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

• NISQ stands for Noisy Intermediate-Scale Quantum. It describes quantum computers with tens to hundreds of qubits that do not yet have error correction. Preskill proposed the term in 2018 Sourced[Refs. 1, 2]
• Because errors build up, long circuits fall apart. So the quantum computer runs only short circuits and a classical computer tunes the knobs. The best-known examples are VQE (energies of molecules) and QAOA (combinatorial optimization) Sourced[Ref. 3]
• There are three walls: noise, barren plateaus (the learning signal vanishes) and the number of measurements. Error suppression and mitigation get past some of them, but theory shows the cost rises exponentially as circuits get deeper Sourced[Refs. 15, 16]

2. Why errors matter: small slips in the arrow add up

The arrow of a qubit is fragile. When a gate rotates it, the arrow may turn slightly too far or not far enough. Heat and electromagnetic noise from the surroundings can make the arrow wobble (this is called decoherence). Measurement can also be misread. One slip is small, but after hundreds of gates the slips add up.

Let us estimate how fast this happens with the simplest possible formula. Suppose each gate fails with probability ε. Then the probability that all N gates succeed is (1−ε)^N.

• Even with a gate error of 0.1% (one in a thousand), stacking 1,000 gates leaves only about a 37% chance that every gate works (0.999^1000 ≈ 0.368) Our calculation
• With a 1% error, 1,000 gates leave about 0.004% (0.99^1000 ≈ 4×10^-5). Cutting the error by ten lets circuits be about ten times longer (N for 50% success: about 69, 693, 6,931) Our calculation
• A real example: one superconducting device reports a two-qubit gate fidelity of 99.62% (Zuchongzhi 3.0, as stated in the abstract) Sourced[Ref. 23]. The error is 0.38%, so success drops to 50% after about 182 gates (ln 0.5 ÷ ln 0.9962 ≈ 182) Our calculation. See the paper for how the figure was measured (average or best, over how many pairs).

3. What NISQ means: noisy and intermediate-scale

Preskill proposed the term NISQ in a 2018 paper. It says quantum computers with 50 to 100 qubits may be able to perform tasks beyond classical computers, while noise in the gates limits the size of the circuits that can be run Sourced[Ref. 1]. A later review describes NISQ machines as made of hundreds of noisy qubits that are not error-corrected and can perform imperfect operations only within a limited coherence time Sourced[Ref. 2].

A summary of the definitions in Preskill and in the review (Refs. 1, 2). The boundary of "intermediate scale" shifts with time (our commentary).

A note on example 2 in Fig. 3. Google reported that in two surface-code memories (distance 7 and distance 5), each increase of the code distance by 2 reduced the logical error rate by a factor of Λ = 2.14 ± 0.02, and that the 101-qubit distance-7 code had 0.143% error per cycle Sourced[Ref. 22]. It shows errors being corrected as the code grows, and marks the doorstep of work beyond the NISQ frame. Our explainer on the surface code covers it in detail.

4. Hybrid computing: how quantum and classical machines share the work

How can a quantum computer that can only run short circuits do useful work? The idea is to give the quantum computer only what it does best and let the classical computer do the rest. Methods of this shape are called variational quantum algorithms (VQAs) Sourced[Ref. 3].

Picture turning the tuning knob of a radio to find the clearest station. The person turning the knob is the classical computer; the radio itself is the quantum computer. Set the knob (θ), listen (the measurement result), turn it a little, and repeat.

The division of roles is this article's commentary on the general structure of VQAs (Ref. 3).

5. VQE: searching for the lowest energy of a molecule

VQE (Variational Quantum Eigensolver) searches for the lowest energy of a system (the ground-state energy). The properties of molecules and materials are set by electron energies, so VQE is an entry point to chemistry and materials calculations. In 2014, Peruzzo et al. reported computing the ground-state energy of HeH⁺ to within chemical accuracy (about 1.6 millihartree) on a small photonic quantum processor Sourced[Ref. 4].

(1) The variational principle: no arrow can give an energy below the true bottom

VQE rests on the variational principle. In quantum mechanics, whatever state you prepare, the average energy you measure is at or above the true lowest energy. So if you adjust the knob θ to lower the energy step by step, you approach the true minimum. Since you cannot punch through the floor from above, simply keep going down.

Let us run the smallest possible example. Take one qubit and a simple energy (H = Z): add +1 if you measure 0 and −1 if you measure 1. If the arrow is tilted by an angle θ from the north pole, the probability of 0 is cos²(θ/2), so the average energy is E(θ) = cos θ. It is +1 at θ = 0° (north pole), 0 at θ = 90°, and −1 at θ = 180° (south pole). The minimum is at θ = 180°.

In practice, measuring near θ = 90° with 100 shots gives an average that scatters by about ±0.10 (standard error = √((1−cos²θ)/100) = 0.10) Our calculation. The classical computer watches these noisy values, judges that "a larger θ seems to lower the energy", and moves the knob.

(2) What grows for a real molecule

For a molecule, the number of qubits grows in proportion to the number of orbitals, and the energy expression (the Hamiltonian) becomes a sum of many terms (Pauli terms). A review states that for Gaussian molecular orbital bases the number of terms grows roughly as the fourth power of the number of orbitals (O(M⁴)) Sourced[Ref. 7]. Each term needs its own measurements, so the number of measurements becomes a heavy burden (chapter 7).

Summary based on the abstracts of Peruzzo et al. (Ref. 4) and Kandala et al. (Ref. 5). Scales are as stated in the papers; comparison with classical computation was not their aim.

6. QAOA: attacking combinatorial optimization with a quantum circuit

QAOA (Quantum Approximate Optimization Algorithm) was proposed by Farhi et al. in 2014 for problems that ask for the best choice among many combinations (combinatorial optimization) Sourced[Ref. 6]. Take the Max-Cut problem as an example: colour the points in two colours so that the number of links joining points of different colours is as large as possible.

With only five points, checking all 2⁵ = 32 colourings reveals the answer (5 links) Our calculation. With 100 points there are 2^100 ≈ 1.3×10^30 colourings, far too many to check. QAOA uses a quantum circuit to make 0/1 strings in which good colourings are more likely to appear, then measures them. The knobs (γ and β) are tuned by a classical computer.

The paper by Farhi et al. showed that for p = 1 on 3-regular graphs (every point has three links), a cut at least 0.6924 times the optimum is always found Sourced[Ref. 6]. This is a lower-bound guarantee ("at worst this good"); it does not mean the method is better than classical algorithms (our commentary). QAOA is a gate-model method whose idea is close to quantum annealing (see our explainer on quantum annealing).

7. Why it is hard: noise, barren plateaus and the number of measurements

(1) Noise: deeper circuits drift toward "nothing in it"

With noise, the deeper the circuit, the more its output drifts toward random results that carry no information. Stilck França and García-Patrón built a framework for comparing classical and quantum algorithms on noisy devices and found that substantial quantum advantages are unlikely for classical optimization unless noise rates fall by orders of magnitude or the problem's topology matches the device Sourced[Ref. 12]. Wang et al. proved that noise itself creates barren plateaus: under local Pauli noise, if the circuit depth grows linearly with the number of qubits n, the gradient vanishes exponentially in n Sourced[Ref. 10].

(2) Barren plateaus: the hint about which way to go disappears

When tuning θ, the classical computer decides which way to move from the slope (gradient) of the energy. But when a circuit is close to random, the landscape can be a wide plain with almost no slope. McClean et al. showed that for a wide class of circuits the probability that the gradient is non-zero shrinks exponentially with the number of qubits, and named this the barren plateau Sourced[Ref. 8]. It is like searching for a valley bottom in fog when the ground is flat everywhere and you cannot tell which way is down.

However, it does not happen for every circuit. Cerezo et al. showed that a cost function that looks at all qubits causes barren plateaus even in shallow circuits, whereas one that looks at only a few qubits gives at worst a polynomially vanishing gradient for depths up to O(log n) Sourced[Ref. 9]. The same group later argued, with case-by-case evidence, that circuits structured to avoid barren plateaus can often also be handled efficiently by classical computers Sourced[Ref. 11]. So there is a tension: the easier a circuit is to train, the less a quantum computer may be needed for it.

(3) Number of measurements: ten times the precision needs a hundred times the shots

The statistical error in an expectation value falls in proportion to the inverse square root of the number of shots M (standard error = σ ÷ √M, where σ is the scatter of single outcomes). Google's experimental paper likewise states that the uncertainty of XEB is 1/√(number of samples) Sourced[Ref. 21]. For molecules there are many terms, each measured separately, so the required number of shots balloons.

M = (σ ÷ precision)², from standard error = σ ÷ √M. Our calculation. σ = 1 is a rough worst case; the real σ depends on the problem.

As a result, one VQE run amounts to "thousands to millions of shots × tens to thousands of terms × the number of knob adjustments". This takes up a large part of the machine time and cost (our commentary).

8. Error suppression, mitigation and correction: three different remedies

Remedies for errors fall into three groups by when and what they do. Error suppression makes errors less likely while the circuit runs. For example, dynamical decoupling applies short pulses while a qubit waits, cancelling the effect of fluctuations from the surroundings. Ezzell et al. compared 60 pulse sequences on three IBM devices and describe dynamical decoupling as the simplest and least resource-intensive error-suppression strategy Sourced[Ref. 17].

Error mitigation does not fix errors; it corrects the averages of many measurements. A review positions it as the way to soften, rather than completely remove, errors on NISQ-era machines Sourced[Ref. 14]. There are two representative methods Sourced[Ref. 13].

• Zero-noise extrapolation (ZNE): deliberately raise the noise (1x, 2x, 3x), measure, and extrapolate to the value at "zero noise". See Fig. 9 for an example.
• Probabilistic error cancellation (PEC): mix in operations that undo the noise at random, so that on average the noise cancels. The needed number of shots grows rapidly with the number of gates.

In ZNE the answer depends on how you extrapolate, and the scatter in the measurements is amplified by the extrapolation (Fig. 9). Kim et al. explain that PEC was too costly at their circuit volumes, so they used the cheaper ZNE, which can leave a bias Sourced[Ref. 18].

The limits of error mitigation have been shown theoretically as well. Takagi et al. showed, for all mitigation protocols in a general class, that for layered circuits with local depolarizing noise the number of samples needed to keep a given accuracy grows exponentially with circuit depth Sourced[Ref. 15]. Quek et al. showed that even at shallow depths comparable to current experiments, a superpolynomial number of samples is needed in the worst case, and that scrambling by noise can set in at exponentially smaller depths than previously thought Sourced[Ref. 16].

9. A real experiment: IBM's 127-qubit run and what followed

Kim et al. ran circuits of up to 60 layers of two-qubit gates (2,880 CNOT gates in total) on a 127-qubit superconducting processor (IBM Eagle) and measured expectation values using error mitigation (ZNE) Sourced[Ref. 18]. They first checked correctness on circuits that classical computers can verify exactly, then moved to circuits beyond brute-force classical simulation. The paper presents the result as evidence for the utility of quantum computing before fault tolerance Sourced[Ref. 18] (a company (IBM) claim).

Soon after the paper appeared, classical researchers showed efficient ways to compute the same problem. Tindall et al. reported a classical simulation, using a tensor network matched to the lattice, that is more accurate and precise than the quantum processor's results Sourced[Ref. 19]. Begušić et al. likewise reported that a method called sparse Pauli dynamics reproduces the results far faster than the reported quantum runtime, even on one laptop core Sourced[Ref. 20]. How to read this dispute (who claimed advantage, on which task, against what) is the subject of our explainer on reading claims of quantum advantage.

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

(1) What we could not confirm

• The text of ISO/IEC 4879:2024 (quantum computing vocabulary) is paywalled, so we could not confirm whether it defines NISQ, variational algorithms or related terms. We confirmed only the number, title, publication month and scope statement on the ISO page.
• IEEE P3185 (hybrid quantum-classical computing), P7130 (definitions) and P7131 (performance metrics and benchmarking) appear on IEEE SA pages as projects in development; we could not confirm them as published standards.
• We could not find a third-party-verified case of VQE or QAOA showing a practical advantage.
• We do not cover company roadmaps (qubit counts, logical qubits, dates).

11. How to read evaluations and standards

The performance of NISQ and hybrid computing is measured with several yardsticks. The error rate (the size of the error per gate, measured by randomized benchmarking) measures the quality of components. Quantum volume folds the number of qubits, errors and connectivity into one number for the quality of the whole machine. Application-oriented benchmarks measure how well real algorithms run. Each measures something different and leaves something else unmeasured. Checked on 16 October 2026.

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

Reading notes for materials engineers

• "How many qubits" alone does not tell you performance: read a NISQ device together with its error rate, circuit length and connectivity. Neither quantum volume nor application-oriented benchmarks summarise everything in one number [Refs. 26, 27, 29]
• Company announcements and third-party checks are different: IBM's 127-qubit experiment was a company claim, followed by reports of classical reproduction [Refs. 18, 19, 20]. The dispute can increasingly be followed in public records such as the quantum advantage tracker [Ref. 34].
• Standards are mostly in development or voluntary: many standards in this field (IEEE P3185, P7130, P7131 and others) are still being drafted, and even once published they generally bind only when cited in law or contracts (our commentary). Distinguish plans (in development) from achievements (published).

12. Glossary

13. References

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19. Tindall J., Fishman M., Stoudenmire E. M., Sels D. "Efficient tensor network simulation of IBM's Eagle kicked Ising experiment", PRX Quantum 5, 010308 (2024), DOI 10.1103/PRXQuantum.5.010308 — doi.org/10.1103/PRXQuantum.5.010308
20. Begušić T., Gray J., Chan G. K.-L. "Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance", Sci. Adv. 10, eadk4321 (2024), DOI 10.1126/sciadv.adk4321 — doi.org/10.1126/sciadv.adk4321
21. Arute F. et al. "Quantum supremacy using a programmable superconducting processor", Nature 574, 505–510 (2019), DOI 10.1038/s41586-019-1666-5 (open access) — nature.com/articles/s41586-019-1666-5
22. Google Quantum AI "Quantum error correction below the surface code threshold", Nature 638, 920–926 (2025), DOI 10.1038/s41586-024-08449-y — nature.com/articles/s41586-024-08449-y
23. Gao D. et al. "Establishing a new benchmark in quantum computational advantage with 105-qubit Zuchongzhi 3.0 processor", Phys. Rev. Lett. 134, 090601 (2025), arXiv:2412.11924 — arxiv.org/abs/2412.11924
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27. Baldwin C. H. et al. "Re-examining the quantum volume test: Ideal distributions, compiler optimizations, confidence intervals, and scalable resource estimations", Quantum 6, 707 (2022), DOI 10.22331/q-2022-05-09-707 — doi.org/10.22331/q-2022-05-09-707
28. Wack A. et al. "Quality, Speed, and Scale: three key attributes to measure the performance of near-term quantum computers", arXiv:2110.14108 (2021) — arxiv.org/abs/2110.14108
29. Lubinski T. et al. "Application-oriented performance benchmarks for quantum computing", IEEE Trans. Quantum Eng. 4, 1–32 (2023), DOI 10.1109/TQE.2023.3253761 — doi.org/10.1109/TQE.2023.3253761
30. ISO/IEC "ISO/IEC 4879:2024 Information technology — Quantum computing — Vocabulary" (ISO catalogue page) — iso.org/standard/80432.html
31. IEC/ISO JTC 3/JSC 3A "Quantum computing" (committee page: scope and working groups WG 1 supply chain, WG 2 benchmarking, WG 3 terminology and quantities) — iso.org/committee/12010875.html
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34. Gambetta J., Davis R. (IBM) "Quantum advantage tracker" (IBM Quantum blog, 23 February 2026) — ibm.com/quantum/blog/quantum-advantage-tracker

14. Claim-to-source audit

Last updated: 16 October 2026. Sources are limited to primary material (peer-reviewed papers, arXiv papers, official ISO and IEEE pages, official company blogs). Numerical examples are our own calculations, not device results. We could not confirm the text of ISO/IEC 4879, the publication status of IEEE P3185, P7130 and P7131, third-party-verified practical advantage of VQE or QAOA, or company roadmaps, 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.