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arXiv:2604.05870

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

Dylan Harley, Robert Koenig

quant-phSubmitted Apr 7, 2026• Announced Apr 8, 2026• 0 citations
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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 $Θ(R)\times Θ(\mathsf{poly}(\log R))$ 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.

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76 pages, 31 figures

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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 RRR. To generate distance-RRR entanglement, a rectangular grid of qubits of dimensions Θ(R)×Θ(poly(log⁡R))Θ(R)\times Θ(\mathsf{poly}(\log R))Θ(R)×Θ(poly(logR)) 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.
Abstract We propose a fault-tolerant scheme for generating long-range entanglement at the ends of a rectangular array of qubits of length R with a square cross-section of m=O(log⁡2R)m=O({\log }^{2}R)m=O(log2R) m = O ( log 2 R ) qubits. It is realized by a constant-depth circuit producing a constant-fidelity Bell-pair (independent of R) for local stochastic noise of strength below an experimentally realistic threshold. The scheme can be viewed as a quantum bus in a quantum computing architecture where qubits are arranged on a rectangular 3D grid, and all operations are between neighboring qubits. Alternatively, it can be seen as a quantum repeater protocol along a line, with neighboring repeaters placed at a short distance to allow constant-fidelity nearest-neighbor operations. To show our protocol uses a number of qubits close to optimal, we show that any noise-resilient distance-R entanglement generation scheme realized by a constant-depth circuit needs at least m=Ω(log⁡R)m=\Omega (\log R)m=Ω(logR) m = Ω ( log R ) qubits per repeater.
Fault-tolerant modular quantum computing requires stabilizer measurements across the modules in a quantum network. For this, entangled states of high quality and rate must be distributed. Currently, two main types of entanglement distribution protocols exist, namely emission-based and scattering-based, each with its own advantages and drawbacks. On the one hand, scattering-based protocols with cavities or waveguides are fast but demand stringent hardware such as high-efficiency integrated circulators or strong waveguide coupling. On the other hand, emission-based platforms are experimentally feasible but so far rely on Bell-pair fusion with extensive use of slow two-qubit memory gates, limiting thresholds to ≈0.16%\approx 0.16\%≈0.16%. Here, we consider a fully distributed surface code using emission-based entanglement schemes that generate GHZ states in a single shot, i.e., without the need for Bell-pair fusions. We show that our optical setup produces Bell pairs, W states, and GHZ states, enabling both memory-based and optical protocols for distilling high-fidelity GHZ states with significantly improved success rates. Furthermore, we introduce protocols that completely eliminate the need for memory-based two-qubit gates, achieving thresholds of ≈0.19%\approx 0.19\%≈0.19% with modest hardware enhancements, increasing to above ≈0.24%\approx 0.24\%≈0.24% with photon-number-resolving detectors. These results show the feasibility of emission-based architectures for scalable fault-tolerant operation.
Measurement-based quantum repeaters employ entanglement distillation and swapping across links using locally prepared resource states of minimal size and local Bell measurements. In this Letter, we introduce a systematic protocol for measurement-based entanglement distillation and its application to repeaters that can leverage any stabilizer code. Given a code, we explicitly define the corresponding resource state and derive an error-recovery operation based on all Bell measurement outcomes. Our approach offers deeper insights into the impact of resource state noise on repeater performance while also providing strategies for efficient preparation and fault-tolerant preservation of resource states. As an application, we propose a measurement-based repeater protocol based on quantum low-density parity-check (QLDPC) codes, enabling constant-yield Bell state distribution over arbitrary distances. Numerical simulations identify a fault-tolerant threshold on the total physical error per repeater segment-including errors on resource states, remotely generated Bell states, and Bell measurements-and confirm that increasing the QLDPC code size further suppresses the logical error while maintaining a fixed encoding rate. This Letter establishes a scalable backbone for future global-scale fault-tolerant quantum networks.
We construct the first known fault-tolerant scheme for injecting states into quantum error-correcting codes with constant space and time overhead. That is, we construct a family of constant-rate quantum error-correcting codes for which a set of bare physical qubits can be injected, i.e. fault-tolerantly encoded, into a code block. Similarly, a code state can be ejected, i.e. fault-tolerantly decoded, back into bare physical qubits. We show that these injection and ejection procedures succeed under circuit-level locally stochastic noise, while incurring just a small constant probability of corrupting each qubit, which is unavoidable for bare physical qubits. We also show how to perform fault-tolerant error correction and code-state preparation under locally stochastic noise. All of our gadgets can be implemented with constant quantum circuit depth (i.e. are single-shot), and with a number of physical qubits growing linearly with the number of logical qubits, assuming the ability to run polynomial-sized noiseless classical circuits on the side. We construct our quantum codes by taking a high-dimensional hypergraph product of classical LDPC codes, which in turn are a simplified version of Spielman's linear-time encodable codes (STOC'95). As our resulting product codes are only resilient to physical errors occurring with non-uniform probabilities across qubits, we then show how to concatenate with inner codes of various sizes to obtain fault-tolerance against uniform noise.
The reliable generation of multi-qubit entanglement is a prerequisite for large-scale quantum information technologies. In particular, W states are a valuable resource owing to their resilience under local loss or measurement. Nevertheless, preparing these states with sequential two-qubit gates often requires substantial time overhead. By contrast, engineered simultaneous interactions enable fast entanglement generation, even in qubit systems with limited nearest-neighbour connectivity. Here, we demonstrate a set of fast and robust operations for coherently distributing a single excitation across a lattice of arbitrary size, thereby directly generating W states from initial product states. In 2D lattices, the excitation propagates along both directions simultaneously, such that the total entanglement time scales only with the largest dimension. We exploit this property to prepare a six-qubit W state in a 3$\times2superconductinglatticewithin99ns,achievingatomographicfidelityof83.92 superconducting lattice within 99 ns, achieving a tomographic fidelity of 83.92superconductinglatticewithin99ns,achievingatomographicfidelityof83.9\pm1.01.0%. We then extend the protocol to create entanglement across chains of up to seven qubits, with the largest W state generated in 264 ns with a fidelity of 79.61.0\pm$1.3%.
A defining feature in the field of quantum computing is the potential of a quantum device to outperform its classical counterpart for a specific computational task.By now, several proposals exist showing that certain sampling problems can be done efficiently quantumly, but are not possible efficiently classically, assuming strongly held conjectures in complexity theory, a feature dubbed quantum speedup.However, the effect of noise on these proposals is not well understood in general, and in certain cases it is known that simple noise can destroy the quantum speedup.Here we develop a fault-tolerant version of one family of these sampling problems, based on nonadaptive measurements on regular graph states composed of m qubits.We show that these sampling problems can be implemented fault tolerantly using quantum circuits of constant depth.We present two constructions, each taking poly(m) physical qubits, some of which are prepared in noisy magic states.The first of our constructions is a constant depth quantum circuit composed of single-and two-qubit nearest-neighbor Clifford gates in four dimensions.This circuit has one layer of interaction with a classical computer before final measurements.Our second construction is a constant depth quantum circuit with single-and two-qubit nearest-neighbor Clifford gates in three dimensions, but with two layers of interaction with a classical computer before the final measurements.For each of these constructions, we show that there is no classical algorithm which can sample according to its output distribution in poly(m) time, assuming two standard complexity theoretic conjectures hold.The noise model we assume is the so-called local stochastic quantum noise.Along the way, we introduce various concepts such as constant depth magic state distillation and constant depth output routing, which arise naturally in measurement based quantum computation, but have no constant-depth analog in the circuit model.