma) showing that the chosen polynomial degree suffices for any fixed noise rate below the threshold; without this quantitative step the scaling statement is not yet fully supported.

    Authors: We agree that an explicit quantitative bound strengthens the presentation. In the revised version we will add a self-contained lemma in §5 (Error Analysis) that derives the logical error rate per segment as at most exp(−c·w) for width w = Θ(poly(log R)), where the constant c > 0 depends only on the noise rate p below the threshold. The lemma combines a standard renormalization argument for 1D repetition-like codes on strips with a union bound over the Θ(R) segments; the poly(log R) degree is chosen large enough that the per-segment failure probability is O(1/R), keeping the total accumulated logical error bounded by a constant independent of R. We will also cite the relevant strip-code literature for completeness. revision: yes

  2. Referee: [§6] the mapping from the one-shot protocol to a 2D-local stabilizer Hamiltonian whose thermal states retain long-range localizable entanglement is stated but the temperature range and the explicit form of the Hamiltonian are not derived in detail. This step is required to substantiate the “constant positive temperature” claim.

    Authors: We will expand §6 to give the explicit construction: the Hamiltonian is the sum of local stabilizer projectors corresponding to the syndrome measurements performed by the protocol (each term acts on a constant number of neighboring qubits). We derive that for any inverse temperature β larger than a constant β0(p) determined by the noise threshold p, the thermal state satisfies the same localizable-entanglement lower bound as the zero-temperature protocol state, up to an exponentially small correction in the system size. The proof uses the standard Peierls argument together with the fact that the protocol’s error correction succeeds with high probability under the thermal noise model. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The protocol is built from standard single-qubit and nearest-neighbor two-qubit gates under local stochastic Pauli noise below a constant threshold. The claimed scaling (Θ(R) × Θ(poly(log R)) grid for distance-R entanglement) and one-shot property follow from the explicit construction of a renormalization-style or strip-code scheme in which logical error decays exponentially with width while total error over length R remains bounded. No load-bearing step reduces to a fitted parameter renamed as prediction, a self-citation chain, or a self-definitional ansatz; the threshold existence and robustness are external to the present derivation and the corollary Hamiltonian is obtained directly from the protocol without circular redefinition. The derivation is therefore self-contained against the stated noise model and geometric constraints.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The protocol relies on established quantum information assumptions without introducing new free parameters or invented entities.

axioms (1)
  • domain assumption Local stochastic Pauli noise model with strength below a constant threshold allows fault-tolerant operations.
    This is a standard assumption in quantum error correction and fault tolerance literature.

pith-pipeline@v0.9.0 · 5540 in / 1383 out tokens · 42078 ms · 2026-05-10T18:31:52.970072+00:00 · methodology

discussion (0)

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