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:2602.22211

Computing with many encoded logical qubits beyond break-even

Shival Dasu, Matthew DeCross, Andrew Y. Guo, Ali Lavasani, Jan Behrends, Asmae Benhemou, Yi-Hsiang Chen, Karl Mayer, Chris N. Self, Selwyn Simsek, Basudha Srivastava, M. S. Allman, Jake Arkinstall, Justin G. Bohnet, Nathaniel Q. Burdick, J. P. Campora, Alex Chernoguzov, Samuel F. Cooper, Robert D. Delaney, Joan M. Dreiling, Brian Estey, Caroline Figgatt, Cameron Foltz, John P. Gaebler, Alex Hall, Craig A. Holliman, Ali A. Husain, Akhil Isanaka, Colin J. Kennedy, Yuga Kodama, Nikhil Kotibhaskar, Nathan K. Lysne, Ivaylo S. Madjarov, Michael Mills, Alistair R. Milne, Brian Neyenhuis, Annie J. Park, Anthony Ransford, Adam P. Reed, Steven J. Sanders, Charles H. Baldwin, David Hayes, Ben Criger, Andrew C. Potter, David Amaro

quant-phSubmitted Feb 25, 2026• Announced Feb 26, 2026• 0 citations
arXivHTMLPDF

Abstract

High-rate quantum error correcting (QEC) codes encode many logical qubits in a given number of physical qubits, making them promising candidates for quantum computation. Implementing high-rate codes at a scale that both frustrates classical computing and improves performance by encoding requires both high fidelity gates and long-range qubit connectivity -- both of which are offered by trapped-ion quantum computers. Here, we demonstrate computations that outperform their unencoded counterparts in the high-rate $[[ k+2,\, k,\, 2 ]]$ iceberg quantum error detecting (QED) and $[[ (k_2 + 2)(k_1 + 2),\, k_2k_1,\, 4 ]]$ two-level concatenated iceberg QEC codes, using the 98-qubit Quantinuum Helios trapped-ion quantum processor. Utilizing new gadgets for encoded operations, we realize this "beyond break-even" performance with reasonable postselection rates across a range of fault-tolerant (FT) and partially-fault-tolerant (pFT) component and application benchmarks with between $48$ and $94$ logical qubits. These benchmarks include FT state preparation and measurement, QEC cycle benchmarking, logical gate benchmarking, GHZ state preparation, and a pFT quantum simulation of the three-dimensional $XY$ model of quantum magnetism. Additionally, we illustrate that postselection rates can be suppressed by increasing the code distance via concatenation. Our results represent state-of-the-art logical component and state fidelities and provide evidence that high-rate QED/QEC codes are viable on contemporary quantum computers for near-term beyond-classical-scale computation.

Related papers

Iceberg Beyond the Tip: Co-Compilation of a Quantum Error Detection Code and a Quantum Algorithm

Yuwei Jin, Zichang He, Tianyi Hao +5 more

The rapid progress in quantum hardware is expected to make them viable tools for the study of quantum algorithms in the near term. The timeline to useful algorithmic experimentation can be accelerated by techniques that use many noisy shots to produce an accurate estimate of the observable of interest. One such technique is to encode the quantum circuit using an error detection code and discard the samples for which an error has been detected. An underexplored property of error-detecting codes is the flexibility in the circuit encoding and fault-tolerant gadgets, which enables their co-optimization with the algorthmic circuit. However, standard circuit optimization tools cannot be used to exploit this flexibility as optimization must preserve the fault-tolerance of the gadget. In this work, we focus on the $[[k+2, k, 2]]$ Iceberg quantum error detection code, which is tailored to trapped-ion quantum processors. We design new flexible fault-tolerant gadgets for the Iceberg code, which we then co-optimize with the algorithmic circuit for the quantum approximate optimization algorithm (QAOA) using tree search. By co-optimizing the QAOA circuit and the Iceberg gadgets, we achieve an improvement in QAOA success probability from $44\%$ to $65\%$ and an increase in post-selection rate from $4\%$ to $33\%$ at 22 algorithmic qubits, utilizing 330 algorithmic two-qubit gates and 744 physical two-qubit gates on the Quantinuum H2-1 quantum computer, compared to the previous state-of-the-art hardware demonstration. Furthermore, we demonstrate better-than-unencoded performance for up to 34 algorithmic qubits, employing 510 algorithmic two-qubit gates and 1140 physical two-qubit gates.

arXiv:2504.21172•Apr 29, 2025•0 citations

Experimental validation of a compact fault-tolerant architecture for trapped ions

Noah Berthusen, Ali Lavasani, Asmae Benhemou +11 more

Quantum error correction (QEC) is beginning to enable logical operations that outperform their unencoded physical counterparts, but useful fault-tolerant computation will require more than low-error quantum memory. An effective architecture must orchestrate efficient logical encoding, low-overhead logical operations, and access to the non-Clifford resources required for universal computation. Here, we introduce and experimentally validate such an architecture based on the $[[20,2,6]]$ $C_4$-Helix code, designed for the early fault-tolerant regime. Using Quantinuum Helios, a 98-qubit trapped-ion quantum processor, we experimentally demonstrate the principal components of this architecture: we perform repeated quantum error correction with an error of $4.6^{+6.2}_{-2.6}\times10^{-5}$ per logical qubit per QEC cycle. We benchmark the complete Clifford group on the two logical qubits of a single codeblock under active error correction, obtaining an error of $2.8^{+1.0}_{-1.6}\times 10^{-4}$ per two-qubit logical Clifford. We further demonstrate a fault-tolerant chain-map interface between $C_4$-Helix and a distance-5 surface code, preparing a heterogeneous three-logical-qubit GHZ state with a fidelity lower bound of $99.925^{+0.068}_{-0.245}\%$. In each case, the encoded implementation outperforms its corresponding unencoded physical baseline without relying on postselection. Circuit-level simulations indicate that improvements in physical fidelity bring the same architecture into the $10^{-6}$-$10^{-8}$ logical-error regime targeted for early fault-tolerant computation. Together, these results establish $C_4$-Helix as a hardware-validated fault-tolerant architecture rather than a bare quantum memory.

quant-pharXiv:2609.03194•Sep 4, 2026•0 citations

Entangling Four Logical Qubits beyond Break-Even in a Nonlocal Code

Yifan Hong, Elijah Durso-Sabina, David Hayes +1 more

Quantum error correction protects logical quantum information against environmental decoherence by encoding logical qubits into entangled states of physical qubits. One of the most important near-term challenges in building a scalable quantum computer is to reach the break-even point, where logical quantum circuits on error-corrected qubits achieve higher fidelity than equivalent circuits on uncorrected physical qubits. Using Quantinuum's H2 trapped-ion quantum processor, we encode the Greenberger-Horne-Zeilinger (GHZ) state in four logical qubits with fidelity 99.5±0.15%≤F≤99.7±0.1% (after postselecting on over 98% of outcomes). Using the same quantum processor, we can prepare an uncorrected GHZ state on four physical qubits with fidelity 97.8±0.2%≤F≤98.7±0.2%. The logical qubits are encoded in a ⟦25,4,3⟧ Tanner-transformed long-range-enhanced surface code. Logical entangling gates are implemented using simple swap operations. Our results are a first step toward realizing fault-tolerant quantum computation with logical qubits encoded in geometrically nonlocal quantum low-density parity check codes.

arXiv:2406.02666•Physical Review Letters•Oct 29, 2024•29 citations

Fault-Tolerant Error Detection Above Break-Even for Multi-Qubit Gates

Colburn Riffel, Reece Robertson, Peter Hendrickson

A fully fault-tolerant implementation of the quantum error-detecting Iceberg $[[2m, 2m-2, 2]]$ code applied to a Toffoli circuit achieved beyond-break-even error detection on a leading trapped-ion quantum computer, where the effect of encoding a circuit with a quantum error-detection code enables increased fidelity compared to an unencoded circuit. This code was also applied to Bell state preparation circuits, where a lean non-fault-tolerant implementation of the Iceberg code enables a fidelity gain as well. This highlights the important point that, at least for small-scale circuits with a substantial portion of error-free runs, it can be effective simply to use error detection to filter out the runs with errors. Furthermore, experiments performed in this work highlight the necessity for judicious compilation of circuits not only for a given hardware but also within a quantum error detection code.

quant-pharXiv:2604.13219•Apr 16, 2026•0 citations

Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer

Nhung H. Nguyen, Muyuan Li, Alaina Green +5 more

Fault-tolerant quantum error correction (QEC) is crucial for unlocking the true power of quantum computers. QEC codes use multiple physical qubits to encode a logical qubit, which is protected against errors at the physical qubit level. Here, we use a trapped-ion system to experimentally prepare $m$-qubit Greenberger-Horne-Zeilinger states and sample the measurement results to construct $m\ifmmode\times\else\texttimes\fi{}m$ logical states of the $[[{m}^{2},1,m]]$ Shor code, up to $m=7$. The synthetic logical fidelity shows how deeper encoding can compensate for additional gate errors in state preparation for larger logical states. However, the optimal code size depends on the physical error rate and we find that $m=5$ has the best performance in our system. We further realize the direct logical encoding of the $[[9,1,3]]$ Shor code on nine qubits in a 13-ion chain for comparison, with $98.8(1)\mathrm{%}$ and $98.5(1)\mathrm{%}$ fidelity for state ${|\ifmmode\pm\else\textpm\fi{}⟩}_{L}$, respectively.

arXiv:2104.01205•Physical Review Applied•Aug 30, 2021•21 citations

Quantum error correction well below the color code threshold

Ali Lavasani, Adam P. Reed, Joan M. Dreiling +20 more

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 $d=3,5,$ and $7$ 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 $5_{-2}^{+4}\times 10^{-6}$, and demonstrate error suppression by over an order of magnitude, $Λ=12_{-2}^{+3}$, 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 $d=3,5$ color codes, achieving logical error rate of $7^{+3}_{-3}\times 10^{-5}$ per logical two-qubit Clifford at distance $5$ with a suppression factor of $Λ=9_{-3}^{+8}$. 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.

quant-pharXiv:2610.24077•Oct 20, 2026•0 citations
High-rate quantum error correcting (QEC) codes encode many logical qubits in a given number of physical qubits, making them promising candidates for quantum computation. Implementing high-rate codes at a scale that both frustrates classical computing and improves performance by encoding requires both high fidelity gates and long-range qubit connectivity -- both of which are offered by trapped-ion quantum computers. Here, we demonstrate computations that outperform their unencoded counterparts in the high-rate [[k+2, k, 2]][[ k+2,\, k,\, 2 ]][[k+2,k,2]] iceberg quantum error detecting (QED) and [[(k2+2)(k1+2), k2k1, 4]][[ (k_2 + 2)(k_1 + 2),\, k_2k_1,\, 4 ]][[(k2​+2)(k1​+2),k2​k1​,4]] two-level concatenated iceberg QEC codes, using the 98-qubit Quantinuum Helios trapped-ion quantum processor. Utilizing new gadgets for encoded operations, we realize this "beyond break-even" performance with reasonable postselection rates across a range of fault-tolerant (FT) and partially-fault-tolerant (pFT) component and application benchmarks with between 484848 and 949494 logical qubits. These benchmarks include FT state preparation and measurement, QEC cycle benchmarking, logical gate benchmarking, GHZ state preparation, and a pFT quantum simulation of the three-dimensional XYXYXY model of quantum magnetism. Additionally, we illustrate that postselection rates can be suppressed by increasing the code distance via concatenation. Our results represent state-of-the-art logical component and state fidelities and provide evidence that high-rate QED/QEC codes are viable on contemporary quantum computers for near-term beyond-classical-scale computation.
The rapid progress in quantum hardware is expected to make them viable tools for the study of quantum algorithms in the near term. The timeline to useful algorithmic experimentation can be accelerated by techniques that use many noisy shots to produce an accurate estimate of the observable of interest. One such technique is to encode the quantum circuit using an error detection code and discard the samples for which an error has been detected. An underexplored property of error-detecting codes is the flexibility in the circuit encoding and fault-tolerant gadgets, which enables their co-optimization with the algorthmic circuit. However, standard circuit optimization tools cannot be used to exploit this flexibility as optimization must preserve the fault-tolerance of the gadget. In this work, we focus on the [[k+2,k,2]][[k+2, k, 2]][[k+2,k,2]] Iceberg quantum error detection code, which is tailored to trapped-ion quantum processors. We design new flexible fault-tolerant gadgets for the Iceberg code, which we then co-optimize with the algorithmic circuit for the quantum approximate optimization algorithm (QAOA) using tree search. By co-optimizing the QAOA circuit and the Iceberg gadgets, we achieve an improvement in QAOA success probability from 44%44\%44% to 65%65\%65% and an increase in post-selection rate from 4%4\%4% to 33%33\%33% at 22 algorithmic qubits, utilizing 330 algorithmic two-qubit gates and 744 physical two-qubit gates on the Quantinuum H2-1 quantum computer, compared to the previous state-of-the-art hardware demonstration. Furthermore, we demonstrate better-than-unencoded performance for up to 34 algorithmic qubits, employing 510 algorithmic two-qubit gates and 1140 physical two-qubit gates.
Quantum error correction (QEC) is beginning to enable logical operations that outperform their unencoded physical counterparts, but useful fault-tolerant computation will require more than low-error quantum memory. An effective architecture must orchestrate efficient logical encoding, low-overhead logical operations, and access to the non-Clifford resources required for universal computation. Here, we introduce and experimentally validate such an architecture based on the [[20,2,6]][[20,2,6]][[20,2,6]] C4C_4C4​-Helix code, designed for the early fault-tolerant regime. Using Quantinuum Helios, a 98-qubit trapped-ion quantum processor, we experimentally demonstrate the principal components of this architecture: we perform repeated quantum error correction with an error of 4.6−2.6+6.2×10−54.6^{+6.2}_{-2.6}\times10^{-5}4.6−2.6+6.2​×10−5 per logical qubit per QEC cycle. We benchmark the complete Clifford group on the two logical qubits of a single codeblock under active error correction, obtaining an error of 2.8−1.6+1.0×10−42.8^{+1.0}_{-1.6}\times 10^{-4}2.8−1.6+1.0​×10−4 per two-qubit logical Clifford. We further demonstrate a fault-tolerant chain-map interface between C4C_4C4​-Helix and a distance-5 surface code, preparing a heterogeneous three-logical-qubit GHZ state with a fidelity lower bound of 99.925−0.245+0.068%99.925^{+0.068}_{-0.245}\%99.925−0.245+0.068​%. In each case, the encoded implementation outperforms its corresponding unencoded physical baseline without relying on postselection. Circuit-level simulations indicate that improvements in physical fidelity bring the same architecture into the 10−610^{-6}10−6-10−810^{-8}10−8 logical-error regime targeted for early fault-tolerant computation. Together, these results establish C4C_4C4​-Helix as a hardware-validated fault-tolerant architecture rather than a bare quantum memory.
Quantum error correction protects logical quantum information against environmental decoherence by encoding logical qubits into entangled states of physical qubits. One of the most important near-term challenges in building a scalable quantum computer is to reach the break-even point, where logical quantum circuits on error-corrected qubits achieve higher fidelity than equivalent circuits on uncorrected physical qubits. Using Quantinuum's H2 trapped-ion quantum processor, we encode the Greenberger-Horne-Zeilinger (GHZ) state in four logical qubits with fidelity 99.5±0.15%≤F≤99.7±0.1% (after postselecting on over 98% of outcomes). Using the same quantum processor, we can prepare an uncorrected GHZ state on four physical qubits with fidelity 97.8±0.2%≤F≤98.7±0.2%. The logical qubits are encoded in a ⟦25,4,3⟧ Tanner-transformed long-range-enhanced surface code. Logical entangling gates are implemented using simple swap operations. Our results are a first step toward realizing fault-tolerant quantum computation with logical qubits encoded in geometrically nonlocal quantum low-density parity check codes.
A fully fault-tolerant implementation of the quantum error-detecting Iceberg [[2m,2m−2,2]][[2m, 2m-2, 2]][[2m,2m−2,2]] code applied to a Toffoli circuit achieved beyond-break-even error detection on a leading trapped-ion quantum computer, where the effect of encoding a circuit with a quantum error-detection code enables increased fidelity compared to an unencoded circuit. This code was also applied to Bell state preparation circuits, where a lean non-fault-tolerant implementation of the Iceberg code enables a fidelity gain as well. This highlights the important point that, at least for small-scale circuits with a substantial portion of error-free runs, it can be effective simply to use error detection to filter out the runs with errors. Furthermore, experiments performed in this work highlight the necessity for judicious compilation of circuits not only for a given hardware but also within a quantum error detection code.
Fault-tolerant quantum error correction (QEC) is crucial for unlocking the true power of quantum computers. QEC codes use multiple physical qubits to encode a logical qubit, which is protected against errors at the physical qubit level. Here, we use a trapped-ion system to experimentally prepare mmm-qubit Greenberger-Horne-Zeilinger states and sample the measurement results to construct $m\ifmmode\times\else\texttimes\fi{}m$ logical states of the [[m2,1,m]][[{m}^{2},1,m]][[m2,1,m]] Shor code, up to m=7m=7m=7. The synthetic logical fidelity shows how deeper encoding can compensate for additional gate errors in state preparation for larger logical states. However, the optimal code size depends on the physical error rate and we find that m=5m=5m=5 has the best performance in our system. We further realize the direct logical encoding of the [[9,1,3]][[9,1,3]][[9,1,3]] Shor code on nine qubits in a 13-ion chain for comparison, with $98.8(1)\mathrm{%}$ and $98.5(1)\mathrm{%}$ fidelity for state ${|\ifmmode\pm\else\textpm\fi{}⟩}_{L}$, respectively.
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.