Fault-Tolerant One-Shot Entanglement Generation with Constant-Sized Quantum Devices in the Plane
Dylan Harley, Robert Koenig
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 . To generate distance- entanglement, a rectangular grid of qubits of dimensions 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.
AI Impact Assessments
(3 models)Scientific Impact Assessment
Core Contribution
This paper solves a fundamental open problem in quantum networking and fault-tolerant quantum information: Can constant-fidelity entanglement be generated in a single shot between arbitrarily distant qubits in a 2D planar network using only constant-size quantum devices at each node? The authors answer affirmatively, providing the first fully rigorous protocol with an analytically proven fault tolerance threshold.
The main result (Theorem 1) establishes that on a rectangular grid of dimensions Θ(R) × Θ(poly(log R)), a constant-depth Clifford circuit can produce a Bell pair between qubits separated by distance R, with success probability ≥ 9/10, even under local stochastic Pauli noise below a constant threshold. This simultaneously addresses two practical settings: (1) planar quantum computing devices with geometrically limited connectivity, and (2) quantum repeater networks where each station controls only O(1) qubits.
The key innovation is leveraging the additional spatial dimension (width) of a 2D network to circumvent fundamental 1D limitations—where constant-size repeaters provably cannot generate distance-independent fidelity entanglement in one shot. The construction proceeds by: (i) building fault-tolerant 1D-local Clifford circuit implementations using concatenated codes on bilinear arrays, (ii) introducing mid-circuit resets to bound qubit lifespans to O(1), and (iii) "unfolding" the 1D time evolution into a 2D spatial arrangement, converting circuit depth into a physical dimension.
Methodological Rigor
The paper is technically thorough, spanning 76 pages with detailed proofs. The construction builds systematically through five clearly delineated steps, each with explicit theorem statements and proofs. Key technical advances include:
1. Robust circuit implementation with noisy encoding/decoding (Section 4): Extending standard fault tolerance beyond prepare-and-measure circuits to circuits with quantum input/output—a non-trivial extension since input/output qubits are unencoded physical qubits. The iterated level reduction (Lemma 4.4.3) with explicit failure probability bounds is carefully derived.
2. Pauli noise propagation through adaptive Clifford circuits (Section 5): The analysis of how local stochastic Pauli noise transforms under circuit modifications (inflation, subcircuit substitution, geometry changes) is rigorous. The preorder relation ≳ on circuits (Definition 5.2.1) provides a clean framework for composing multiple transformations while tracking noise parameters.
3. The space-time transformation (Section 7.3): Converting the 1D circuit with bounded qubit lifespan into a 2D constant-depth circuit is elegant and well-justified.
One limitation is that the threshold estimates are highly conservative—the authors acknowledge this explicitly, noting that analytical thresholds are typically orders of magnitude below actual thresholds. The construction relies on the [[7,1,3]] Steane code via the Stephens-Fowler-Hollenberg bilinear array scheme, and specific numerical threshold values are not computed.
Potential Impact
Quantum Networks: This work provides theoretical foundation for quantum repeater architectures that do not require scaling local quantum processors with communication distance—a major practical constraint. The one-shot nature eliminates distance-dependent latency, critical for quantum internet applications.
Quantum Computing: The robust 2D-local implementation of 1D circuits (Theorem 7.3.1) could serve as a building block for making general fault tolerance schemes geometrically local in 2D, as the authors note regarding parallel repetition.
Many-body Physics: The paper provides the first example of a 2D short-range entangled state with long-range localizable entanglement robust to local stochastic Pauli noise. The corollary constructing a 2D stabilizer Hamiltonian whose Gibbs states possess long-range localizable entanglement at constant positive temperature (Corollary 8.3.1) is a notable contribution to quantum statistical mechanics, complementing the 3D cluster state results of Raussendorf-Bravyi-Harrington.
Computational Complexity: The authors suggest their construction could establish that noisy 2D constant-depth quantum circuits are computationally more powerful than classical circuits, extending the Bravyi-Gosset-König-Tomamichel result from 3D to 2D.
Timeliness & Relevance
This work is highly timely. Near-term quantum architectures are moving toward modular, distributed designs where entanglement distribution between modules is a bottleneck. The paper directly addresses this with practical geometric constraints (planar, nearest-neighbor). It also connects to the recent surge of interest in fault-tolerant quantum I/O (Christandl-Fawzi-Goswami 2026, Belzig-Yamasaki 2026), constant-rate fault tolerance (Gidney-Bergamaschi 2025), and thermal entanglement properties (Bakshi et al. 2024).
Strengths & Limitations
Strengths:
Limitations:
Generated Apr 8, 2026
Comparison History (62)
Paper 2 presents a fundamental breakthrough in quantum network scalability and fault tolerance by enabling one-shot, long-distance entanglement using constant-sized 2D devices. Additionally, its corollary proving long-range localizable entanglement at positive temperatures in a 2D Hamiltonian resolves important questions in quantum many-body physics. While Paper 1 offers a highly practical approach for biased-noise systems, Paper 2's theoretical advancements span across quantum computing, networking, and many-body physics, promising a broader and deeper foundational impact.
Paper 2 offers broadly applicable advances in fault-tolerant quantum networking/computation: a constant-threshold, one-shot protocol achieving long-distance entanglement with only polylog overhead in one dimension, plus a stabilizer-Hamiltonian corollary at finite temperature. This is methodologically aligned with standard noise models and impacts multiple areas (quantum repeaters, MBQC, many-body physics, Hamiltonian complexity). Paper 1, while potentially extremely impactful if correct (breaking standardized PQC under quantum attack), is narrower in domain and would require exceptionally strong validation given extraordinary claims; its impact is also contingent on feasibility of the stated quantum resources.
Paper 1 appears more impactful due to a clear breakthrough-style result: single-shot, constant-time, fault-tolerant long-range Bell-pair generation using only constant-sized local devices under a noise threshold, with favorable resource scaling. It also connects to multiple areas (quantum networking/repeaters, fault tolerance, many-body entanglement, and stabilizer Hamiltonians at finite temperature), suggesting broad downstream applications and conceptual significance. Paper 2 is solid and timely for nonequilibrium open quantum systems, but its contributions (scaling extension + numerics) are more specialized and likely narrower in cross-field technological impact.
Paper 2 presents a fundamental breakthrough in quantum networking and many-body physics by enabling fault-tolerant, one-shot long-range entanglement in constant time using constant-sized devices. This solves a major scalability bottleneck for quantum repeaters and offers profound theoretical insights into localizable entanglement. In contrast, Paper 1 offers a practical but incremental optimization for data encoding in near-term quantum machine learning models, making Paper 2 significantly more impactful in advancing the core foundations of scalable quantum technologies.
Paper 1 presents a breakthrough in fault-tolerant quantum entanglement generation with constant-sized devices in 2D, solving a fundamental open problem at the intersection of quantum error correction, quantum networks, and condensed matter physics. It introduces the first example of a short-range entangled 2D state with noise-robust long-range localizable entanglement and constructs a novel 2D Hamiltonian with long-range entanglement at positive temperature. This has broad implications for quantum networking, fault-tolerant quantum computing, and many-body physics. Paper 2 makes important but more incremental theoretical contributions to entanglement distillation theory, extending single-letter formulas to new state families—significant but narrower in scope and practical impact.
Paper 2 addresses a fundamental problem in quantum information—fault-tolerant long-distance entanglement generation—with novel theoretical results that have broad implications. It achieves constant-time, one-shot entanglement generation using only constant-sized devices in 2D, overcoming key limitations of prior protocols. The results span quantum networking, fault-tolerant quantum computing, and condensed matter (2D Hamiltonian with long-range localizable entanglement at finite temperature). Its methodological contributions are more foundational and broadly applicable compared to Paper 1's specific quantum simulation proposal for Motzkin chains, which, while interesting, is more niche.
Paper 1 addresses a fundamental bottleneck in quantum networking and computing: fault-tolerant, scalable entanglement generation. By achieving one-shot, constant-time entanglement over arbitrary distances with constant-sized devices and a known error threshold, it offers a major theoretical breakthrough with broad applications for the quantum internet and scalable quantum architectures. Paper 2, while proposing an interesting experimental realization of the Motzkin spin chain, is more narrowly focused on simulating a specific mathematical model, making its potential impact more confined to condensed matter and many-body physics.
Paper 2 has higher potential impact due to a more foundational advance in fault-tolerant long-distance entanglement generation with constant-sized planar devices, addressing a central bottleneck for scalable quantum networks and distributed quantum computing. Its constant-threshold, one-shot, constant-time protocol and the corollary about 2D-local stabilizer Hamiltonians with long-range localizable entanglement at finite temperature suggest broad implications across quantum information, condensed matter, and quantum networking. Paper 1 is a valuable methodological extension of TEMPO for open-system simulation, but is more specialized and primarily impacts numerical modeling.
Paper 1 presents a fundamental breakthrough in quantum information: the first fault-tolerant, single-shot entanglement generation protocol using constant-sized devices in 2D, with implications spanning quantum networking, quantum error correction, and condensed matter physics (Gibbs states with long-range entanglement). It resolves open questions about localizable entanglement under noise and establishes new theoretical results. Paper 2, while practical and useful for quantum chemistry, is more incremental—combining existing techniques (ADAPT-VQE, Majorana propagation) into a classical algorithm. Paper 1's broader theoretical impact across multiple fields gives it the edge.
Paper 2 solves a fundamental open problem in quantum information: generating long-range entanglement in constant time with constant-sized devices under realistic noise, using only planar 2D connectivity. This has profound implications for quantum networks, fault-tolerant quantum computing, and quantum repeater design. The result that constant-size quantum devices suffice (with only polylog width) is surprising and practically transformative. The corollary about 2D Gibbs states with long-range localizable entanglement at constant temperature is also a notable theoretical contribution. While Paper 1 provides elegant theoretical scaling laws for neural quantum states, Paper 2 addresses more fundamental constraints with broader impact across quantum computing, networking, and condensed matter physics.