How will quantum computers provide an industrially relevant computational advantage in quantum chemistry?

A. Bochevarov; B.W. Broer; J. Gavartin; K. P. Lorton; M.D. Halls; M. Webber; V.E. Elfving

REVIEW 4 major objections 4 minor 1 cited by

How will quantum computers provide an industrially relevant computational advantage in quantum chemistry?

T0 review · 4 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read Quantum computers cross the classical chemistry speed limit near 26 active orbitals

desk verdict A careful Cr2 resource estimate with a plausible crossover range, undercut by an overbroad claim against 'any classical computer' that omits DMRG and selected-CI. read the letter →

arxiv 2009.12472 v1 pith:5OCWO5E2 submitted 2020-09-25 quant-ph physics.chem-ph

V.E. Elfving , B.W. Broer , M. Webber , J. Gavartin , M.D. Halls , K. P. Lorton , A. Bochevarov This is my paper · ORCID

classification quant-phphysics.chem-ph

keywords quantumadvantagechemistryfault-tolerantcomputingresourceestimationcompleteactivespacechromiumdimerFeMo-cofactorsurfacecode

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks where, if anywhere, quantum computers will beat classical computers at a chemistry calculation that industrial users care about. It argues that the popular targets are the wrong targets: tiny molecules like H2 are already solved classically, while the nitrogen-fixing enzyme cofactor FeMo-co is so hard that no near-term quantum computer can treat its full electronic structure. The paper's central quantitative claim is that for the chromium dimer, a fault-tolerant quantum computer should outrun a classical computer once the complete active space reaches roughly (19,19) to (34,34), much smaller than the (54,54) active spaces often cited for quantum advantage. It then proposes that the first industrially relevant advantage will come from medium-sized strongly correlated systems, such as homogeneous catalysts and biomimetic metal complexes, with active spaces of roughly 26 to 60 electrons and orbitals, where non-dynamic correlation is treated on the quantum computer while dynamic correlation is handled classically.

What carries the argument

The central object is the crossover curve: the active-space size N at which the estimated wall-clock time of a fault-tolerant quantum phase-estimation calculation falls below the estimated runtime of a classical CASCI/CASSCF calculation. The paper constructs this by combining determinant-count and TFLOP estimates for classical configuration interaction on desktop and supercomputer hardware, T-gate and Toffoli counts for two Hamiltonian-simulation oracles, and surface-code error-correction overhead expressed in qubit-seconds from magic-state factory analyses. The plot of wall-clock time versus N for the chromium dimer carries the argument; the load-bearing assumption inside it is that the required Trotter number for Cr2 follows the same shifted scaling with system size as in the earlier FeMo-co resource study.

What would settle it

Compute the exact Trotter error for the Cr2 Hamiltonian at 1.68 Å for a (26,26) active space in cc-pVTZ for several Trotter orders and step sizes; if the Trotter number needed for chemical precision differs by more than a small factor from the imported FeMo-co scaling, the claimed T-gate counts and the 19–34 crossover range shift accordingly.

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Extended reading notes

Core claim

Using resource estimates for two fault-tolerant quantum phase-estimation implementations, Trotter-Suzuki decomposition and sparse qubitization, with surface-code error correction, the authors claim that the crossover for CASCI/CASSCF-type calculations of the chromium dimer as a function of active-space size N occurs around 19 ≤ N ≤ 34, with N ≈ 26 as a representative threshold. At that crossover, Trotterization would still need hundreds of years of runtime and roughly $10^{5}$ physical qubits, while sparse qubitization reduces the runtime to days with somewhat more physical qubits. The same estimates imply that FeMo-co, often discussed as the flagship quantum-chemistry application, is farther from reach than previously claimed: the active space needed for converged non-dynamic correlation is probably much larger than (54,54), perhaps around (113,76), and the basis sets needed for dynamic correlation are too large for near-term devices. The paper's positive proposal is that the 'sweet spot' is medium-sized multireference systems, such as homogeneous catalysts and biomimetic complexes, with active spaces of roughly 26 to 60 electrons and orbitals, where quantum computers can supply what classical methods cannot and the results bear on industrial chemistry.

Load-bearing premise

The whole crossover estimate rests on the assumption that the number of small time steps used to approximate the chromium dimer's time evolution grows with system size in the same way it does in the earlier FeMo-co study, which the paper itself calls only an approximation.

Editorial extensions

If this is right

The first useful quantum computational chemistry will be for strongly correlated multireference systems, not small light-atom molecules, because dynamic-correlation-dominated problems need basis sets and qubit counts beyond near-term devices.
For chromium-dimer-style CAS(N,N) problems, a fault-tolerant machine crosses the classical runtime curve around N = 19–34, with N ≈ 26 as a representative threshold; past N = 34 all assessed quantum algorithms should be faster than any available classical computer.
At that crossover, Trotterization still requires centuries of runtime, whereas sparse qubitization runs in days at similar physical-qubit counts, so qubitization-like algorithms are the more plausible path to early practical advantage.
FeMo-co is not a good near-term target because its non-dynamic correlation is probably not converged until an active space near (113,76), and its dynamic correlation needs basis sets too large for near-term quantum hardware.
A workable route to industrial relevance is benchmarking and correcting DFT on transition states and catalytic steps using quantum-computed multireference energies for active spaces of 26–60 electrons and orbitals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: if the Cr2 crossover generalizes to other transition-metal dimers with metal-metal multiple bonds, repeating the resource estimate for Mo2 or W2 would be a natural test of whether the 19–34 active-space window is a general feature.
Editorial inference: the paper's reliance on an imported Trotter-number scaling makes the crossover range a prediction to be refined; computing actual Trotter errors for Cr2 at each N would either confirm the 19–34 window or shift it, and would be a concrete next step.
Editorial inference: the 'sweet spot' logic treats the classical baseline as fixed, but classical approximate methods keep improving, so the industrially relevant crossover may need to be re-evaluated as those methods gain accuracy.
Editorial inference: the two-electron-integral data-loading bottleneck the paper flags suggests that quantum chemistry advantage may depend as much on classical I/O and compression as on gate counts, so co-designing integral handling with quantum oracles could be a decisive engineering constraint.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper is a serious resource-estimation study with a new crossover estimate: for Cr2 CAS(N,N), the quantum/classical crossover lands at N≈19–34 under surface-code assumptions. That is the number people will quote, and it is lower than what previous papers implied for FeMo-co. The fresh part is the Cr2 comparison (Trotterization vs qubitization vs classical CASCI extrapolation) and the gap analysis showing that industrially relevant homogeneous-catalyst active spaces sit around N≈60, well above the crossover. The FeMo-co critique is mostly borrowed from prior work, but the authors use it to motivate a sensible middle-ground target.

The paper is honest. It repeatedly states assumptions that could shift the crossover: the Trotter number scaling is borrowed from Ref. 4, the classical timing model is an extrapolation, and the authors explicitly say the crossover point is hard to pin down. That transparency is a real strength. What I would hold against it is the absence of DMRG and selected-CI from the classical baseline. The crossover comparison matches quantum CASCI against exact diagonalization; DMRG is the standard classical method for strongly correlated active spaces, and the paper itself cites DMRG on FeMo-co with a (113,76) active space and bond dimension 6000. If a (26,26) or (34,34) Cr2 active space can be converged by DMRG on a workstation or the 105x HPC baseline in less time than the quantum estimate, the claim that 'for N > 34 any assessed quantum algorithm should be faster than any available classical computer' is overstated. That is a load-bearing soft spot, not a minor omission. It does not kill the paper, but it should be fixed before publication, or at least the claim should be softened to 'faster than exact CASCI.'

A second soft spot is reproducibility: the fitted curves are not backed by published code or data, and the fitted parameters are undisclosed. That is a minor issue for a physics paper, but it matters here because the curve fitting is the bridge between the resource estimates and the crossover number.

Overall, this is a careful, useful contribution for anyone working on fault-tolerant quantum chemistry. It deserves a serious referee, but I would not accept it as is. The authors should either add DMRG/selected-CI timings to the crossover analysis or change the wording from 'any classical computer' to 'exact CASCI.'

Referee Report

4 major / 4 minor

Summary. The paper reviews the prospects for industrially relevant quantum advantage in quantum chemistry, argues that the field should distinguish chemical accuracy from chemical precision, surveys past quantum chemistry experiments, and critically examines FeMo-co as a target. Its main new quantitative contribution is a set of end-to-end resource estimates for fault-tolerant quantum simulation of H2 and Cr2, comparing Trotterization and qubitization under surface-code error correction. For Cr2 CASCI/N(N,N) active spaces, the authors extrapolate both quantum runtimes and classical exact-diagonalization runtimes and conclude that a quantum computer would become faster than a classical desktop or HPC at around N = 19–34, with the caveat that many assumptions enter the estimate. They then speculate that homogeneous catalysts with CAS sizes 26–60 are promising near-term industrial targets.

Significance. If the numerical crossover estimate were robust, it would be a valuable, sobering benchmark: it gives a concrete active-space size for early fault-tolerant quantum advantage in strongly correlated systems and challenges prior claims that much larger active spaces are required. The paper is also useful for its honest treatment of uncertainty, its comparison of Trotterization and qubitization including error-correction overhead, and its explicit discussion of chemical accuracy versus chemical precision. The central claim, however, rests on an extrapolation from fitted curves and on a classical baseline that excludes widely used approximate methods; the manuscript's own caveats indicate that the crossover range should be read as conditional rather than as a firm quantitative prediction.

major comments (4)

[Section III and Section VI.E, Fig. 3] The paper defines quantum advantage as beating 'any classical computer' and later states that for N > 34 any assessed quantum algorithm should be faster than any available classical computer. However, the classical timing curves in Fig. 3 compare only against exact CASCI/CASSCF diagonalization on a desktop or HPC. DMRG, selected-CI, and FCIQMC—methods cited elsewhere in the paper for FeMo-co and benzene (Refs. 66, 80–83)—are absent from the classical baseline. Since these methods approximate strongly correlated active spaces with polynomial resources in the bond dimension or selected-space size, the claim to outperform 'any available classical computer' is not supported. The authors should either include realistic DMRG/selected-CI timing estimates for Cr2 CAS(N,N) or explicitly restrict the claim to exact CASCI/CASSCF.
[Section VI.D.1] The Trotterization T-gate counts in Figs. 2 and 3 use the Trotter number scaling borrowed from Ref. 4, as stated in Section VI.D.1, where the authors call this 'merely an approximation.' This assumption is load-bearing because the Trotter number directly multiplies the T-gate cost and therefore sets the quantum runtime curves from which the N = 19–34 crossover is derived. The manuscript should test the sensitivity of the crossover to this assumption, for example by computing Trotter-error bounds for the Cr2 Hamiltonian or by varying the assumed exponent and shift over a plausible range and reporting how the crossover moves.
[Section VI.E, Fig. 3 and Table II] The crossover range is obtained by extrapolating fitted curves, but the manuscript provides no fitting function, number of data points, or error bars for the classical or quantum curves. This matters because Table II shows that the total runtime varies by orders of magnitude with error rate and with space- versus time-optimized distillation strategies; a sensitivity analysis over these hardware parameters is needed to determine whether the 19–34 range is robust or is, for example, 15–50 under different but still reasonable assumptions.
[Section VII] The extrapolation from the Cr2 crossover to the claim that homogeneous catalysts with CAS sizes 26–60 are promising early targets is asserted rather than demonstrated; the resource estimates were computed for Cr2 only. The paper labels this as speculation, which is appropriate, but the transition from the quantitative crossover to the proposed application window should be framed more carefully so that the speculative generalization is not read as part of the central result.

minor comments (4)

[References [54] and [84]] These references have incomplete bibliographic information ('Journal of Chemical Theory and Computation 0, 0, null'); they should be completed before publication.
[Fig. 3 caption] The caption should state the number of classical data points, the range of N used in the fits, and the functional form of the fitted curves so that readers can assess the extrapolation.
[Table II] The terms 'space-optimized' and 'time-optimized' are used without definition in the caption or text; please define them or refer to the specific strategy in Ref. 130.
[Section VI.D.1] The text says the T-gate scaling is 'to less than seventh order' but does not give the fitted polynomial coefficients; reporting the fitted curves would improve reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the crossover estimate is an extrapolative comparison built from external algorithm cost models and classical timing data; only a minor non-central self-citation appears.

full rationale

The central claim (Sec. VI, Fig. 3) is a crossover of two independently constructed timing curves. The quantum side uses Trotterization cost formulas from Reiher et al. (Ref. 4) and qubitization from Berry et al. (Ref. 125), with error-correction overhead from Gidney/Fowler (Ref. 122) and Litinski (Ref. 130). These are external, published cost models with stated assumptions, not quantities defined by this paper's own outputs. The classical side is fit to actual CASCI timing data and extrapolated to larger active spaces; the crossover is the intersection of the fits, not a parameter fitted to the claimed result. The paper explicitly flags its weakest point: the Trotter-number scaling for Cr2 is assumed to follow Fig. 4 of Ref. 4's Supplementary material, calling it 'merely an approximation that should generally be investigated case-specifically.' That is an unvalidated external assumption and a correctness risk, but it is not circular because the assumed scaling is not derived from, nor defined in terms of, the 19-34 crossover claim. The only self-citations (Table I, Ref. 25; the passing remark in Sec. II about prior chemical-accuracy claims) play no role in the resource estimates or the crossover calculation. The omission of DMRG/selected-CI from the classical benchmark curves is a benchmark-completeness concern, not a circularity: it weakens the practical force of the claim without making the quantum timing curves equivalent to the classical comparison by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central estimates rely on published resource models and on fitted extrapolations, but the fitted values are not disclosed in the paper, making the estimates difficult to audit.

free parameters (3)

Trotter number scaling shift/exponent (from Ref. 4) = not disclosed; assumed equal to Ref. 4 Fig. 4
Used to estimate Trotterization T-gate costs for H2 and Cr2 as a function of basis/orbital count; directly affects runtime curves in Fig. 2 and crossover in Fig. 3.
Classical CASCI runtime fitting parameters = not disclosed; shown as curves for 1.2 TFLOPS and 125 PFLOPS
Classical timings for Cr2 CAS(N,N) are fitted and extrapolated to large N; the crossover N depends on this extrapolation.
Quantum resource scaling curve parameters for Trotterization and qubitization = not disclosed; fitted curves shown in Fig. 2
T-gate and Toffoli counts as functions of N are fitted to polynomial curves; extrapolation beyond computed points determines the crossover.

assumptions (4)

domain assumption The surface-code fault-tolerance cost models of Refs. 122 and 130 are accurate for future hardware.
Used in Section VI.E to convert logical gate counts into physical qubits and runtimes; variations in this model change resource estimates by orders of magnitude.
domain assumption The Trotterization and qubitization resource estimation frameworks from Refs. 4 and 125 apply to the studied molecules.
The paper explicitly borrows Trotter-number scaling from Ref. 4 and the qubitization sparse method from Ref. 125; if these do not transfer to Cr2, the estimates break.
domain assumption Chemical accuracy for the studied systems requires the chosen basis set extrapolations and active spaces (e.g., cc-pVTZ for Cr2).
The definition of the computational target (chemical accuracy) determines the number of spin-orbitals and active space sizes, hence all resource estimates.
standard math Standard results in quantum computation (QPE, Jordan-Wigner mapping, phase estimation) are correct.
Background for the algorithms; not disputed.

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Pith. "Pith review of How will quantum computers provide an industrially relevant computational advantage in quantum chemistry?." pith.science (2026). https://pith.science/paper/5OCWO5E2

@misc{pith2026200912472,
  author       = {Pith},
  title        = {Pith review of: How will quantum computers provide an industrially relevant computational advantage in quantum chemistry?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5OCWO5E2}},
  note         = {Machine review of arXiv:2009.12472}
}

abstract

Numerous reports claim that quantum advantage, which should emerge as a direct consequence of the advent of quantum computers, will herald a new era of chemical research because it will enable scientists to perform the kinds of quantum chemical simulations that have not been possible before. Such simulations on quantum computers, promising a significantly greater accuracy and speed, are projected to exert a great impact on the way we can probe reality, predict the outcomes of chemical experiments, and even drive design of drugs, catalysts, and materials. In this work we review the current status of quantum hardware and algorithm theory and examine whether such popular claims about quantum advantage are really going to be transformative. We go over subtle complications of quantum chemical research that tend to be overlooked in discussions involving quantum computers. We estimate quantum computer resources that will be required for performing calculations on quantum computers with chemical accuracy for several types of molecules. In particular, we directly compare the resources and timings associated with classical and quantum computers for the molecules H$_2$ for increasing basis set sizes, and Cr$_2$ for a variety of complete active spaces (CAS) within the scope of the CASCI and CASSCF methods. The results obtained for the chromium dimer enable us to estimate the size of the active space at which computations of non-dynamic correlation on a quantum computer should take less time than analogous computations on a classical computer. Using this result, we speculate on the types of chemical applications for which the use of quantum computers would be both beneficial and relevant to industrial applications in the short term.

Figures

Figures reproduced from arXiv: 2009.12472 by the authors.

**Figure 2.** FIG. 2. (Top) number of T gates for a Trotterization im [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3. Wallclock time scaling (top) and total physical qubit [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

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We aim to ﬁnd the optimal resource estimates for ground state energy simulation to chemical precision within the chosen basis set

Trotterization In this section we provide resource estimates for the iterative QPE algorithm with oracles based on a Trotter- Suzuki decomposition 123 of the direct Hamiltonian sim- ulation operator ˆU =e−i ˆHt. We aim to ﬁnd the optimal resource estimates for ground state energy simulation to chemical precision within the chosen basis set. As we con- sid...

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sweet spot

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