TECHNOLOGY EXPLAINER
Industrial Applications: What Is Realistic
What is expected, what is still unknown, and the difference between a PoC and production
"Quantum computers will optimise logistics", "finance risk calculations in an instant", "smarter AI". How should such talk be taken? This article checks, with small numerical examples, the kind of theoretical speedup, the input/output wall and the slowness of error correction for applications such as optimization, finance, machine learning and linear equations (the base of fluid and structural calculations). Quantum machine learning is split into "what is known and what is not", and the article ends with the difference between a proof of concept (PoC) and production and questions for reading requests and announcements. The aim is to exaggerate neither hope nor disappointment.
- The short version, what to know first, and where this fits in the series
- Four questions for judging an application
- Kinds of speedup and the weight of error correction: checking by calculation
- Optimization: high hopes, weak guarantees of speed
- Finance: a quadratic speedup and the estimated scale needed
- Linear and differential equations: reading the fine print
- Quantum machine learning: what is known and what is not
- Telling a PoC from production: questions for reading requests and announcements
- A materials engineer's view
- Limits, what we could not confirm, and a summary
- How to read evaluations and standards
- Glossary
- References
- Claim-to-source audit
Sourced = stated in a published source (linked)
Our calculation = a value computed in this article from the stated assumptions
Not yet confirmed = plans, targets, drafts, work in progress, outlooks: not settled
Our own reading for materials engineers is marked as commentary. Inference is shown in bold. Company announcements are labelled as company claims.
1. The short version, what to know first, and where this fits in the series
- Practicality of quantum applications depends heavily on the kind of speedup. Exponential speedups are promising but come with strict conditions, while quadratic (square-root) speedups are easily eaten by the slowness of error correction, according to the analyses Sourced[Refs. 2, 3]
- Optimization, finance, machine learning and linear equations each have expected uses and open questions. In quantum machine learning there are cases reproduced classically (recommendation systems) and small-scale comparisons in which classical methods came out ahead Sourced[Refs. 18, 20]
- A PoC that works does not mean production will deliver value. Use the question list in this article (chapter 8) to read requests and announcements. DARPA uses "computational value exceeds cost" as its standard of utility scale Sourced[Ref. 22]
We use the picture of a qubit state as an arrow (north pole = 0, south pole = 1, in between = superposition) that settles on 0 or 1 when measured (see "Quantum Computing Fundamentals"). This article is readable on its own; it explains what is needed.
Two terms first. A classical computer is an ordinary computer, supercomputers included. A heuristic is a method that often works but carries no guarantee of speed or correctness.
A good order is Quantum Algorithms → NISQ and Hybrid Computing → Reading Claims of Quantum Advantage → this article. Chemistry and materials are in the quantum chemistry explainer, cryptography in the post-quantum cryptography explainer, sensing in the quantum sensing explainer and national strategies in the strategies explainer, so this article centres on optimization, finance, machine learning and linear equations.
2. Four questions for judging an application
When you hear a claim about a quantum application, check the following four points in order, and the size of the real expectation becomes clearer.
The survey by Dalzell et al. says anticipated applications of quantum computers span science and industry (quantum chemistry, many-body physics, optimization, finance, machine learning and more), and cautions that each proposal combines several primitives and must incorporate error correction and fault tolerance, so it is hard to assess how much an application benefits unless it is evaluated by its end-to-end complexity Sourced[Ref. 4]. So the starting point is to count not only the theoretical speed of an algorithm but all the effort needed to run it on a real machine.
3. Kinds of speedup and the weight of error correction: checking by calculation
There are kinds of "quantum speedup". To search N candidates, a classical computer needs time proportional to N, whereas Grover's search needs about √N operations (a quadratic speedup) Sourced[Ref. 1]. For N = one million, a million steps become 1,000, which looks like a big gap.
But on an error-corrected quantum computer, one logical operation is far slower than one classical step. Babbush et al. computed run times including error correction using surface-code constructions and concluded that quadratic speedups will not give quantum advantage on early fault-tolerant devices unless error correction improves greatly, while quartic speedups look more practical Sourced[Ref. 2]. The next figure checks this logic with made-up numbers.
| Quantum step time (r) | Break-even N* against one core (N* = r²) | Time at break-even | N* against 1,000 classical cores |
|---|---|---|---|
| 10³ ns (1 µs) | 10⁶ | 1 ms | 10¹² |
| 10⁶ ns (1 ms) | 10¹² | 1,000 s (about 17 min) | 10¹⁸ (time about 278 h) |
| 10⁹ ns (1 s) | 10¹⁸ | 10⁹ s (about 32 years) | 10²⁴ |
Our calculation. Classical assumed 1 ns per candidate, quantum √N × r ns (quadratic speedup). At break-even N* we have √N × r = N × 1 ns. With K classical cores in parallel, N* = (K × r)² (if the quantum side cannot be parallelized). Illustrative values, not real machine speeds.
- If a quantum step is a million times slower than a classical one (r = 10⁶), the classical machine is faster for any problem with N below 10¹² (a trillion). Even at N = 10¹² the break-even time is about 17 minutes Our calculation
- If the classical side uses 1,000 cores in parallel, break-even moves to N = 10¹⁸ with a time of about 278 hours (about 12 days) Our calculation. Even if "quantum wins for large N", the region where it wins is pushed toward very large problems or very long times
- Hoefler et al. argue that practical quantum advantage needs small-data problems and super-quadratic speedups; most proposed applications do not meet this, but there is great potential in materials science and chemistry Sourced[Ref. 3]
A square-root speedup applies only to problems with particular structure (Grover's search, the error reduction of Monte Carlo methods and the like), and it is a statement about theoretical step counts. Actual time is set by the slowness of each step, the burden of error correction and data loading (see the calculation above). The right reading: "fewer steps" and "actually faster" are different things.
4. Optimization: high hopes, weak guarantees of speed
Combinatorial optimization, such as logistics, scheduling, portfolios and design, is the most frequently cited application of quantum computers. The review by Abbas et al. separates optimization into "provably exact" and "heuristic" settings and uses complexity theory to sort out where quantum advantage is possible in each. It also stresses the importance of showing metrics for fair comparison with classical optimization methods and names finance and sustainability as areas where real-world impact could be tested Sourced[Ref. 5].
| Approach | Guarantee of speed | Status (primary sources) | What is unknown |
|---|---|---|---|
| Grover type (exact search speedup) | A guaranteed quadratic (√N) speedup Sourced[Ref. 1] | With error correction, break-even moves to very large problems (chapter 3) [Ref. 2] | Whether real industrial problems exceed the break-even size |
| QAOA and VQE (gate-model heuristics) | None (try and compare) | Experiments on small problems; limits under noise are debated (see the NISQ explainer) | A real case beating the best classical solvers |
| Quantum annealing | None (try and compare) | Advantage claims followed by classical results (see the quantum annealing explainer) | Whether it wins on total time and cost for real optimization problems |
The comparison of approaches is this article's commentary (based on the statements in the references).
As an example, Neukart et al. mapped a traffic-flow optimization problem to quantum annealing and, because of the device's size and coupling constraints, solved it with a combination of quantum and classical methods Sourced[Ref. 6]. It is a PoC showing that a real-world problem can be cast into the shape of a quantum device, and not a demonstration of advantage over the best classical methods (our commentary; chapter 8 covers how to read PoCs).
5. Finance: a quadratic speedup and the estimated scale needed
In finance, the typical candidate is the Monte Carlo method (trying many random draws and averaging) used for derivative pricing and risk. Montanaro presented a quantum algorithm (amplitude estimation) that reduces the error of Monte Carlo methods with a near-quadratic speedup over the best classical methods Sourced[Ref. 7]. Stamatopoulos et al. used it to price options, ran small examples on IBM hardware (IBM Q Tokyo) and reduced errors with error mitigation Sourced[Ref. 8].
How much is needed for practical scale? Chakrabarti et al. gave the first complete resource estimates for useful quantum derivative pricing. For two benchmark derivatives (autocallable and TARF) they find about 8,000 logical qubits and a T-depth of 54 million are required, and that advantage would need this to run on the order of a second Sourced[Ref. 9].
- The paper itself says the required resources are out of reach of current systems and offers them as a roadmap for improvements in algorithms, implementations and planned hardware Sourced[Ref. 9]
- In the rough estimate in the figure, the required logical-operation rate is about 5,000 times the one assumed by Hoefler et al. Our calculation. It shifts with the assumptions, but it shows that the speed of logical operations, not only the qubit count, is a major wall
- A review of quantum computing for finance (Herman et al.) sorts out challenges and outlook Sourced[Ref. 10]. For announcements by financial firms, check with the chapter 8 questions whether they are PoCs or production
6. Linear and differential equations: reading the fine print
Fluid flow, structural deformation, electromagnetic fields: many large engineering calculations reduce to very large systems of linear equations. Harrow, Hassidim and Lloyd (HHL) showed a quantum algorithm that, for a sparse N×N matrix with condition number κ, estimates not the solution itself but a quantity built from it (an expectation value) in time polynomial in log N and κ. The best classical methods take time proportional to N√κ, so this is an exponential improvement in N Sourced[Ref. 11].
However, in a commentary titled "Read the fine print", Aaronson argued that for quantum algorithms claiming exponential speedups for machine learning and data analysis, the real speedup cannot be judged without examining the details Sourced[Ref. 12]. The fine print amounts, roughly, to the following four points (our summary).
| Fine print | What it means | Why it matters |
|---|---|---|
| 1. Input | The right-hand side (data) must be loaded as a quantum state | If loading is slow, the speedup disappears (Hoefler et al.: I/O bandwidth limits) [Ref. 3] |
| 2. Condition number | A large κ (how hard the matrix is to solve) increases the time | Real engineering matrices often have large κ (our commentary) |
| 3. Output | What comes out is not the whole solution but one quantity derived from it (an expectation value) | Unsuited to uses that need every component of the solution [Ref. 11] |
| 4. The comparison | The comparison must be with the best classical methods (including ones specialised for sparse problems) | Classical re-evaluation has erased claimed advantages (quantum machine learning, chapter 7) [Ref. 18] |
The reading of Aaronson's commentary combined with HHL and Hoefler et al. is this article's commentary (we did not check the full text of the commentary).
Differential equations (fluid, structural and electromagnetic calculations) are studied within the same framework. The survey by Dalzell et al., which includes such applications, says they should be evaluated by end-to-end complexity including error correction Sourced[Ref. 4]. For which concrete engineering problem a quantum computer would actually beat classical methods we could not confirm within the scope of this article.
7. Quantum machine learning: what is known and what is not
Machine learning is the showpiece of "quantum applications", but also the field with the most misunderstandings. The review by Biamonte et al. argues that because quantum systems produce patterns believed not to be efficiently produced classically, quantum computers may outperform classical ones in machine learning, and says hardware and software challenges remain considerable Sourced[Ref. 13]. For clarity, we split it into "what is known" and "what is not".
| Category | Content (as the papers state) | Reading |
|---|---|---|
| Known 1: a proven advantage (artificial task) | Liu et al. constructed datasets for supervised classification with only classical data access on which, assuming hardness of the discrete logarithm, no classical learner does inverse-polynomially better than guessing, while a fault-tolerant quantum classifier achieves high accuracy [Ref. 17]Sourced | The task is artificial and assumes a fault-tolerant machine; not an advantage on real data |
| Known 2: learning from quantum data | Huang et al. proved tasks in which a quantum machine learns from exponentially fewer experiments when physical systems are transduced to a stable quantum memory, and showed a substantial advantage in experiments with up to 40 superconducting qubits and 1,300 gates [Ref. 16]Sourced | Applies when the data are outputs of quantum experiments, not ordinary images or tables |
| Known 3: reproduced classically | Tang gave a classical algorithm only polynomially slower than the quantum recommendation-system algorithm (previously a strong candidate for provable exponential speedup), so that algorithm gives no exponential speedup [Ref. 18]Sourced | A claim of quantum speedup overturned by a new classical method |
| Known 4: classical ahead in small benchmarks | Bowles et al. tested 12 quantum machine learning models on 160 datasets from 6 binary classification tasks and found that out-of-the-box classical models outperform the quantum classifiers, and removing entanglement often gives equal or better performance [Ref. 20]Sourced | Results on small tasks and simulations, not on real devices or at scale |
| Known 5: the power of data | Huang et al. showed that some problems hard to compute classically can be easily predicted by classical machine learning from data, and developed a method to assess potential quantum advantage; numerical tests up to 30 qubits showed a prediction advantage on engineered datasets [Ref. 15]Sourced | "Hard to compute classically" and "hard to predict by classical learning" are different |
| Unknown 1 | Whether quantum beats classical on real data at real scale (images, text, materials data), and whether a noisy device can do practical learning [Refs. 19, 20]Not yet confirmed | Schuld and Killoran propose reconsidering the narrative that makes advantage the goal |
| Unknown 2 | The argument that circuits free of barren plateaus can often be handled efficiently by classical computers too: does the more trainable a circuit is, the less a quantum computer is needed? [Ref. 21]SourcedNot yet confirmed | See the NISQ explainer; the evidence is case by case and the conclusion unsettled |
| Unknown 3 | Whether QRAM, which loads large data into a quantum state, can be built at realistic cost [Ref. 3]Not yet confirmed | Assumes a fault-tolerant implementation with fundamental bandwidth limits (Hoefler et al.) |
A summary of what each paper states (checked on 16 October 2026). The "known" items are conditional results.
Havlíček et al. implemented two methods on a superconducting processor and demonstrated classification using "quantum-enhanced feature spaces" Sourced[Ref. 14]. The experiment is small-scale and does not show an advantage over classical methods (our commentary).
As the table shows, proven advantages are limited to artificial tasks and quantum data. For ordinary data there are cases reproduced classically and small comparisons where classical methods won, so an advantage on real data is unconfirmed. There is also the wall of data loading (the entrance and exit in the figure). The right reading: quantum machine learning is a promising research field, not a technology that replaces AI today (our commentary).
8. Telling a PoC from production: questions for reading requests and announcements
Announcements about quantum computers often take the form "we ran a proof of concept (PoC) together". A PoC checks "does it work?" on a small problem; it does not check whether value arises in production. We sort this out in three steps.
| Step | What it shows | What it does not show | Example reading (cases in this article) |
|---|---|---|---|
| PoC | That the problem can be mapped onto a quantum device and the device runs | Advantage over the best classical methods, performance at real scale, total cost | The traffic-flow example of Neukart et al. (a quantum-classical combination) [Ref. 6] |
| Pilot | Results on real data and scale, compared with classical under the same conditions | Long-term operation, maintenance, extension to other problems | We found no third-party-verified pilot among the cases in this article |
| Production | Value arising with total cost, availability and maintenance included | — | DARPA's "utility scale" = computational value exceeds cost [Ref. 22] |
The three steps are this article's commentary (not a standardised classification).
For numbers in company announcements, check what was counted. For example, D-Wave's May 2025 announcement cites more than 20.6 million customer problems run on Advantage2 prototypes, a proof of concept by Japan Tobacco using quantum and AI in drug discovery, and 99.9% availability Sourced[Ref. 31] (a company announcement). The number of problems run measures usage, not value created, and "proof of concept" means it was not production (our commentary).
(1) Was it compared with the best classical methods (mixed-integer programming, annealing, machine learning) under the same conditions? (2) Is it a real-data, real-size problem or a small sample? (3) Is the time and cost the total including preprocessing and communication, or only the chip run? (4) Is it reported in a form a third party can check (papers, open data, code)? (5) Can operational conditions (availability, safety, maintenance, people) be met?
These are this article's own summary and are not a standardised criterion. Third-party frameworks include IBM's advantage tracker Sourced[Ref. 23], a proposed operational definition of advantage Sourced[Ref. 24] and DARPA's QBI Sourced[Ref. 22] (chapter 11).
9. A materials engineer's view
Sorting the applications near the materials and manufacturing floor with this article's framework gives the following (our commentary).
· Electronic-structure calculations (catalysts, battery materials, magnetism): a "small data, big computation" type, a region Hoefler et al. also call promising Sourced[Ref. 3]. But the scale tried on real devices is still small; see the quantum chemistry and materials explainer.
· Optimization of formulations, processes and schedules: classical optimization solvers and Bayesian optimization are strong, and we did not confirm any third-party example of quantum advantage. The break-even reasoning of chapters 3 and 4 applies.
· Machine learning on experimental data: the data are often small and the advantage of quantum machine learning is unconfirmed (chapter 7). AI-driven materials discovery is led by classical machine learning (see the materials informatics explainers).
· Simulation of flow, structure and heat transfer: the fine print of linear equations (chapter 6) must be checked first.
Whether or not quantum computers become practical, quantum devices ask for special materials and fabrication: superconducting thin films and junctions, dielectric loss of substrates, isotope-purified silicon, diamond, materials for optical fibres, lasers and detectors, cryogenic wiring and heat conduction, and refrigerators (our commentary; see our explainers on superconducting qubits, silicon spin qubits, cryogenics and control, and materials and fabrication).
For materials companies this is a story of demand and technical challenges that is separate from whether quantum computers prove useful for computation. This article does not forecast the size or timing of demand (we do not use market-size estimates), and it makes no assessment of specific companies or investments.
10. Limits, what we could not confirm, and a summary
(1) What we could not confirm
- We did not check the full text of Aaronson's commentary (only the summary on its page). The "four points of fine print" in the table are this article's summary based on the HHL paper and Hoefler et al.
- We found no third-party-verified case showing a concrete engineering problem on which quantum beats classical methods for linear or differential equations.
- For optimization, finance and machine learning we found no third-party-verified pilot or production case on real data at real scale where value arose.
- We do not cover company roadmaps (qubit counts, dates) or market-size estimates.
- The text of ISO/IEC 4879:2024 is paywalled, so we could not confirm whether it defines application-related terms or PoC. IEEE P7131 and similar efforts were confirmed only as working-group activities in development.
- Judge an application by the kind of speedup, input/output and the slowness of error correction. If a quantum step is a million times slower, a quadratic speedup loses to classical computers below N ≈ 10¹² (a calculation with made-up numbers) Our calculationSourced
- Optimization has high hopes, but heuristics carry no guarantee of speed and real-world examples are at the PoC stage Sourced
- Finance has a quadratic speedup and an estimate of about 8,000 logical qubits, a T-depth of 54 million and about a second; the required logical-operation rate is roughly 5,000 times the assumed one (a rough estimate) SourcedOur calculation
- Linear equations carry fine print behind their exponential speedup. In quantum machine learning, proven advantages are limited to artificial tasks and quantum data, and are unconfirmed on real data Sourced
- A successful PoC does not imply value in production. Use the five questions to check for same-condition comparison with the best classical method, total cost and third-party verification Sourced
11. How to read evaluations and standards
The yardsticks for applications are the theoretical complexity of the algorithm, resource estimates including error correction (logical qubits, gates, run time) and comparison with classical methods under the same conditions. In addition, application-oriented benchmarks (QED-C, BACQ) have been built. Checked on 16 October 2026.
| Item | Representative benchmarks and methods (name, source) | Standards and guidance (ISO/IEC, IEEE, DARPA, etc.) | Reading notes |
|---|---|---|---|
| End-to-end complexity | A survey of end-to-end complexities by application (Dalzell et al.); run times including error correction (Babbush et al.) [Refs. 4, 2]Sourced | Not found | Estimates depend strongly on assumptions (cycle time, error rate, code) |
| Resource-estimation tools | The Azure Quantum Resource Estimator (van Dam et al.) [Ref. 27]Sourced | Not found | A tool for comparing designs, not a measure of real machine performance |
| Application-oriented benchmarks | The QED-C benchmarks (Lubinski et al. 2023); France's BACQ (2024) [Refs. 25, 26]Sourced | IEEE P7131 (proposed standard for performance metrics and benchmarking; working group active) [Ref. 30]; WG 2 (benchmarking) of ISO/IEC JTC 3/JSC 3A [Ref. 28]SourcedNot yet confirmed | Results on small problems; not a direct measure of practical value |
| Tracking and defining advantage | The quantum advantage tracker (IBM and others) [Ref. 23]; a proposed operational definition (Lanes et al., preprint) [Ref. 24]SourcedNot yet confirmed | Not found | The organisers include companies; read with independent checks |
| Standard of utility scale | DARPA QBI: utility scale = computational value exceeds cost; third-party verification and validation [Ref. 22]SourcedNot yet confirmed | A US government programme (not an international standard) | 2033 is DARPA's goal and estimate |
| Terminology | — | ISO/IEC 4879:2024 (quantum computing vocabulary, published May 2024) [Ref. 29]Sourced | The text is paywalled; whether it covers terms such as PoC is unconfirmed |
The table's organisation is Our calculation (a summary). Numbers, titles and status were checked on the official ISO, IEEE SA 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 decide whether an application works: as in the finance example, what matters is the combination of logical qubits, logical-operation speed and run time [Refs. 9, 3]
- Company announcements and third-party checks are different: PoC announcements, counts of problems run and availability figures are company statements [Ref. 31]. Disputes over advantage can increasingly be followed in public venues such as the tracker [Ref. 23].
- Separate plans from achievements: DARPA's 2033, company roadmaps and resource estimates are goals and estimates. Many standards are in development, and even once published they generally bind only when cited in law or contracts (our commentary) [Refs. 22, 30].
12. Glossary
- PoC (proof of concept)
- A test on a small problem of whether something works; not proof of value in production.
- Pilot
- A trial on real data and scale, compared with classical methods under the same conditions.
- Production
- The state in which value arises with total cost, availability and maintenance included.
- Heuristic
- A method that often works but has no guarantee of speed or correctness.
- Quadratic (square-root) speedup
- A speedup turning N classical steps into about √N (e.g. Grover's search).
- Exponential speedup
- A speedup in which time shrinks exponentially with problem size; the conditions are strict.
- Break-even
- The problem size at which quantum and classical times are equal.
- Logical qubit / logical operation
- A qubit protected by error correction and its operations; slower than physical operations.
- T-depth
- A measure of circuit length after error correction (layers of T gates).
- Resource estimation
- Estimating from a design the logical qubits, gates and time a calculation needs.
- Monte Carlo method
- Getting an answer by averaging many random trials; widely used in finance and physics.
- Amplitude estimation
- A quantum algorithm that reduces Monte Carlo error with a quadratic speedup.
- HHL
- A quantum algorithm said to estimate quantities from linear-equation solutions exponentially faster; carries much fine print.
- Condition number
- How hard a matrix is to solve; a large value lengthens HHL and similar methods.
- QRAM
- A mechanism for loading large data into a quantum state; assumes a fault-tolerant implementation.
- Quantum machine learning
- The research field of doing machine learning on quantum computers.
- Barren plateau
- A flattening of the circuit landscape that removes the signal for training.
- Dequantization
- Reproducing a quantum algorithm's speedup with a classical algorithm.
- Utility scale
- In DARPA's definition, a scale at which computational value exceeds cost.
- Quantum advantage tracker
- A public record that collects advantage candidates and compares them with the best classical methods.
- Application-oriented benchmark
- A test that measures a device by how well it runs real algorithms and applications (QED-C, BACQ).
13. References
- Grover L. K. "A fast quantum mechanical algorithm for database search", arXiv:quant-ph/9605043 (1996) — arxiv.org/abs/quant-ph/9605043
- Babbush R. et al. "Focus beyond quadratic speedups for error-corrected quantum advantage", PRX Quantum 2, 010103 (2021), DOI 10.1103/PRXQuantum.2.010103 — doi.org/10.1103/PRXQuantum.2.010103
- 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
- Dalzell A. M. et al. "Quantum algorithms: a survey of applications and end-to-end complexities", Cambridge University Press (2025), arXiv:2310.03011 — arxiv.org/abs/2310.03011
- Abbas A. et al. "Challenges and opportunities in quantum optimization", Nat. Rev. Phys. 6, 718–735 (2024), DOI 10.1038/s42254-024-00770-9 — doi.org/10.1038/s42254-024-00770-9
- Neukart F. et al. "Traffic flow optimization using a quantum annealer", Front. ICT 4, 29 (2017), DOI 10.3389/fict.2017.00029 — doi.org/10.3389/fict.2017.00029
- Montanaro A. "Quantum speedup of Monte Carlo methods", Proc. R. Soc. A 471, 20150301 (2015), DOI 10.1098/rspa.2015.0301 — doi.org/10.1098/rspa.2015.0301
- Stamatopoulos N. et al. "Option pricing using quantum computers", Quantum 4, 291 (2020), DOI 10.22331/q-2020-07-06-291 — doi.org/10.22331/q-2020-07-06-291
- Chakrabarti S. et al. "A threshold for quantum advantage in derivative pricing", Quantum 5, 463 (2021), DOI 10.22331/q-2021-06-01-463 — doi.org/10.22331/q-2021-06-01-463
- Herman D. et al. "Quantum computing for finance", Nat. Rev. Phys. 5, 450–465 (2023), DOI 10.1038/s42254-023-00603-1 — doi.org/10.1038/s42254-023-00603-1
- Harrow A. W., Hassidim A., Lloyd S. "Quantum algorithm for linear systems of equations", Phys. Rev. Lett. 103, 150502 (2009), DOI 10.1103/PhysRevLett.103.150502 — doi.org/10.1103/PhysRevLett.103.150502
- Aaronson S. "Read the fine print", Nat. Phys. 11, 291–293 (2015), DOI 10.1038/nphys3272 — nature.com/articles/nphys3272
- Biamonte J. et al. "Quantum machine learning", Nature 549, 195–202 (2017), DOI 10.1038/nature23474 — doi.org/10.1038/nature23474
- Havlíček V. et al. "Supervised learning with quantum-enhanced feature spaces", Nature 567, 209–212 (2019), DOI 10.1038/s41586-019-0980-2 — doi.org/10.1038/s41586-019-0980-2
- Huang H.-Y. et al. "Power of data in quantum machine learning", Nat. Commun. 12, 2631 (2021), DOI 10.1038/s41467-021-22539-9 (open access) — doi.org/10.1038/s41467-021-22539-9
- Huang H.-Y. et al. "Quantum advantage in learning from experiments", Science 376, 1182–1186 (2022), DOI 10.1126/science.abn7293 — doi.org/10.1126/science.abn7293
- Liu Y., Arunachalam S., Temme K. "A rigorous and robust quantum speed-up in supervised machine learning", Nat. Phys. 17, 1013–1017 (2021), DOI 10.1038/s41567-021-01287-z — doi.org/10.1038/s41567-021-01287-z
- Tang E. "A quantum-inspired classical algorithm for recommendation systems", Proc. 51st ACM STOC, 217–228 (2019), DOI 10.1145/3313276.3316310 — doi.org/10.1145/3313276.3316310
- Schuld M., Killoran N. "Is quantum advantage the right goal for quantum machine learning?", PRX Quantum 3, 030101 (2022), DOI 10.1103/PRXQuantum.3.030101 — doi.org/10.1103/PRXQuantum.3.030101
- Bowles J., Ahmed S., Schuld M. "Better than classical? The subtle art of benchmarking quantum machine learning models", arXiv:2403.07059 (2024) — arxiv.org/abs/2403.07059
- Cerezo M. et al. "Does provable absence of barren plateaus imply classical simulability?", Nat. Commun. 16, 7907 (2025), DOI 10.1038/s41467-025-63099-6 — doi.org/10.1038/s41467-025-63099-6
- DARPA "Quantum Benchmarking Initiative (QBI)" (program page; list current as of 8 October 2026) — darpa.mil/research/programs/quantum-benchmarking-initiative
- Gambetta J., Davis R. (IBM) "Quantum advantage tracker" (IBM Quantum blog, 23 February 2026) — ibm.com/quantum/blog/quantum-advantage-tracker
- Lanes O. et al. "A framework for quantum advantage", arXiv:2506.20658 (2025, preprint) — arxiv.org/abs/2506.20658
- 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
- Barbaresco F. et al. "BACQ — Application-oriented benchmarks for quantum computing", arXiv:2403.12205 (2024) — arxiv.org/abs/2403.12205
- 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
- IEC/ISO JTC 3/JSC 3A "Quantum computing" (committee page: scope and working groups, including WG 2 benchmarking) — iso.org/committee/12010875.html
- ISO/IEC "ISO/IEC 4879:2024 Information technology — Quantum computing — Vocabulary" (ISO catalogue page) — iso.org/standard/80432.html
- IEEE SA "Quantum Benchmarking Working Group P7131" (working group site) — sagroups.ieee.org/7131/
- D-Wave Quantum Inc. "D-Wave Announces General Availability of Advantage2 Quantum Computer, Its Most Advanced and Performant System" (press release, 20 May 2025) — dwavequantum.com/company/newsroom/press-release/d-wave-announces-general-availability-of-advantage2-quantum-computer-its-most-advanced-and-performant-system/
14. Claim-to-source audit
| Claim in the text | Basis | Category |
|---|---|---|
| Grover's search needing about √N operations while classical algorithms must look at at least N/2 (a quadratic speedup) | Reference 1 https://arxiv.org/abs/quant-ph/9605043 | Sourced |
| Quadratic speedups not enabling advantage on early fault-tolerant devices without significant improvement in error correction, with quartic speedups looking more practical | Reference 2 https://doi.org/10.1103/PRXQuantum.2.010103 | Sourced |
| Practical advantage needing small-data problems and super-quadratic speedups; most applications not meeting this but materials science and chemistry having potential; the fundamental I/O bandwidth limit, QRAM's fault-tolerant requirement and the 10.5 k operations/s assumption (10 µs cycle, 200-cycle latency) | Reference 3 https://doi.org/10.1145/3571725 | Sourced |
| The survey of quantum-algorithm applications: the need to evaluate by end-to-end complexity including error correction | Reference 4 https://arxiv.org/abs/2310.03011 | Sourced |
| The optimization review: provably exact and heuristic settings, the importance of benchmarking, and finance and sustainability as test domains | Reference 5 https://doi.org/10.1038/s42254-024-00770-9 | Sourced |
| Mapping traffic-flow optimization to quantum annealing and solving it with a quantum-classical combination (Neukart et al.) | Reference 6 https://doi.org/10.3389/fict.2017.00029 | Sourced |
| The near-quadratic speedup of Monte Carlo (Montanaro); option pricing and a small run on IBM Q Tokyo with error mitigation (Stamatopoulos et al.) | Reference 7 https://doi.org/10.1098/rspa.2015.0301 Reference 8 https://doi.org/10.22331/q-2020-07-06-291 | Sourced |
| The resource estimate for derivative pricing: about 8,000 logical qubits, a T-depth of 54 million, about a second, out of reach of current systems | Reference 9 https://doi.org/10.22331/q-2021-06-01-463 | Sourced |
| The review of quantum computing for finance (Herman et al.) | Reference 10 https://doi.org/10.1038/s42254-023-00603-1 | Sourced |
| HHL: for a sparse matrix with condition number κ, estimating a quantity built from the solution in time polynomial in log N and κ; the best classical time proportional to N√κ | Reference 11 https://doi.org/10.1103/PhysRevLett.103.150502 | Sourced |
| Aaronson's commentary: exponential-speedup claims for machine learning and data analysis cannot be judged without the details | Reference 12 https://www.nature.com/articles/nphys3272 | Sourced |
| The review of quantum machine learning (Biamonte et al.); Havlíček et al.: classification with quantum-enhanced feature spaces on a superconducting processor | Reference 13 https://doi.org/10.1038/nature23474 Reference 14 https://doi.org/10.1038/s41586-019-0980-2 | Sourced |
| Liu et al.: a rigorous quantum speedup for classification with classical data access, assuming hardness of the discrete logarithm | Reference 17 https://doi.org/10.1038/s41567-021-01287-z | Sourced |
| Huang et al. (2022): exponentially fewer experiments needed when learning from quantum data, experiments on 40 qubits and 1,300 gates; Huang et al. (2021): problems predictable by classical learning from data, numerical tests up to 30 qubits | Reference 16 https://doi.org/10.1126/science.abn7293 Reference 15 https://doi.org/10.1038/s41467-021-22539-9 | Sourced |
| Tang: a classical algorithm only polynomially slower than the quantum recommendation-system algorithm | Reference 18 https://doi.org/10.1145/3313276.3316310 | Sourced |
| Bowles et al.: classical models outperforming quantum classifiers across 12 models, 6 tasks and 160 datasets; Schuld and Killoran: questioning the narrative of advantage as the goal; Cerezo et al.: the argument that barren-plateau-free circuits can often be handled classically | Reference 20 https://arxiv.org/abs/2403.07059 Reference 19 https://doi.org/10.1103/PRXQuantum.3.030101 Reference 21 https://doi.org/10.1038/s41467-025-63099-6 | Sourced |
| D-Wave's announcement of 20 May 2025: more than 20.6 million customer problems, a Japan Tobacco proof of concept and 99.9% availability (company announcement) | Reference 31 https://www.dwavequantum.com/company/newsroom/press-release/d-wave-announces-general-availability-of-advantage2-quantum-computer-its-most-advanced-and-performant-system/ | Sourced |
| DARPA QBI's definition of utility scale (value exceeds cost) and its 2033 estimate; IBM's advantage tracker; the advantage framework of Lanes et al. (preprint) | Reference 22 https://www.darpa.mil/research/programs/quantum-benchmarking-initiative Reference 23 https://www.ibm.com/quantum/blog/quantum-advantage-tracker Reference 24 https://arxiv.org/abs/2506.20658 | Sourced |
| The QED-C application-oriented benchmarks, BACQ and the Azure Quantum Resource Estimator | Reference 25 https://doi.org/10.1109/TQE.2023.3253761 Reference 26 https://arxiv.org/abs/2403.12205 Reference 27 https://arxiv.org/abs/2311.05801 | Sourced |
| WG 2 (benchmarking) of ISO/IEC JTC 3/JSC 3A; ISO/IEC 4879:2024 number, title and May 2024 publication; the technical scope of the IEEE P7131 working group | Reference 28 https://www.iso.org/committee/12010875.html Reference 29 https://www.iso.org/standard/80432.html Reference 30 https://sagroups.ieee.org/7131/ | Sourced |
| The break-even table (r = 10³, 10⁶, 10⁹ giving N* = 10⁶, 10¹², 10¹⁸ and times of 1 ms, 1,000 s and 10⁹ s; with 1,000 cores N* = 10¹², 10¹⁸, 10²⁴ and 278 h) and the finance rough estimate (5.4×10⁷ ÷ 1.05×10⁴ ≈ 5,000) | Our calculation (Python; arithmetic with made-up numbers for illustration, not real machine speeds) | Our calculation |
| Third-party-verified cases beating classical methods on concrete differential-equation or engineering problems; third-party-verified pilot or production cases on real data at real scale for optimization, finance and machine learning; the full text of Aaronson's commentary; application terms in ISO/IEC 4879; and the publication status of IEEE P7131 | Not confirmed in the primary sources reachable for this article, so not stated | Commentary |
| The readings: the "four questions", the four-point arrangement of HHL fine print, the three steps and five questions from PoC to production, the sorting of applications on the materials floor, and "quantum technology as a new customer for materials" | This article's commentary based on published material; not views stated by the institutions | Commentary |
| Fig. 3 and Fig. 5 are illustrations (calculations) rather than real data; Fig. 2, Fig. 4, Fig. 6 and Fig. 7 are concept diagrams; Fig. 1 and Fig. 8 are ImageGen-generated conceptual illustrations | A note in this article | Commentary |
Last updated: 16 October 2026. Sources are limited to primary material (peer-reviewed papers, arXiv papers, D-Wave's announcement, official DARPA, ISO and IEEE pages). Numerical examples are our own calculations, not device results. We could not confirm the full text of Aaronson's commentary, third-party-verified pilot or production cases on real data at real scale, cases beating classical methods on concrete engineering problems, application terms in ISO/IEC 4879, or the publication status of IEEE P7131, so we do not state them. All figures are illustrative concept diagrams or drawings that include our calculations; Fig. 1 and Fig. 8 are ImageGen-generated conceptual illustrations.