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arXiv:2412.14256 · Nature

Scaling and logic in the colour code on a superconducting quantum processor

Nathan Lacroix, Alexandre Bourassa, Francisco J. H. Heras, L. M. Zhang, Johannes Bausch, Andrew Senior, Thomas Edlich, Noah Shutty, Volodymyr Sivak, Andreas Bengtsson, Matt McEwen, Oscar Higgott, Dvir Kafri, Jahan Claes, Alexis Morvan, Z. Chen, Adam Zalcman, Sid Madhuk, R. Acharya, Laleh Aghababaie Beni, Georg Aigeldinger, R. Alcaraz, Trond I. Andersen, M. Ansmann, Frank Arute, Kunal Arya, Abraham T. Asfaw, Juan Atalaya, Ryan Babbush, B. Ballard, Joseph C. Bardin, Alexander Bilmes, Susan A. Blackwell, Jenna Bovaird, D. Bowers, L. Brill, Michael Broughton, D. A. Browne, Brett Buchea, Bob B. Buckley, Tim Burger, Brian Burkett, Nicholas Bushnell, A. Cabrera, Juan Campero, Hung-Shen Chang, Benjamin Chiaro, Liang-Ying Chih, Agnetta Y. Cleland, J. G. Cogan, R. Collins, P. Conner, William Courtney, A. L. Crook, Ben Curtin, Sayan Das, Sean Demura, Luis Lorenzo, Agustín Di Paolo, P. Donohoe, Ilya Drozdov, A. Dunsworth, Alec Eickbusch, Aviv Moshe Elbag, Mahmoud Elzouka, Catherine Erickson, V. S. Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, Suhas Ganjam, G. Garcia, R. Gasca, Élie Genois, W. Giang, Dar Gilboa, Raja Gosula, Alejandro Grajales Dau, Dietrich Graumann, Alex Greene, Jonathan A. Gross, T. Ha, Steve Habegger, M. Hansen, Matthew P. Harrigan, Sean David Harrington, Stephen Heslin, Paula Heu, Reno Hiltermann, Jeremy Hilton, Sabrina Hong, Hsin-Yuan Huang, Ashley Huff, William J. Huggins, Evan R. Jeffrey, Jiang Zhang, X. Jin, C. Joshi, Pavol Juhás

Published May 26, 2025• 51 citations
arXivHTMLPDF

Abstract

Quantum error correction [1–4] is essential for bridging the gap between the error rates of physical devices and the extremely low error rates required for quantum algorithms. Recent error-correction demonstrations on superconducting processors [5–8] have focused primarily on the surface code [9], which offers a high error threshold but poses limitations for logical operations. The color code [10] enables more efficient logic, but it requires more complex stabilizer measurements and decoding. Measuring these stabilizers in planar architectures like superconducting qubits is challenging, and realizations of color codes [11–19] have not addressed performance scaling with code size on any platform. Here, we present a comprehensive demonstration of the color code on a superconducting processor [8]. Scaling the code distance from three to five suppresses logical errors by a factor of Λ 3/5 = 1.56(4). Simulations indicate this performance is below the threshold of the color code, and the color code may become more efficient than the surface code following modest device improvements. We test transversal Clifford gates with logical randomized benchmarking [20] and inject magic states [21], a key resource for universal computation, achieving fidelities exceeding 99 % with post-selection. Finally, we teleport logical states between color codes using lattice surgery [22]. This work establishes the color code as a compelling research direction to realize fault-tolerant quantum computation on superconducting processors in the near future.

DOI
10.1038/s41586-025-09061-4

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Quantum error correction [1–4] is essential for bridging the gap between the error rates of physical devices and the extremely low error rates required for quantum algorithms. Recent error-correction demonstrations on superconducting processors [5–8] have focused primarily on the surface code [9], which offers a high error threshold but poses limitations for logical operations. The color code [10] enables more efficient logic, but it requires more complex stabilizer measurements and decoding. Measuring these stabilizers in planar architectures like superconducting qubits is challenging, and realizations of color codes [11–19] have not addressed performance scaling with code size on any platform. Here, we present a comprehensive demonstration of the color code on a superconducting processor [8]. Scaling the code distance from three to five suppresses logical errors by a factor of Λ 3/5 = 1.56(4). Simulations indicate this performance is below the threshold of the color code, and the color code may become more efficient than the surface code following modest device improvements. We test transversal Clifford gates with logical randomized benchmarking [20] and inject magic states [21], a key resource for universal computation, achieving fidelities exceeding 99 % with post-selection. Finally, we teleport logical states between color codes using lattice surgery [22]. This work establishes the color code as a compelling research direction to realize fault-tolerant quantum computation on superconducting processors in the near future.
Fault-tolerant logical operations are fundamental for scalable quantum computation. Here, we report the experimental realization of lattice-surgery operations between a pair of distance-three surface-code logical qubits on a planar superconducting processor. During repeated syndrome extraction cycles, the logical qubits exhibit per-cycle error rates of 0.0365(2)0.0365(2)0.0365(2) and 0.0282(1)0.0282(1)0.0282(1), respectively, after leakage events are rejected. By leveraging joint initialization and lattice splitting, we deterministically prepare a logical Bell state, confirming genuine bipartite entanglement via the error-corrected logical state fidelity. We further execute a two-qubit Deutsch-Jozsa algorithm at the logical level to demonstrate algorithmic utility in a fault-tolerant framework. Finally, to achieve universal control, we implement magic-state injection and gate teleportation to realize continuous non-Clifford rotations about the logical XXX axis. For the logical RX(π/4)R_{X}(π/4)RX​(π/4) gate, we achieve a logical gate fidelity of 0.943−9+100.943_{-9}^{+10}0.943−9+10​ conditioned on the absence of detected errors. These results establish lattice surgery as a practical and versatile paradigm for logical computation in near-term surface-code architectures, representing a critical milestone toward scalable fault-tolerant quantum advantage in superconducting circuits.
Quantum error correction enables scalable fault-tolerant quantum computation by suppressing logical errors exponentially with code distance once physical error rates fall below a critical threshold. Here we demonstrate below-threshold operation of triangular color codes with distances d=3,5,d=3,5,d=3,5, and 777 on Quantinuum's Helios trapped-ion quantum processor. Leveraging both circuit- and hardware- level leakage reduction schemes, we achieve logical error rates as low as 5−2+4×10−65_{-2}^{+4}\times 10^{-6}5−2+4​×10−6, and demonstrate error suppression by over an order of magnitude, Λ=12−2+3Λ=12_{-2}^{+3}Λ=12−2+3​, with each successive increase in code distance. Moreover, we demonstrate below threshold scaling of error rates for Clifford two-qubit randomized benchmarking circuits when encoded in d=3,5d=3,5d=3,5 color codes, achieving logical error rate of 7−3+3×10−57^{+3}_{-3}\times 10^{-5}7−3+3​×10−5 per logical two-qubit Clifford at distance 555 with a suppression factor of Λ=9−3+8Λ=9_{-3}^{+8}Λ=9−3+8​. We compare logical performance for existing near-optimal algorithmic decoders, and both convolutional neural network and transformer-based neural network decoders capable of real-time decoding an order of magnitude faster than the syndrome extraction rate. These results approach the logical error rates required for megaquop-scale quantum computation and indicate that scaled trapped-ion processors could reach application-relevant performance with substantially smaller code blocks than indicated by previous results in other qubit modalities.
Fault-tolerant quantum computation requires logical operations that manipulate encoded information while preserving quantum error-correction protection. In planar surface-code architectures, code deformation and lattice surgery provide a local, measurement-based route to such operations. Here we experimentally realize key elements of patch-based surface-code logical processing on a 107-qubit superconducting quantum processor. We first implement a reusable primitive layer comprising merge and split, patch expansion and shrinkage, and deformations mediated by domain walls and twist defects. We then compose these primitives to realize logical state routing, the logical controlled-NOT gate, and the single-qubit Hadamard and phase gates, which together form a Clifford-generating set. All operations are implemented on distance-three rotated surface-code patches with multi-round syndrome extraction and neural-network decoding, without post-selection. Our results advance superconducting surface-code experiments from protected logical memory to active, patch-based fault-tolerant logical operations.
Color codes are promising alternatives to surface codes for universal fault-tolerant quantum computing due to their simpler lattice-surgery protocols and the transversal implementation of logical Clifford gates. However, their practical deployment has been limited by slower decoding algorithms and worse logical failure rates and thresholds compared to surface codes. Although AI-based logical-flip decoders have recently been proposed to address these challenges, no clear framework currently exists for implementing such decoders within the parallel block-wise decoding schemes in both space and time required for large-scale fault-tolerant computation. AI-based pre-decoders offer a scalable alternative due to their local nature. By performing spacelike corrections on physical qubits and timelike corrections on stabilizer measurements, pre-decoders are naturally compatible with parallel block-wise decoding schemes and lattice-surgery protocols. In this work, we introduce AI-based pre-decoders for triangular color codes. We present a novel neural-network architecture for their implementation and develop methods to simplify the complex training data generated by color-code syndrome-extraction circuits containing feedforward operations. Remarkably, we find that both logical failure rates (LERs) and runtimes improve relative to raw Chromobius decoding as the code distance increases. For example, at code distance d=31 and physical error rate p=0.3%p=0.3\%p=0.3%, our pre-decoder + Chromobius pipeline improves the logical failure rate by a factor of 347x while reducing runtime by 7.33x compared to raw Chromobius decoding alone. These results demonstrate that AI-based pre-decoding can substantially narrow the performance gap between color codes and surface codes, bringing color codes closer to practical large-scale fault-tolerant quantum computation.
Current quantum processors are fragile, noisy and fairly limited in both quantity and quality with tens of qubits and physical error rates of around 10^-3. To realize practical quantum applications, however, error rates need to be below 10^-15 across millions of qubits. To bridge this gap and fully harness the potential of quantum computers, quantum error correction (QEC) is essential. QEC codes are designed to protect quantum information by redundantly encoding it onto multiple physical qubits. This encoding allows for the detection and correction of local errors affecting individual qubits, e.g., through stabilizer measurements. Importantly, if the physical error rates are below a specific threshold, QEC codes can exponentially suppress logical error rates by increasing the number of physical qubits involved. This is essential for achieving fault-tolerant computations, which are key to unlocking the full potential of quantum computers. The work presented in this thesis focuses on the implementation and optimization of small-scale QEC experiments using the surface code and flux-tunable superconducting qubits (Transmons). It addresses several key challenges: enhancing two-qubit gate fidelity in Surface-4 (Chapter 2), implementing an error-detection code with Surface-7 (Chapter 3), automating the calibration and benchmarking of the building blocks in Surface-17 (Chapter 4), reducing leakage into higher excited states with leakage reduction units (Chapter 5), assessing and enhancing the performance of logical qubits (Chapters 7 and 8).
Color codes are a promising topological code for fault-tolerant quantum computing. Insufficient research on color codes has delayed their practical application. In this work, we address several key issues to facilitate practical fault-tolerant quantum computing based on color codes. First, by introducing decoding graphs with error-rate-related weights, we obtain the threshold of 0.47 % of the 6.6.6 triangular color code under the standard circuit-level noise model, narrowing the gap to that of the surface code. Second, our work first investigates the circuit-level decoding of color code lattice surgery, then gives an efficient decoding algorithm, which is crucial to perform logical operations in a quantum computer with two-dimensional architectures. Last, a state injection protocol of the triangular color code is proposed, reducing the output magic state error rate in one round of 15 to 1 distillation by two orders of magnitude compared to a previous rough protocol. We also prove that our protocol offers the lowest logical error rates for state injection among all possible codes. Published by the American Physical Society 2024