Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.26736

A Syndrome-Extraction Framework for Distributed Lattice Surgery on Arbitrary Rotated Surface-Code Layouts

Pritesh Thakur, Daniel Dilley, Zain Saleem

quant-phSubmitted Sep 22, 2026• Announced Sep 23, 2026• 0 citations
arXivHTMLPDF

Abstract

Modular quantum computing is a leading candidate for scalable fault tolerant quantum computation. Performing lattice surgery using surface code patches in different modules requires the use of inter-module gates that are noisier than the local gates. Furthermore, the merged layouts during lattice surgery expose multiple directions in which hook errors can propagate to reduce the fault distance. Existing hook-avoiding schedules either increase the syndrome extraction circuit depth, or let link faults propagate into the bulk. We introduce a seam construction that allows splitting arbitrary merged rotated surface code layout into smaller patches with uniform boundaries that could keep their own hook-avoiding N/Z shaped schedule. At the seam we split weight-4 stabilizers into pairs of weight-2 stabilizers that are mediated through a Bell pair and measured within a CNOT depth of 4. We also employ the use of three-qubit $\texttt{CXX}$ gates for parity measurements at the seam to maintain the minimum CNOT depth. Through this seam construction, we confine the elevated link error at the seam, protecting the bulk from higher error rate. We also give a custom detector annotator for such seams that can find the detector space, annotate detectors in the Stim circuits, and ensure that they are suitable to use by matching decoders. We implement our construction for the logical Pauli-$XX$ measurement lattice surgery and a cross-shaped spatial junction and sweep the link error rate from $10$ times to $100$ times a fixed bulk error rate, and show that our construction gives the lowest and most stable logical error rate than the other frameworks compared here.

Related papers

No More Hooks in the Surface Code: Distance-Preserving Syndrome Extraction for Arbitrary Layouts at Minimum Depth

Yuga Hirai, Shota Ikari, Yosuke Ueno +1 more

Hook errors are a major challenge in implementing logical operations with the surface code, because they can reduce the fault distance below the code distance. This motivates syndrome-extraction circuits that suppress hook-error effects for the stabilizer layouts that appear during logical operations. However, the existing methods either increase circuit depth or require simultaneous execution of measurements and CNOT gates, both of which introduce additional overheads and degrade the threshold. We propose the ZX interleaving syndrome extraction, which preserves the full fault distance $d$ for any surface-code layout with regular stabilizer tiles at minimum depth, i.e., four layers of CNOT gates, without requiring additional circuit depth or simultaneous execution of measurements and CNOT gates. The key idea is to interleave the Z and X stabilizer tiles so that hook-error edges in the decoding graph are shortened and effectively eliminated. Numerical simulations under uniform depolarizing noise for memory and lattice-surgery experiments confirm that the proposed method achieves a full fault distance of $d$, whereas the best existing minimum-depth approach achieves $d-1$. Since the full fault distance is achievable for any regular tiling layout of the surface code, the proposed method may serve as an indispensable technique for practical fault-tolerant quantum computation.

quant-pharXiv:2603.01628•Mar 3, 2026•0 citations

Surface code off-the-hook: diagonal syndrome-extraction scheduling

Gilad Kishony, Austin Fowler

In the rotated surface code, hook errors (errors on auxiliary qubits midway through syndrome extraction that propagate to correlated two-qubit data errors) can reduce the circuit-level code distance by a factor of two if the extraction schedule is poorly chosen. The traditional approach uses N-shaped and Z-shaped schedules, selecting the orientation in each plaquette to avoid hook errors aligned with logical operators. However, this becomes increasingly complex within lattice surgery primitives with varied boundary geometries, and requires a 7-step schedule to avoid gate collisions. We propose the diagonal schedule, which orients hook errors along the diagonal of each plaquette. These diagonal errors crucially never align with logical operators regardless of boundary orientation, achieving full code distance. The diagonal schedule is globally uniform: all X-type plaquettes use one schedule and all Z-type plaquettes use another, eliminating geometry-dependent planning. On hardware supporting parallel measurement, reset, and gate operations, the schedule achieves a minimal period of 6 time steps, compared to 7 for the traditional approach. We demonstrate effectiveness for memory experiments, spatial junctions, spatial Hadamard gates, and patch rotation, showing equivalent or improved logical error rates while simplifying circuit construction.

quant-pharXiv:2602.09099•Feb 11, 2026•0 citations

Patchwork: automated lattice surgery and space-time partitioning for real-time decoding

Christopher Chamberland, Muyuan Li, Scott Thornton +3 more

We introduce patchwork, a lattice-surgery compiler for arbitrary multi-qubit Pauli measurements on a prescribed narrow-corridor two-dimensional surface-code layout. Systematic geometric rules simplify stabilizer placement and gate scheduling, including for logical $Y$ measurements implemented using twist defects. We introduce measurement circuits and schedules that, together with protruding lobes of $Y$-extension patches, preserve the full effective code distance for all numerically tested code distances. To support parallel decoding, we develop a wavefront algorithm that partitions merged lattice-surgery patches into space-time commit, buffer, and cleanup regions. Simulations of three sequential multi-qubit Pauli measurements acting on ten logical qubits show that staged decoding achieves total failure probabilities close to those of monolithic decoding over the parameters tested. These methods provide an automated framework for fault-tolerant lattice surgery and a route toward scalable, real-time decoding.

quant-pharXiv:2610.22112•Oct 16, 2026•0 citations

Lattice surgery with Bell measurements: Modular fault-tolerant quantum computation at low entanglement cost

Trond Hjerpekjøn Haug, Timo Hillmann, Anton Frisk Kockum +1 more

Modular architectures are a promising approach to scaling quantum computers to fault tolerance. Small, low-noise quantum processors connected through relatively noisy quantum links are capable of fault-tolerant operation as long as the noise can be confined to the interface. Finding protocols that implement the quantum links between modules as efficiently as possible is essential because inter-module entanglement is challenging to produce at a similar rate and fidelity as local entanglement. We introduce a protocol for lattice surgery on surface codes in which all non-local operations are Bell measurements. The protocol simultaneously confines the link noise and requires only half as many module-crossing gates as previously proposed protocols. To mitigate distance-reducing hook errors, we introduce a strategy of alternating the gate sequence between rounds of syndrome measurement, which prevents multiple hooks from simultaneously aligning with a logical operator in the code. We evaluate our protocol's performance when two logical qubits on separate modules are prepared in a logical Bell state. Circuit-level simulations under depolarizing noise show that the logical error suppression for a given entanglement rate between modules is consistently stronger compared to the best-performing alternative protocols for a wide range of link noise, with a typical 40% entanglement resource saving for a constant logical error rate. Our approach to protocol design is applicable to any quantum circuit that must be divided across processor modules and can therefore guide development of resource-efficient modular quantum computation beyond the surface code.

arXiv:2510.13541•Oct 15, 2025•1 citation

Error Suppression in Distributed Quantum Computing with Heterogeneous-Distance Lattice Surgery

Daniel Dilley, Anastashia Jebraeilli, Rayat Roy +3 more

Distributed quantum computing requires fault-tolerant operations across inter-QPU links that can be substantially noisier than local gates. Uniformly increasing code distance provides additional protection but also enlarges data patches used for local storage and computation. Here, we introduce distributed heterogeneous-distance lattice surgery using an eight-data-patch ancilla-mediated (8-DAM) architecture, which will be useful for near-term quantum devices with less qubit overhead. In this architecture, the central ancilla spanning the inter-QPU boundary is enlarged while the data patches retain distance $d$. The protocol uses traveling stabilizers to suppress hook errors during merge and split operations between these unequal-distance patches. Circuit-level simulations of rotated surface codes at fixed local depolarizing noise show that logical-readout error rates depend only weakly on link noise. The resulting advantage over conventional lattice surgery grows as link errors increase. Comparisons with uniform distance implementations demonstrate comparable logical error suppression with reduced physical-qubit overhead. We also demonstrate how 8-DAM layouts support two simultaneous distributed logical CNOT operations between four logical data qubits using a single enlarged ancilla. At higher link noise, this construction yields lower logical error rates and more stability than two independent distributed logical CNOTs. These results support selective ancilla enlargement as a resource-efficient approach to fault-tolerant distributed quantum computing.

quant-pharXiv:2609.26784•Sep 23, 2026•0 citations

Boundary-Aware Stabilizer Scheduling for Distributed Quantum Error Correction

Sanidhya Gupta, Sanidhay Bhambay, Narges Alavisamani +2 more

Future quantum architectures are expected to be modular, with quantum processors connecting multiple quantum processing units (QPUs) via photonic interconnects. In topological quantum error correction, such as color codes, this creates seam boundaries where parity checks require remote CNOT operations using heralded Bell pairs. These non-local checks are slower and noisier than bulk local checks because entanglement generation is probabilistic, causing data qubits to accumulate idle noise while waiting for remote operations. A natural way to reduce this overhead is to skip some seam measurements; however, doing so makes seam syndrome information stale and can degrade decoding. The central scheduling problem is therefore to determine how frequently seam checks should be measured so as to balance remote-operation and waiting noise against syndrome staleness. To address this trade-off, we develop a scheduling module that integrates directly into standard syndrome-extraction circuits. We consider two policies: Skip-Seam-$τ$ (SS-$τ$), which measures all bulk checks every round while measuring seam checks once every $τ$ rounds and copying the most recent syndrome in skipped rounds, and Adaptive Skip-$τ$ (AST), which selects $τ$ as a function of code distance and entanglement generation rate (EGR). We evaluate these policies on triangular color codes under circuit-level noise in Stim, including idling errors induced by Bell-pair generation delays. Our simulations show that SS-tau and AST reduce remote-operation overhead and can lower the logical error rate (LER) relative to the Measure-All (MA) baseline. For physical error rate $p = 10^{-3}$, we identify an EGR regime in which both SS-$τ$ and AST exhibit behavior consistent with fault-tolerant scaling, with LER decreasing as code distance increases. Across these regimes, SS-$τ$ and AST outperform MA.

quant-pharXiv:2604.22471•Apr 27, 2026•0 citations
Modular quantum computing is a leading candidate for scalable fault tolerant quantum computation. Performing lattice surgery using surface code patches in different modules requires the use of inter-module gates that are noisier than the local gates. Furthermore, the merged layouts during lattice surgery expose multiple directions in which hook errors can propagate to reduce the fault distance. Existing hook-avoiding schedules either increase the syndrome extraction circuit depth, or let link faults propagate into the bulk. We introduce a seam construction that allows splitting arbitrary merged rotated surface code layout into smaller patches with uniform boundaries that could keep their own hook-avoiding N/Z shaped schedule. At the seam we split weight-4 stabilizers into pairs of weight-2 stabilizers that are mediated through a Bell pair and measured within a CNOT depth of 4. We also employ the use of three-qubit CXX\texttt{CXX}CXX gates for parity measurements at the seam to maintain the minimum CNOT depth. Through this seam construction, we confine the elevated link error at the seam, protecting the bulk from higher error rate. We also give a custom detector annotator for such seams that can find the detector space, annotate detectors in the Stim circuits, and ensure that they are suitable to use by matching decoders. We implement our construction for the logical Pauli-XXXXXX measurement lattice surgery and a cross-shaped spatial junction and sweep the link error rate from 101010 times to 100100100 times a fixed bulk error rate, and show that our construction gives the lowest and most stable logical error rate than the other frameworks compared here.
Hook errors are a major challenge in implementing logical operations with the surface code, because they can reduce the fault distance below the code distance. This motivates syndrome-extraction circuits that suppress hook-error effects for the stabilizer layouts that appear during logical operations. However, the existing methods either increase circuit depth or require simultaneous execution of measurements and CNOT gates, both of which introduce additional overheads and degrade the threshold. We propose the ZX interleaving syndrome extraction, which preserves the full fault distance ddd for any surface-code layout with regular stabilizer tiles at minimum depth, i.e., four layers of CNOT gates, without requiring additional circuit depth or simultaneous execution of measurements and CNOT gates. The key idea is to interleave the Z and X stabilizer tiles so that hook-error edges in the decoding graph are shortened and effectively eliminated. Numerical simulations under uniform depolarizing noise for memory and lattice-surgery experiments confirm that the proposed method achieves a full fault distance of ddd, whereas the best existing minimum-depth approach achieves d−1d-1d−1. Since the full fault distance is achievable for any regular tiling layout of the surface code, the proposed method may serve as an indispensable technique for practical fault-tolerant quantum computation.
In the rotated surface code, hook errors (errors on auxiliary qubits midway through syndrome extraction that propagate to correlated two-qubit data errors) can reduce the circuit-level code distance by a factor of two if the extraction schedule is poorly chosen. The traditional approach uses N-shaped and Z-shaped schedules, selecting the orientation in each plaquette to avoid hook errors aligned with logical operators. However, this becomes increasingly complex within lattice surgery primitives with varied boundary geometries, and requires a 7-step schedule to avoid gate collisions. We propose the diagonal schedule, which orients hook errors along the diagonal of each plaquette. These diagonal errors crucially never align with logical operators regardless of boundary orientation, achieving full code distance. The diagonal schedule is globally uniform: all X-type plaquettes use one schedule and all Z-type plaquettes use another, eliminating geometry-dependent planning. On hardware supporting parallel measurement, reset, and gate operations, the schedule achieves a minimal period of 6 time steps, compared to 7 for the traditional approach. We demonstrate effectiveness for memory experiments, spatial junctions, spatial Hadamard gates, and patch rotation, showing equivalent or improved logical error rates while simplifying circuit construction.
We introduce patchwork, a lattice-surgery compiler for arbitrary multi-qubit Pauli measurements on a prescribed narrow-corridor two-dimensional surface-code layout. Systematic geometric rules simplify stabilizer placement and gate scheduling, including for logical YYY measurements implemented using twist defects. We introduce measurement circuits and schedules that, together with protruding lobes of YYY-extension patches, preserve the full effective code distance for all numerically tested code distances. To support parallel decoding, we develop a wavefront algorithm that partitions merged lattice-surgery patches into space-time commit, buffer, and cleanup regions. Simulations of three sequential multi-qubit Pauli measurements acting on ten logical qubits show that staged decoding achieves total failure probabilities close to those of monolithic decoding over the parameters tested. These methods provide an automated framework for fault-tolerant lattice surgery and a route toward scalable, real-time decoding.
Modular architectures are a promising approach to scaling quantum computers to fault tolerance. Small, low-noise quantum processors connected through relatively noisy quantum links are capable of fault-tolerant operation as long as the noise can be confined to the interface. Finding protocols that implement the quantum links between modules as efficiently as possible is essential because inter-module entanglement is challenging to produce at a similar rate and fidelity as local entanglement. We introduce a protocol for lattice surgery on surface codes in which all non-local operations are Bell measurements. The protocol simultaneously confines the link noise and requires only half as many module-crossing gates as previously proposed protocols. To mitigate distance-reducing hook errors, we introduce a strategy of alternating the gate sequence between rounds of syndrome measurement, which prevents multiple hooks from simultaneously aligning with a logical operator in the code. We evaluate our protocol's performance when two logical qubits on separate modules are prepared in a logical Bell state. Circuit-level simulations under depolarizing noise show that the logical error suppression for a given entanglement rate between modules is consistently stronger compared to the best-performing alternative protocols for a wide range of link noise, with a typical 40% entanglement resource saving for a constant logical error rate. Our approach to protocol design is applicable to any quantum circuit that must be divided across processor modules and can therefore guide development of resource-efficient modular quantum computation beyond the surface code.
Distributed quantum computing requires fault-tolerant operations across inter-QPU links that can be substantially noisier than local gates. Uniformly increasing code distance provides additional protection but also enlarges data patches used for local storage and computation. Here, we introduce distributed heterogeneous-distance lattice surgery using an eight-data-patch ancilla-mediated (8-DAM) architecture, which will be useful for near-term quantum devices with less qubit overhead. In this architecture, the central ancilla spanning the inter-QPU boundary is enlarged while the data patches retain distance ddd. The protocol uses traveling stabilizers to suppress hook errors during merge and split operations between these unequal-distance patches. Circuit-level simulations of rotated surface codes at fixed local depolarizing noise show that logical-readout error rates depend only weakly on link noise. The resulting advantage over conventional lattice surgery grows as link errors increase. Comparisons with uniform distance implementations demonstrate comparable logical error suppression with reduced physical-qubit overhead. We also demonstrate how 8-DAM layouts support two simultaneous distributed logical CNOT operations between four logical data qubits using a single enlarged ancilla. At higher link noise, this construction yields lower logical error rates and more stability than two independent distributed logical CNOTs. These results support selective ancilla enlargement as a resource-efficient approach to fault-tolerant distributed quantum computing.
Future quantum architectures are expected to be modular, with quantum processors connecting multiple quantum processing units (QPUs) via photonic interconnects. In topological quantum error correction, such as color codes, this creates seam boundaries where parity checks require remote CNOT operations using heralded Bell pairs. These non-local checks are slower and noisier than bulk local checks because entanglement generation is probabilistic, causing data qubits to accumulate idle noise while waiting for remote operations. A natural way to reduce this overhead is to skip some seam measurements; however, doing so makes seam syndrome information stale and can degrade decoding. The central scheduling problem is therefore to determine how frequently seam checks should be measured so as to balance remote-operation and waiting noise against syndrome staleness. To address this trade-off, we develop a scheduling module that integrates directly into standard syndrome-extraction circuits. We consider two policies: Skip-Seam-τττ (SS-τττ), which measures all bulk checks every round while measuring seam checks once every τττ rounds and copying the most recent syndrome in skipped rounds, and Adaptive Skip-τττ (AST), which selects τττ as a function of code distance and entanglement generation rate (EGR). We evaluate these policies on triangular color codes under circuit-level noise in Stim, including idling errors induced by Bell-pair generation delays. Our simulations show that SS-tau and AST reduce remote-operation overhead and can lower the logical error rate (LER) relative to the Measure-All (MA) baseline. For physical error rate p=10−3p = 10^{-3}p=10−3, we identify an EGR regime in which both SS-τττ and AST exhibit behavior consistent with fault-tolerant scaling, with LER decreasing as code distance increases. Across these regimes, SS-τττ and AST outperform MA.