1. The short version, what to know first, and where this fits in the series

• The properties of molecules and materials are set by the behaviour of electrons (quantum mechanics). As electrons are added, the number of values needed for an exact calculation grows combinatorially, and about 22 electrons in 22 orbitals is considered the limit for classical computers. Since Feynman (1982) there has been the expectation that "quantum systems call for quantum computers" Sourced[Refs. 1, 17]
• The molecules tried on real devices are small, mostly of a size classical methods solve accurately. Even the largest example (77 qubits) works by having the quantum part supply configuration samples while a classical supercomputer does the rest. The estimated scale needed (about four million physical qubits for FeMoco) assumes a future error-corrected machine Sourced[Refs. 17, 21]
• Researchers disagree on whether quantum will beat classical. This article sets expectations beside the present state and gives no verdict. A small number of strongly correlated orbitals (the active space) is seen as the most promising target Sourced[Refs. 26, 23, 27]

2. Why compute molecules and materials on a quantum computer?

Whether a chemical reaction happens, whether a catalyst works well, whether a material becomes a magnet: such properties are set by the behaviour of electrons in atoms. The state of the electrons is governed by the Schrödinger equation. If one knows the lowest energy of a molecule (the ground-state energy), one can predict its stable structure and how readily it reacts.

The required accuracy is demanding. The review aims at energies within chemical accuracy (1.6×10⁻³ hartree), which lets one predict room-temperature reaction rates to within an order of magnitude Sourced[Ref. 4]. In other units, 1.6 mHa ≈ 1.0 kcal/mol ≈ 4.2 kJ/mol ≈ 0.044 eV Our calculation.

In 1982 Feynman argued that simulating quantum systems on classical computers is inherently hard and that it is natural to use a computer with quantum properties Sourced[Ref. 1]. Lloyd (1996) showed that a quantum computer can efficiently imitate quantum systems (universal quantum simulators) Sourced[Ref. 2], and Aspuru-Guzik et al. (2005) proposed computing molecular energies on a quantum computer Sourced[Ref. 3].

3. The quantum nature of electrons: why the numbers explode

Electrons do not move independently one by one; they move while avoiding each other. When one comes near, another moves away. This "relationship between electrons" is called electron correlation. If electrons were independent one could compute them one at a time and add up the results, but with correlation the states of all electrons must be handled at once.

The Hartree-Fock method follows the left picture (mean field) and neglects dynamic and static correlation. Static correlation arises when several electron arrangements are equally important, as in bond breaking and in systems with transition metals. Strongly correlated systems have wavefunctions with a high degree of entanglement, and catalysts and high-temperature superconductors are believed to be examples Sourced[Ref. 4].

(1) Counting every arrangement: the full-CI dimension

How many numbers does an exact calculation need? The number of ways to put n/2 spin-up and n/2 spin-down electrons into n orbitals (the "rooms" electrons occupy) is C(n, n/2)² (the dimension of full configuration interaction, FCI).

• 10 electrons in 10 orbitals is 6.4×10⁴ arrangements, instant on a laptop. 18 electrons in 18 orbitals gives about 2.4×10⁹ (about 19 GB), and 22 in 22 gives about 5.0×10¹¹ (about 4 TB) Our calculation
• Robledo-Moreno et al. state that the limit of full CI is about 22 electrons in 22 orbitals (22e,22o) and 26 electrons in 23 orbitals (26e,23o) Sourced[Ref. 17]. This agrees in order of magnitude with our calculation above
• For 50 electrons in 50 orbitals it is about 1.6×10²⁸, and for the FeMoco active space (54 electrons in 54 orbitals) about 3.8×10³⁰ Our calculation. Since each added orbital multiplies the number by roughly a hundred, faster computers barely move the limit of exact calculation

On a quantum computer each spin-orbital is assigned one qubit, with "0 = no electron in that orbital, 1 = an electron" (the basic Jordan-Wigner mapping Sourced[Ref. 4]). For n orbitals, 2n qubits can hold 2^(2n) states (in superposition). The number of states grows exponentially for a quantum computer too, but the number of qubits needed is only 2n. That is why quantum computers are thought to suit calculations of quantum systems.

4. Classical methods: what they cover and where they struggle

Before turning to quantum computers, let us sort out the classical methods. Classical methods are very powerful. Most molecular and materials calculations are still done classically, with no role for a quantum computer.

A summary of each method. Cost formulas are given only where we could confirm them (CCSD(T)); we did not confirm how the costs of the other methods are expressed.

5. How a calculation runs on a quantum computer

Putting a molecular calculation on a quantum computer takes several preparations. A classical computer first computes, from the molecular geometry and basis set, the coefficients (integrals) of the energy expression. Then only the strongly correlated orbitals are chosen (the active space), mapped to qubits, an initial state is prepared, and finally the energy is read out.

There are three broad ways to read out the energy (step 6). They differ in their dependence on error correction and in how reliable the result is.

The comparison of methods is this article's commentary (based on the statements in the references).

(1) Intuition for phase estimation: reading how fast the arrow turns

Phase estimation is easiest to grasp as reading the energy from how fast a clock hand turns. The arrow of a qubit turns over time at a speed proportional to the energy: the higher the energy, the faster the hand. Even two states with only slightly different energies can be told apart if run for long enough, because the angle between their hands grows.

6. Molecules tried on real devices: scale and accuracy, without exaggeration

Here is how far the reasoning above has been tested on real devices, in chronological order. To state the conclusion first: the molecules tried on real devices are small, and most are of a size classical methods solve accurately. This is not a disappointment; it is fair to read it as a step-by-step check of device quality.

A summary of what each paper states (checked on 16 October 2026). Scales are as stated in the papers; approaches, error rates and the classical share differ, so do not rank them against each other.

Opinions on SQD (the sample-based method) are divided. Robledo-Moreno et al. state that even at current error rates, a quantum-centric supercomputing architecture can tackle chemistry problems beyond the size amenable to exact diagonalization Sourced[Ref. 17]. Reinholdt et al., on the other hand, analyse N₂ and [2Fe-2S] numerically and conclude that sample-based methods (QSCI and SQD) waste effort by repeatedly sampling configurations already seen, and fall behind more efficient classical methods Sourced[Ref. 18]. We do not judge which is right. Whether the method becomes practical needs results with the conditions matched against the best classical methods.

7. Estimated scale needed: falling year by year, but still the future

How large a machine would a "practical molecule" need? Researchers have produced resource estimates (numbers of logical qubits and gates, and run time) assuming phase estimation on an error-corrected machine. The estimates have fallen year by year as algorithms improved. But an estimate is a calculation premised on a future machine, not the performance of any existing one.

A summary of what each paper states. Resource estimates depend strongly on assumptions (cycle time, error rate, code). Checked on 16 October 2026.

That the estimates are falling is itself encouraging. But the timing of realisation is not settled. When and by which approach large error-corrected machines will arrive is outside this article and is covered in our explainers on logical qubits and fault tolerance and on national strategies Not yet confirmed.

8. Comparison with classical methods: opinions are divided

Researchers' views differ on whether quantum computers will beat classical ones in chemistry. The main positions are set out side by side.

A summary of what each paper states. It does not decide among them.

9. Materials simulation: lattice models and how they differ from molecules

The properties of materials (solids) are set by the behaviour of electrons just as in molecules, but because atoms sit in a periodic arrangement, they are often studied with a simple model of "electrons on a lattice". The best-known is the Hubbard model: electrons occupy sites and are governed by just two quantities, how easily they hop to a neighbouring site (t) and the repulsion when two share a site (U). The contest between the two is believed to give rise to behaviour such as magnetism and superconductivity.

• Lattice models map onto qubits more easily than molecular electronic structure and are listed among promising candidates for near-term experiments (the 2D Fermi-Hubbard model, spin models and others) Sourced[Refs. 5, 6]. In 2020 Google observed charge and spin spreading at different speeds in a 1D Hubbard model, using 16 qubits and over 600 two-qubit gates Sourced[Ref. 28].
• In 2026 Quantinuum reported in Nature that it observed the time evolution of an Ising model out to times that severely challenge classical simulation methods and computed a diffusion constant (two-qubit partial-entangler fidelity 99.94(1)%) Sourced[Ref. 29] (a company report).
• Both are experiments on how a model behaves, not measurements of an actual material's properties. How well the model represents a real material is a separate question (our commentary).

10. A materials engineer's view

11. Limits, what we could not confirm, and a summary

(1) What we could not confirm

• We did not confirm the specific figures of the FeMoco estimate of Reiher et al. (2017); the abstract gives none.
• We do not give details such as qubit counts for Peruzzo et al. (2014) and Lanyon et al. (2010), as the abstracts do not state them.
• We did not confirm the size of the energy errors in the SQD experiments (the gap to the best classical results).
• We do not give cost formulas for DFT, DMRG and others beyond what we confirmed (about M⁷ for CCSD(T)).
• We found no third-party-verified case in which a real device computed a material's properties more accurately than the best classical methods.
• The text of ISO/IEC 4879:2024 is paywalled, so we could not confirm whether it covers quantum-chemistry terms. We did not find a standard defining the performance of quantum computers for chemistry and materials.

12. How to read evaluations and standards

The first yardstick of quantum chemistry is energy accuracy (chemical accuracy, 1.6 mHa). Next come which classical method it is compared with (full CI, CCSD(T), DMRG and so on) and the resources needed (logical qubits, gates, run time). The chemical-accuracy target exists so that energy differences can be tied to predictions of reaction rates Sourced[Ref. 4]. Checked on 16 October 2026.

The table's organisation is Our calculation (a summary). Numbers, titles and status were checked on the official ISO, NIST and DARPA pages (16 October 2026). Standards texts are paywalled, so we state only scope and no requirements.

Reading notes for materials engineers

• "How many qubits" does not tell you the molecule's size: the meaning of the count changes with the number of orbitals, the method and the classical share. The 77 qubits of the largest SQD run belong to an approach in which the quantum part is small [Ref. 17]
• Put error bars next to chemical accuracy: in the error-corrected H₂ example the error bar (±0.013 Ha) is larger than chemical accuracy (0.0016 Ha), so one cannot say the accuracy reached chemical accuracy [Refs. 19, 4]
• Separate plans from achievements: resource estimates, DARPA's 2033 and company roadmaps are goals and estimates. Many standards are still in development, and even once published they generally bind only when cited in law or contracts (our commentary) [Refs. 21, 34].

13. Glossary

14. References

1. Feynman R. P. "Simulating physics with computers", Int. J. Theor. Phys. 21, 467–488 (1982), DOI 10.1007/BF02650179 — doi.org/10.1007/BF02650179
2. Lloyd S. "Universal quantum simulators", Science 273, 1073–1078 (1996), DOI 10.1126/science.273.5278.1073 — doi.org/10.1126/science.273.5278.1073
3. Aspuru-Guzik A. et al. "Simulated quantum computation of molecular energies", Science 309, 1704–1707 (2005), DOI 10.1126/science.1113479 — doi.org/10.1126/science.1113479
4. McArdle S. et al. "Quantum computational chemistry", Rev. Mod. Phys. 92, 015003 (2020), DOI 10.1103/RevModPhys.92.015003 — doi.org/10.1103/RevModPhys.92.015003
5. Cao Y. et al. "Quantum chemistry in the age of quantum computing", Chem. Rev. 119, 10856–10915 (2019), DOI 10.1021/acs.chemrev.8b00803 — doi.org/10.1021/acs.chemrev.8b00803
6. Bauer B., Bravyi S., Motta M., Chan G. K.-L. "Quantum algorithms for quantum chemistry and quantum materials science", Chem. Rev. 120, 12685–12717 (2020), DOI 10.1021/acs.chemrev.9b00829 — doi.org/10.1021/acs.chemrev.9b00829
7. Hohenberg P., Kohn W. "Inhomogeneous electron gas", Phys. Rev. 136, B864–B871 (1964), DOI 10.1103/PhysRev.136.B864 — doi.org/10.1103/PhysRev.136.B864
8. Kohn W., Sham L. J. "Self-consistent equations including exchange and correlation effects", Phys. Rev. 140, A1133–A1138 (1965), DOI 10.1103/PhysRev.140.A1133 — doi.org/10.1103/PhysRev.140.A1133
9. Cohen A. J., Mori-Sánchez P., Yang W. "Challenges for density functional theory", Chem. Rev. 112, 289–320 (2012), DOI 10.1021/cr200107z — doi.org/10.1021/cr200107z
10. White S. R. "Density matrix formulation for quantum renormalization groups", Phys. Rev. Lett. 69, 2863–2866 (1992), DOI 10.1103/PhysRevLett.69.2863 — doi.org/10.1103/PhysRevLett.69.2863
11. Kitaev A. Yu. "Quantum measurements and the Abelian stabilizer problem", arXiv:quant-ph/9511026 (1995) — arxiv.org/abs/quant-ph/9511026
12. Abrams D. S., Lloyd S. "Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors", Phys. Rev. Lett. 83, 5162–5165 (1999), DOI 10.1103/PhysRevLett.83.5162 — doi.org/10.1103/PhysRevLett.83.5162
13. Lanyon B. P. et al. "Towards quantum chemistry on a quantum computer", Nat. Chem. 2, 106–111 (2010), DOI 10.1038/nchem.483 — doi.org/10.1038/nchem.483
14. Peruzzo A. et al. "A variational eigenvalue solver on a quantum processor", Nat. Commun. 5, 4213 (2014), DOI 10.1038/ncomms5213 (open access) — doi.org/10.1038/ncomms5213
15. Kandala A. et al. "Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets", Nature 549, 242–246 (2017), DOI 10.1038/nature23879 — doi.org/10.1038/nature23879
16. Google AI Quantum and collaborators "Hartree-Fock on a superconducting qubit quantum computer", Science 369, 1084–1089 (2020), DOI 10.1126/science.abb9811 — doi.org/10.1126/science.abb9811
17. Robledo-Moreno J. et al. "Chemistry beyond the scale of exact diagonalization on a quantum-centric supercomputer", Sci. Adv. 11, eadu9991 (2025), arXiv:2405.05068 — arxiv.org/abs/2405.05068
18. Reinholdt P. et al. "Critical limitations in quantum-selected configuration interaction methods", J. Chem. Theory Comput. 21, 6811–6822 (2025), arXiv:2501.07231 — arxiv.org/abs/2501.07231
19. Yamamoto K. et al. (Quantinuum) "Quantum error-corrected computation of molecular energies", PRX Quantum (2026), arXiv:2505.09133 — arxiv.org/abs/2505.09133
20. Reiher M. et al. "Elucidating reaction mechanisms on quantum computers", Proc. Natl. Acad. Sci. USA 114, 7555–7560 (2017), DOI 10.1073/pnas.1619152114 — doi.org/10.1073/pnas.1619152114
21. Lee J. et al. "Even more efficient quantum computations of chemistry through tensor hypercontraction", PRX Quantum 2, 030305 (2021), DOI 10.1103/PRXQuantum.2.030305 — arxiv.org/abs/2011.03494
22. von Burg V. et al. "Quantum computing enhanced computational catalysis", Phys. Rev. Research 3, 033055 (2021), DOI 10.1103/PhysRevResearch.3.033055 — doi.org/10.1103/PhysRevResearch.3.033055
23. Goings J. J. et al. "Reliably assessing the electronic structure of cytochrome P450 on today's classical computers and tomorrow's quantum computers", Proc. Natl. Acad. Sci. USA 119, e2203533119 (2022), DOI 10.1073/pnas.2203533119 — doi.org/10.1073/pnas.2203533119
24. Caesura A. et al. "Faster quantum chemistry simulations on a quantum computer with improved tensor factorization and active volume compilation", PRX Quantum 6, 030337 (2025), arXiv:2501.06165 — arxiv.org/abs/2501.06165
25. Low G. H. et al. "Fast quantum simulation of electronic structure by spectrum amplification", Phys. Rev. X 15, 041016 (2025), arXiv:2502.15882 — arxiv.org/abs/2502.15882
26. Lee S. et al. "Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry", Nat. Commun. 14, 1952 (2023), DOI 10.1038/s41467-023-37587-6 (open access) — doi.org/10.1038/s41467-023-37587-6
27. Hoefler T., Häner T., Troyer M. "Disentangling hype from practicality: on realistically achieving quantum advantage", Commun. ACM 66(5), 82–87 (2023), DOI 10.1145/3571725 — doi.org/10.1145/3571725
28. Google AI Quantum and collaborators "Observation of separated dynamics of charge and spin in the Fermi-Hubbard model", arXiv:2010.07965 (2020) — arxiv.org/abs/2010.07965
29. Haghshenas R. et al. (Quantinuum) "Digital quantum magnetism on a trapped-ion quantum computer", Nature 653, 56–62 (2026), DOI 10.1038/s41586-026-10445-3 — nature.com/articles/s41586-026-10445-3
30. Google Quantum AI "A verifiable quantum advantage" (Google Research blog, 22 October 2025) — research.google/blog/a-verifiable-quantum-advantage/
31. NIST "Computational Chemistry Comparison and Benchmark DataBase (CCCBDB)", NIST Standard Reference Database 101 — cccbdb.nist.gov
32. Sawaya N. P. D. et al. "HamLib: a library of Hamiltonians for benchmarking quantum algorithms and hardware", Quantum 8, 1559 (2024), DOI 10.22331/q-2024-12-11-1559 — doi.org/10.22331/q-2024-12-11-1559
33. van Dam W., Mykhailova M., Soeken M. "Using Azure Quantum Resource Estimator for assessing performance of fault tolerant quantum computation", arXiv:2311.05801 (2023) — arxiv.org/abs/2311.05801
34. DARPA "Quantum Benchmarking Initiative (QBI)" (program page; list current as of 8 October 2026) — darpa.mil/research/programs/quantum-benchmarking-initiative
35. IEC/ISO JTC 3/JSC 3A "Quantum computing" (committee page: scope and working groups, including WG 2 benchmarking) — iso.org/committee/12010875.html
36. ISO/IEC "ISO/IEC 4879:2024 Information technology — Quantum computing — Vocabulary" (ISO catalogue page) — iso.org/standard/80432.html

15. Claim-to-source audit

Last updated: 16 October 2026. Sources are limited to primary material (peer-reviewed papers, arXiv papers, Google's official blog, official NIST, DARPA and ISO pages). Numerical examples are our own calculations, not device results. We could not confirm the specific figures of the estimate by Reiher et al., the size of the SQD energy errors, cost formulas for DFT and DMRG, third-party-verified quantum advantage on material properties, or quantum-chemistry terms in ISO/IEC 4879, so we do not state them. All figures are illustrative concept diagrams or drawings that include our calculations; Fig. 1 and Fig. 10 are ImageGen-generated conceptual illustrations.