Quantinuum and SoftBank Corp. published a joint white paper on July 21, 2026, in which 20 authors matched applications in quantum chemistry and topological data analysis to four generations of Quantinuum’s trapped-ion computers. The paper, “Quantum Computing Frontiers,” has 16 authors from Quantinuum and four from SoftBank.
The authors placed chemically accurate simulation of more than 100 spin orbitals on Lumos, the large-scale fault-tolerant generation they anticipate in the 2030s. For Helios, the current machine, and for Sol, anticipated for 2027, they listed about 10 spin orbitals in error-corrected circuits.
Chemists measure the size of a molecular simulation by its number of spin orbitals, the states that the molecule’s electrons can occupy. Chemical accuracy is an energy error of about 1.6 millihartree, or 1 kilocalorie per mole, a conventional target for predicting how molecules react. Trapped-ion computers use electrically charged atoms, held in place by electromagnetic fields, as qubits.
Four hardware generations
The authors described Helios as the “current generation” and gave anticipated dates of 2027 for Sol, 2029 for Apollo, and the 2030s for Lumos. For error-corrected chemistry circuits, they listed these figures.
| Generation | Year in the paper | Spin orbitals | Logical error rate | Code distance |
|---|---|---|---|---|
| Helios | Current (2025) | About 10 | About 1 in 1,000 | 3–5 (Steane code) |
| Sol | 2027 | About 10 | About 1 in 10,000 | 5–7 |
| Apollo | 2029 | About 100 | About 1 in 10 million | 7–9 |
| Lumos | 2030+ | More than 100, at chemical accuracy | 1 in 10 billion or lower | Not given |
A logical error rate is the chance that an error-corrected qubit fails during an operation.
Code distance is the smallest number of physical errors that can change the encoded information undetected. Larger distances require more physical qubits for each logical qubit. The Steane code uses seven physical qubits to protect one logical qubit.
For circuits on physical qubits or with quantum error detection, the authors listed about 50 spin orbitals on Helios and about 100 on Sol and Apollo. Quantum error detection flags errors and discards the runs that contain them.
For topological data analysis, the authors expected graph sizes to grow from fewer than 90 nodes on Helios to 100 on Sol, 100 to 300 on Apollo, and hundreds to thousands on Lumos.
The chemistry experiment
The team ran quantum phase difference estimation, a variant of the phase estimation algorithm, on ethylene at a geometry where two of its electronic states meet. They used a minimal model with two electrons in two orbitals and encoded the qubits in the Steane code.
The authors listed Helios-1, Helios-1E, and H2-2 for the encoded circuits and H2-2E for “the logical circuits.” The E suffix marks Quantinuum’s emulators, classical simulations of its machines, so the Helios-1E and H2-2E results did not run on quantum hardware. The authors did not describe the two targets as emulators.
From 1,400 circuit samples on Quantinuum’s H2-2 system, the team estimated an energy difference of −0.0025561 hartree, the atomic unit of energy. The authors wrote that the estimate agrees with their approximate energy difference of −0.002559 hartree.
In a separate comparison, the authors wrote that “the physical implementation outperformed the logical one” and that “break-even with respect to the physical baseline was not reached” for the encoded circuits. Break-even is the point at which an error-corrected circuit gives better results than the same calculation done directly on physical qubits.
The authors attributed the shortfall mainly to memory errors that accumulate while the ions are idle during encoding, decoding, and error correction. “Revisiting this experiment on next-generation hardware will serve as a natural benchmark for progress toward algorithm-level fault tolerance,” they wrote.
Topological data analysis
Topological data analysis describes the shape of data, such as loops and voids in a network, through counts called Betti numbers. The team’s quantum algorithm estimates normalized Laplacian moments, quantities that approach normalized Betti numbers as the depth of the calculation grows.
The team validated the algorithm on a noisy emulator of Helios, not on the machine itself, using a 20-vertex graph and 1,000 runs. The exact answers for such graphs are known, so running the algorithm on them “would not be a demonstration of quantum advantage,” the authors wrote.
In an application study on BUPT, a benchmark dataset for detecting telecommunications fraud, the team computed the Betti numbers and Laplacian moments exactly with classical methods because the graph “is relatively sparse, with low edge density,” the authors wrote.
When the team added the moments to a graph neural network, a machine-learning model for network data, macro recall rose from 0.8317 to 0.8449 across 10 random seeds, with a p-value of 0.026. Macro recall averages the share of cases a model identifies correctly in each class, fraudulent or legitimate.
At a fixed precision of 0.9, the authors converted that gain into about $220 million a year in avoided fraud losses. They assumed a $10 billion pool of losses, about a quarter of the Communications Fraud Control Association’s $38.95 billion estimate for 2023.
How the companies describe the timeline
“[T]his roadmap is framed as an assumption-driven envelope rather than a deterministic forecast,” the authors wrote. They tied each generation’s capacity to projected qubit counts and physical error rates, as well as to error-correction overhead. “Uncertainty naturally increases toward later milestones,” they wrote.
In their announcement, the companies wrote that the analysis “does not represent financial guidance or forecasts” and called the scenarios illustrative. Duncan Jones, general manager of Quantinuum’s applications group, said organizations “do not need to wait for large-scale, fault-tolerant systems.” He said companies can use today’s machines to develop and benchmark applications.
Ryuji Wakikawa, SoftBank’s chief technology officer, said the open question is which problem classes become workable at each stage of hardware maturity. Sol, anticipated for 2027, is the next generation on the paper’s timeline.
Update, November 12, 2026: Quantinuum said in its November 5 third-quarter results that Sol remains on track for the second half of 2027 and Apollo for 2029, without mentioning Lumos.
