The Quantum Utility Ladder: What Fault-Tolerant Quantum Computers Will Actually Be Used For

This article is the technical foundation of my Quantum Utility Map Deep Dive series, a seven-part investigation into what fault-tolerant quantum computers will actually be used for and which industries they will transform. Across the series I also set out what organizations and nations should do about it. The series includes: Quantum Computing by 2033 (competitive analysis by industry), Quantum Sovereignty and the Utility Trap (geopolitical implications), Why Quantum Won’t Save Wall Street (finance deep dive), Quantum Chemistry’s Honest Ledger (materials and chemistry assessment), The Error Correction Revolution (technology acceleration), and The Narrow Advantage (capstone synthesis).

What the Utility Ladder Maps

Every quantum computing headline fixates on the same question: when will a quantum computer break encryption? It’s the wrong question, or at least, it’s the wrong first question. Long before any machine accumulates the roughly 850 to 1,400 logical qubits (estimate today, might change tomorrow) needed to threaten RSA or elliptic curve cryptography, fault-tolerant quantum computers will start doing something far more interesting: solving problems in chemistry, materials science, and physics that no classical supercomputer on Earth can touch.

The fault-tolerant era will be defined by what quantum computers build: new catalysts, new batteries, new drugs, new materials. And for the first time, we have enough concrete resource estimates to map exactly what becomes possible at each rung of the logical-qubit ladder.

I tried to write this article a few years ago. But things have changed. A lot. I’ve spent the past few weeks assembling what I believe is the most comprehensive catalog of fault-tolerant quantum algorithms mapped to their resource requirements (logical qubits, T-gate counts, estimated runtimes) and the real-world problems they solve.

The requirements are more uneven than the smooth staircase vendor marketing suggests, in which every extra 50 qubits brings a new killer app. The utility landscape is lumpy. Some problems become tractable at 130 logical qubits. Others need 100,000. And a surprising number of the applications people talk about most (finance, logistics, machine learning) face a structural barrier that no amount of better hardware can fix.

This article is my attempt to lay it all out: what fault-tolerant quantum computers will actually do, when, and what the evidence supports. It’s long. There’s a comprehensive table. And I’ve tried to be honest about where the evidence is strong, where it’s speculative, and where the hype has outrun the science.

Logical Width, Logical Depth, and T-Count

Before climbing the ladder, we need to understand what determines whether a quantum algorithm can run on a given machine. It isn’t just the number of logical qubits, though that gets all the attention. Three resource dimensions interact to determine feasibility:

Logical width is the number of simultaneously active logical qubits, including ancillas for arithmetic, phase estimation, and magic state production. People who say “you need X qubits” usually mean width.

Logical depth is the total number of sequential non-Clifford operations (T-gates or Toffolis) in the critical path of the circuit. This determines runtime. If your algorithm needs 10¹⁵ T-gates and your machine produces one T-gate per microsecond, you’re looking at 30 years. Depth is often the binding constraint, not width.

T-count is the total number of non-Clifford gates consumed across the entire computation, which determines the spatial footprint of “magic state factories”: the dedicated regions of the quantum processor that continuously manufacture the high-fidelity ancillary states needed to execute each T-gate. A higher T-count means either more factories (more physical qubits) or longer runtime (more depth).

These three dimensions interact in ways that make simple “qubit count” comparisons misleading. An algorithm requiring 300 logical qubits and 10⁸ T-gates is a very different engineering challenge from one requiring 300 logical qubits and 10¹⁵ T-gates. The first might run in minutes. The second might take centuries on the same hardware.

Three dimensions covered every end-to-end estimate in the literature when I first wrote this. A fourth has since been added, and it behaves unlike the others.

IBM’s roadmap makes this explicit: their Starling system, targeted for 2029, aims for 200 logical qubits executing 100 million gates. Their Blue Jay, projected for 2033 or beyond, targets 2,000 logical qubits executing 1 billion gates. The qubit count is the headline, but the gate budget is what determines which algorithms actually run.

Those budgets are vendor-defined, and until recently there was no common unit to compare them in. A 30-author collaboration led from Sandia, with Timothy Proctor as first author and Robin Blume-Kohout as senior author, proposed one in September 2026: QUOPS, for quantum universal operation performance system, which counts the quantum operations a device can execute with the error behavior of the whole machine accounted for and reports the rate alongside the count. On physical qubits, Google’s Willow scored 216 QUOPS at 2.0×10⁷ per second, IBM’s ibm_boston 204 at 3.1×10⁵ per second, and Quantinuum’s Helios-1 1,504 at 303 per second. Those are physical-computation scores and should not be read as logical-operation budgets; the paper assesses encoded performance separately, on up to eight [[7,1,3]]-encoded logical qubits on Helios-1, where the encoded architecture scored 40. The paper’s abstract puts the distance still to cover at five orders of magnitude of measured capability before machines reach the workloads the field is aiming at. Every row below is on the far side of that gap.

Memory Footprint, the Fourth Dimension

Memory footprint is the total space a computation consumes, counting the classical storage the quantum approach forces alongside the qubits themselves. It is not the same as logical width. A poly(log N)-qubit algorithm that depends on QRAM has a tiny qubit count and an O(N) total footprint, because the N bits specifying the loading gates have to be stored somewhere. Width is the number of qubits on the chip, and footprint adds the classical storage the computation requires.

This axis was added to the map in April 2026 by Zhao, Zlokapa, Neven, Babbush, Preskill, McClean, and Huang, a Caltech, Google Quantum AI, and MIT team. They prove that, on streamed classical data, a quantum computer of poly(log N) size can perform binary classification, dimension reduction, and sparse linear-system solving at sample budgets where classical machines cannot. For the linear-system task, a classical machine limited to O(N^0.99) size needs superpolynomially more samples. For classification and dimension reduction the bound is O(D^0.99) in the feature dimension.

The linear-system result is narrower than its name suggests. The task is estimating a quadratic form of the solution to a sparse, well-conditioned system, from streamed samples of its entries, rather than general sparse linear algebra. Each sample is read once and discarded, and there is no QRAM. On IMDb sentiment classification and PBMC single-cell RNA sequencing, they measure four to six orders of magnitude less memory at under 60 logical qubits.

The proof is information-theoretic. It allows the classical machine unlimited time and still applies if BPP = BQP. Unconditional separations of this kind are not new. Hidden Matching, further down this ladder, is one, and the streaming and communication-complexity literature has had them for two decades. What is new is that the tasks are recognizable ones, on real datasets, with memory figures attached. The chemistry advantages depend on the empirical failure of classical heuristics, a basis Lee et al. showed to be weaker than it appears. Shor’s advantage assumes that factoring is classically hard. The memory separation requires two things: that quantum mechanics is correct, and that the data arrive through the streaming access model the theorem assumes, where each sample is read once and discarded.

The algorithm needs only poly(log N) qubits and an enormous amount of gate work. Zhao and colleagues prove that Õ(N) gate complexity is optimal in the space-efficient regime, because every sample must be touched by a gate and no compiler removes that floor. Their own discussion extrapolates the memory scaling while, in their words, “ignoring the exponential runtime overhead.”

The only fault-tolerant cost anyone has published in this family belongs to a different algorithm. Niroula and colleagues, a JPMorganChase and Quantinuum team, costed Hidden Matching. On Helios, the 98-qubit trapped-ion system published in Nature in June 2026, they ran the quantum pair sketch against a real-time external data stream for graphs of 4 to 32 nodes, then compiled the algorithm to surface and bivariate bicycle codes to estimate the problem size at which a real advantage would begin. For Hidden Matching, the canonical streaming-separation problem, they place break-even with the best classical method at n ≈ 10¹², which takes 581 logical qubits and about 53,000 physical qubits under a two-gross bivariate bicycle code at a physical error rate of 10⁻⁴. Under a rotated surface code at 10⁻³ the same row needs 1.4 million.

The gate count at that row is 8.68×10¹⁴ Toffolis. Serialized at one per microsecond that is 27.5 years, and at 10 kHz it is 2,750 years. Those figures assume no parallelism at all, which is the pessimistic bound rather than the expected one, but they establish the scale of what has to be engineered away. Triangle counting and maximum directed cut, two other applications of the same primitive, cost more still. And nothing at n=10¹² has been executed. Execution stopped at 32 nodes, because the processor’s fidelity is not sufficient for an unconditional space advantage without fault tolerance.

The Niroula figures apply only to Hidden Matching. Niroula and colleagues posted their preprint on 5 November 2025, five months before Zhao and colleagues posted theirs, so it is not and cannot be a compilation of the oracle-sketching algorithm above. The two use related but differently named primitives, Niroula’s pair sketch and Zhao’s oracle sketching, and Zhao’s paper gives no fault-tolerant resource estimate of its own. Nobody has compiled the classification, dimension-reduction or linear-system algorithms to a fault-tolerant architecture. Hidden Matching is the nearest costed relative in the family and the only one with a number, which is why it has a row in this table. The August version of this article introduced these figures as the gate bill for Zhao’s advantage, and they are not.

Width and T-count trade against each other continuously, as the FeMoco history in the next section shows. Memory trades against total gate work instead, and there the floor is proven. Every sample must be touched by a gate, so the algorithm cannot do less than Õ(N) gate operations in the space-efficient regime.

Whether that gate work becomes wall-clock time is a separate question, and an open one. Zhao and colleagues point out that their algorithm is largely built from commuting operations, which leaves room for parallel scheduling they have not yet worked out. Data-delivery bandwidth, parallel processors, and error-correction architecture all determine how long a given gate count takes to execute. The memory advantage is bought with an enormous amount of gate work whose serialization is undetermined. That is a weaker claim than a depth theorem, and a considerably harder one to plan around.

Nobody has published a crossover for a real task, meaning the specific N and D at which the memory advantage begins to pay, the Toffoli count that implies, and the logical clock rate required to execute it. Joseph Bowles, whose 2024 benchmarking paper deflated much of the earlier quantum machine learning literature, put that question to the authors on SciRate. Lead author Haimeng Zhao agreed the numbers would be valuable and declined to produce them, on the grounds that doing so would mean committing to a specific fault-tolerant architecture, which a general algorithms paper deliberately avoids. That is a reasonable position for a theory paper and an unhelpful one for anyone doing capital planning. Bowles’s own read was that to reach a regime where the space advantage matters, “you’d need huge runtimes, which puts practicality into question“. Until someone publishes that crossover, I am moving machine learning on the ladder but leaving the hardware timeline as it was.

The Resource Estimate Revolution

One of the most remarkable, and underreported, stories of the past decade is how dramatically fault-tolerant resource estimates have fallen. Every major quantum algorithm has seen its requirements drop by orders of magnitude through better compilation, smarter Hamiltonian decomposition, and architectural innovations.

The canonical example is the FeMo-cofactor of nitrogenase, the enzyme that fixes atmospheric nitrogen at ambient temperature, a trick that the industrial Haber-Bosch process replicates only at 450°C and 200 atmospheres while ammonia production consumes around 2% of total final energy consumption worldwide, or 8.6 exajoules, on the IEA’s accounting. In 2017, Reiher, Wiebe, Svore, Wecker, and Troyer published the first serious resource estimate for simulating FeMoco on a quantum computer: approximately 111 logical qubits, but an astonishing 10¹⁴ to 10¹⁵ T-gates using second-order Trotterization, for target accuracies of 1 and 0.1 millihartree. The paper put those runs at 12 and 130 days by assuming a logical T gate every 10 nanoseconds; at one per microsecond, the same counts take about three years and 35 years.

Then the algorithms got better. Berry, Gidney, and colleagues introduced sparse qubitization in 2019, bringing the T-count down to ~10¹⁰. Lee et al. applied tensor hypercontraction in 2021, reaching 2,142 logical qubits and 5.3×10⁹ Toffoli gates, roughly four days at 4 million physical qubits. Rocca et al. (2024) halved that further with symmetry-compressed double factorization. And in 2025, Low, Berry, Rubin et al. reported speedups of 4× to 195× over the previous state of the art for ground-state energy estimation, through spectrum amplification, while Caesura et al. (PsiQuantum/Boehringer Ingelheim) estimated a 278× reduction in quantum runtime with BLISS-THC factorization on a photonic active-volume architecture. Both factors compare new quantum compilations with earlier quantum compilations; neither involves a classical computer.

Five orders of magnitude in eight years. The algorithm didn’t change; it’s still quantum phase estimation applied to an electronic Hamiltonian. What changed was how cleverly the Hamiltonian was decomposed and how efficiently the non-Clifford operations were compiled.

This pattern repeats across every major application. And it means any “utility ladder” I present today will look conservative by 2030. Resource estimates are still falling.

Rung 1: The Scientific Beachhead at 25–100 Logical Qubits

The first clear scientific advantages from fault-tolerant quantum computing need only tens of logical qubits, paired with the ability to execute millions to trillions of gates without losing coherence.

A 2025 perspective paper published in the Journal of Chemical Theory and Computation explicitly analyzed the 25–100 logical qubit regime and identified it as the threshold where early fault-tolerant machines can tackle scientifically meaningful electronic-structure problems that are qualitatively beyond what classical solvers handle reliably. The key targets are the bread-and-butter problems where classical quantum chemistry hits a wall: multireference charge-transfer states, conical intersections in photochemistry, and small catalytic fragments with strong electron correlation.

Glycine ground-state energy has been costed in the QuRE resource estimation framework at 55 logical qubits using quantum phase estimation, making it a concrete early target. The width is modest; the challenge is the gate count, which the 2013 study places at 10¹⁰–10¹¹ logical operations before synthesis. That is a historical estimate on an older cost model, and it is not directly comparable to the Toffoli counts from modern qubitization work elsewhere on this ladder.

Fermi-Hubbard models on classically intractable lattices represent perhaps the strongest near-term candidate for a scientifically meaningful quantum computation. Campbell’s plaquette Trotterization (2022) needs approximately 10⁶ Toffoli-equivalent gates for 2D Fermi-Hubbard QPE at L=8 and u/t=4, a lattice beyond exact classical diagonalization, on 162 logical qubits: 128 for the system and 34 for phasing ancillas and phase estimation. Kan (PsiQuantum) and Symons (Oxford) pushed further in 2025, achieving ~8×10⁵ Toffolis for Fermi-Hubbard and ~5×10⁶ for single-orbital cuprate models using catalyzed Hamming-weight phasing.

Google Quantum AI’s Stage-IV quantum utility compendium lists three quench simulations as concrete early targets: a 121-site 2D Ising model at 140 logical qubits, a 70-site Heisenberg model at 131, and a 128-spin-orbital 2D Hubbard model at 160. These are condensed matter model dynamics, a long way from the chemistry stories that excite pharmaceutical executives. They are nonetheless among the strongest candidates for the first classically intractable logical-qubit computation.

In March 2026, IBM and the DOE Quantum Science Center hit a related milestone: a 50-qubit Heron processor simulation of KCuF₃ magnetic material that reproduced experimental inelastic-neutron-scattering spectra, which the authors describe as the first quantum benchmark validated against a physical experiment rather than a classical computation.

In May 2026, Q-CTRL pushed the pre-fault-tolerant frontier further with a 120-qubit digital simulation of the 1D Fermi-Hubbard model on IBM Heron hardware. Their compilation pipeline achieved over 9,000 two-qubit gates with ~1% RMSE accuracy against classical tensor-network benchmarks. The simulation took under three minutes on the hardware; their classical baseline (ITensor TDVP at bond dimension χ=4096) required over 100 hours to match it.

Two classical responses followed within weeks. Algorithmiq’s Majorana Propagation method matched single-site results in minutes on a laptop (covered in my editor’s note on the original Q-CTRL article), and Multiverse Computing (June 2026) used GPU-accelerated TDVP at bond dimension χ ≈ 62,000 to fully converge the entire simulation window, including the high-entanglement regime the original classical baseline had left unresolved, and on their accounting cut the reported 3,000× advantage to about 36×. The result is best read as a milestone in deep-circuit execution rather than a settled quantum advantage claim. The broader case that condensed-matter simulation is the natural beachhead for quantum computational advantage does not depend on any single experiment.

For this rung: The first fault-tolerant quantum advantage will likely appear in condensed-matter model dynamics and carefully curated active-space chemistry. It will be scientifically important but won’t transform industries. Think of it as the Wright Brothers moment. Proving the thing flies came first; airlines came later.

Rung 2: First Industrial Signals at 100–300 Logical Qubits

Several independent resource estimates, set out below, place the first results that could feed real industrial workflows in this range.

Rovibrational Spectroscopy

One of the most striking concrete examples comes from a 2025 algorithm for high-accuracy nuclear-motion Hamiltonians using a Walsh-Hadamard QROM architecture. For a 30-dimensional model system representing a 12-atom molecule with six-body coupled potential, classical calculation of spectroscopic-accuracy energy levels would require over 30,000 years on the world’s fastest supercomputer under the paper’s classical cost model. The optimized quantum algorithm requires fewer than 300 logical qubits and completes in approximately three months on a 1 MHz fault-tolerant processor. This reduces the required quantum computational volume by a factor of 10⁵ to 10⁶ compared to earlier quantum approaches.

Astrochemistry, atmospheric modeling, and molecular spectroscopy all depend on precise knowledge of how molecules vibrate and rotate.

NMR Spectral Prediction

A 2024 resource estimation paper established zero-to-ultralow-field NMR spectral simulation as an early FTQC candidate. Natural products and small proteins could be simulated with a few hundred logical qubits and fewer than 10¹² T-gates, corresponding to runtimes of days on plausible hardware. This use case has a powerful advantage: the results can be directly verified against laboratory NMR experiments, providing a natural “proof of quantum utility” channel.

OLED Materials Discovery

Organic light-emitting diode emitters built around heavy transition metals (platinum, iridium) bring electron correlation and relativistic effects into the design problem. Classical quantum-inspired methods such as Ryabinkin et al.’s iQCC approach (2025) have reached an 80-qubit Hamiltonian for Ir(F₂ppy)₃, an industrially relevant iridium emitter, running entirely on classical computers. Nobody has published a fault-tolerant resource estimate for these complexes, so OLED emitters belong on this ladder as an uncosted candidate, not at a measured rung.

Emitter design keeps appearing in quantum roadmaps because the display market is large; what it lacks is a compiled fault-tolerant estimate and a matched classical benchmark.

Photodynamic Cancer Therapy

Photosensitizer design for photodynamic cancer therapy (PDT) requires exact modeling of excited electronic states, particularly the intersystem crossing from singlet to triplet states that generates the reactive oxygen species that destroy tumors. Classical methods struggle with the heavily correlated excited states of advanced BODIPY-derivative photosensitizers.

In late 2025, Xanadu researchers published fault-tolerant resource estimates specifically targeting cumulative absorption and intersystem crossing rates in these BODIPY systems. Their estimates indicate that simulating active spaces ranging from 11 to 45 spatial orbitals requires 180–350 logical qubits and Toffoli gate depths between 10⁷ and 10⁹.

A cancer drug design calculation with Toffoli depths from tens of millions to a billion, within algorithmic reach of hardware projected for 2029–2031, is not the usual hand-wave. The caveat is that a logical-width estimate does not establish the physical-qubit count, factory demand, repetitions, or wall-clock runtime, so this is a comparatively favorable algorithmic resource profile rather than a delivery date.

What This Range Does Not Yet Do

A machine with 100–300 logical qubits cannot simulate FeMoco at full scale. It cannot model complete catalytic cycles. It cannot run quantum amplitude estimation for derivative pricing, which requires roughly 4,700 to 8,000 logical qubits. It cannot solve industrial-scale optimization problems.

What it can do is produce scientific results that are (a) beyond classical exact methods, (b) verifiable against laboratory experiments, and (c) embedded in workflows that inform real industrial decisions about molecules and materials. That is a genuine inflection point: classical computing augmented at specific bottlenecks where it cannot reach the required accuracy alone.

Rung 3: The Early Industrial Era at 300–1,000 Logical Qubits

Between 300 and 1,000 logical qubits, the character of accessible problems changes. We move from benchmark-scale active spaces to systems that start looking like real industrial targets.

Battery Materials

The battery industry represents one of the most compelling near-term quantum computing applications, because the key bottleneck, understanding degradation mechanisms at the atomic level, involves exactly the kind of strongly correlated electronic states that classical DFT handles poorly.

Lithium-rich NMC (Nickel Manganese Cobalt) cathodes theoretically offer massive energy densities but suffer from structural degradation and voltage fade. Understanding this requires simulating Resonant Inelastic X-ray Scattering (RIXS) spectra, a second-order spectroscopic technique that probes the correlated electronic excited states responsible for degradation.

Earlier plane-wave estimates for related battery-cathode ground-state problems ran to more than 2,000 logical qubits and 10¹³ Toffoli gates, a completely impractical requirement. Xanadu has changed the spectroscopy picture with two sets of partners. For X-ray absorption spectroscopy (XAS), Xanadu and Volkswagen applied compressed double-factorized Hamiltonians and Lorentzian-kernel sampling to bring a Li₄Mn₂O cluster down to 100 workspace logical qubits and under 4×10⁸ T gates for the longest single circuit; sampling the full spectrum for that active space costs about 4.58×10¹¹ Toffolis. For the more complex RIXS spectra, a February 2026 preprint from Xanadu with the National Research Council of Canada and university partners reached roughly 414 logical qubits and 2.0×10¹⁰ Toffoli gates per shot. That makes battery degradation spectra a concrete algorithmic target, and still a long way from an industrial crossover.

Separately, a first-quantization approach to the LiNiO₂ cathode problem uses a system register of only 1,380 logical qubits, compared to roughly 75,000 in second-quantized encoding, a dramatic illustration of how representation choice can shift a problem by nearly two orders of magnitude. The lower-Toffoli configuration adds roughly 8,000 ancillas on top of that register, so the total logical width is closer to 9,400.

Gene Regulatory Networks

In systems biology, genetic regulatory networks modeled as Boolean networks require finding structural attractors that dictate cellular phenotypes. Rossini and colleagues’ attractor-search algorithm uses a register of (T+1)·n qubits for T state transitions of an n-gene network, and its worked examples have four and five genes. Applied to contemporary models of 40 to 50 genes followed for ten time steps, that formula gives roughly 440 to 550 logical qubits. That is my width-only extrapolation, not an end-to-end resource estimate. The search is Grover-inspired, so it inherits the quadratic-speedup problem discussed below.

Thermal State Preparation and Surface Chemistry

Open-system quantum simulation of strongly correlated materials at finite temperature is one of the clearest use cases for this qubit range. Mozgunov, Marshall, and Anand (2024) estimated that one run of a 10×10 Hubbard model, of the kind a metal-insulator-transition materials search would need many of, requires roughly 600 logical qubits and 3.3×10⁹ T-gates, about 11 hours on 2.9 million physical qubits. Their Gibbs state preparation discussion identifies surface chemistry and magnetic materials as the most promising targets but defers specific resource estimates to companion publications.

Chemical Dynamics

A 2026 paper on pre-Born-Oppenheimer quantum dynamics provides detailed resource estimates, at widths just above this rung, for simulating real chemical reactions with full electron-nuclear coupling. NH₃ + BF₃ requires 1,362 logical qubits (312 ancilla) and 8.72×10⁹ Toffolis per femtosecond of simulated time. The paper gives similar widths for 2NO₂ (1,419 LQ), C₂H₄ + O₃ (1,341 LQ), and C₂H₄ + O₂ (1,453 LQ), at 8.05×10⁹ to 7.07×10¹⁰ Toffolis per femtosecond. The per-femtosecond qualifier is central rather than incidental, since a chemically useful trajectory needs many femtoseconds. This is the regime where quantum computers start modeling actual chemical reactions as they unfold rather than as static snapshots of molecular ground states.

Tensor PCA for Machine Learning

An end-to-end algorithmic framework for Tensor Principal Component Analysis and planted kXOR problems demonstrates that 900 logical qubits can solve problems requiring approximately 10²³ classical FLOPs, a superquadratic quantum speedup for a planted problem under the paper’s stated parameter and classical-baseline assumptions. The quantum algorithm demands approximately 10¹⁵ total gates and a gate depth of 10¹², which is staggering but still dramatically better than the classical alternative. Without recent compilation improvements, the quantum gate count would have exceeded 10¹⁹.

When I first published this article, Tensor PCA was the only example outside chemistry, physics and cryptanalysis of a fault-tolerant quantum algorithm with a provable advantage that held up under end-to-end resource analysis. It now has company. The space-advantage results described above need under 60 logical qubits instead of 900, and unlike Tensor PCA their separation is unconditional. What they share with Tensor PCA is the gate bill.

Rung 4: Grand Challenge Chemistry at 1,000–5,000 Logical Qubits

The chemistry problems in vendor pitch decks (FeMoco, cytochrome P450, CO₂ catalysts) begin at this rung, and each now has a published resource estimate.

The FeMoco Problem at Scale

Google Quantum AI’s Stage-IV compendium places the full FeMoco nitrogen fixation calculation at approximately 1,500 logical qubits for a 76-orbital problem, with a quoted runtime of about 9 hours under their stated assumptions. The tensor hypercontraction estimates from Lee et al. (2021) settle at 2,142 logical qubits and 5.3×10⁹ Toffoli gates, running in under four days, and remain the most complete published physical-resource analysis even though the algorithmic frontier has moved past them.

Understanding how nitrogenase fixes nitrogen at ambient conditions could change ammonia production, which the IEA puts at around 2% of total final energy consumption and 450 million tonnes of direct CO₂ emissions a year. The Bellonzi et al. study on homogeneous nitrogen fixation catalysts is one of the few papers that connects physics, economics, and runtime. It models the highest-utility calculation at approximately $200,000 of economic value, against an estimated 139,000 QPU-hours and roughly 400,000 CPU-hours for the classical calculations; QPU-hours and CPU-hours are unlike resources, so the two figures do not divide into a speed or cost ratio.

Cytochrome P450

Cytochrome P450 enzymes metabolize the majority of marketed drugs, with CYP3A4 alone commonly estimated at around half, through extreme spin-state fluctuations at their heme cores. Goings et al. (PNAS 2022) compiled the tensor-hypercontraction configuration to ~7.8×10⁹ Toffoli gates and 73 hours of runtime on roughly 4.6 million physical qubits, and the comparison table in Caesura et al. (2025) lists that configuration at 1,434 logical memory qubits. The same table sets it against double factorization at about 4,922 qubits and 1.9×10¹⁰ Toffolis and against Caesura’s own BLISS-THC factorization at about 999 qubits and 1.7×10⁹ Toffolis. Caesura’s BLISS-THC runtime comes out 234× below the paper’s own THC reference run on a baseline architecture, combining 25.18× from compiling for the active-volume architecture, 8.23× from BLISS-THC and 1.12× from circuit improvements (the paper’s Table I). Google’s compendium lists a 58-orbital configuration at about 1,200 logical qubits and 5×10⁸ Toffolis. Those figures make P450 a plausible target for machines in the 2030–2035 timeframe.

Ruthenium CO₂ Catalyst

Von Burg et al. estimated that a ruthenium catalyst for CO₂-to-methanol conversion requires approximately 4,000 logical qubits and ~10¹⁰ Toffoli gates, which at 10 μs per Toffoli comes to 28 hours. Google’s compendium lists a later 56-orbital configuration at roughly 900 logical qubits and 2×10⁸ Toffolis, so this should be read as a historical end-to-end estimate, not the current frontier. This is the kind of computation that could accelerate the design of carbon utilization catalysts, a climate application with obvious urgency.

The Chemistry Advantage Debate

Any honest assessment of this rung must contend with the Lee et al. analysis (Nature Communications 14:1952, 2023), which examined the empirical performance of classical methods (DMRG, CCSD(T), FCIQMC, selected CI) across chemical space and concluded that evidence for exponential quantum advantage in ground-state chemistry has yet to be found. Features that enable efficient quantum state preparation tend to also enable efficient classical heuristics.

Quantum chemistry simulations remain useful. The analysis weakens the case for assuming a generic exponential advantage and shifts attention toward system-specific polynomial advantages and accuracy regimes where classical heuristics fail, and the “killer app” framing of 2017 has quietly been revised to match. The quantum advantage in chemistry is likely to be real but narrower than the early hype suggested: specific strongly correlated systems where classical methods cannot achieve the required accuracy, embedded in larger hybrid quantum-classical workflows.

Rung 5: Materials at Scale and Fundamental Physics at 5,000–100,000+ Logical Qubits

Past 5,000 logical qubits, the problems are materials at industrial scale and fundamental physics, on timelines that start in 2033.

Inertial Confinement Fusion

Rubin, Berry, Baczewski et al. (Google-Sandia, PNAS 2024) costed α-particle stopping power in warm-dense matter, directly relevant to National Ignition Facility target design. Their own framing is roughly as many logical qubits as state-of-the-art FeMoco and P450 simulations and about a hundred times more Toffoli gates. Google’s compendium lists the 218-electron calculation at about 6,200 logical qubits and 10¹⁵ Toffolis. That makes it the clearest fusion-energy quantum application on width and one of the most expensive on gate count. Tokamak MHD, by contrast, has no credible FTQC resource estimate.

Battery Electrolytes at Scale

Google’s compendium places a LiPF₆ battery electrolyte problem at roughly 18,000 logical qubits and 9×10¹¹ Toffoli gates, the scale where quantum simulation moves from heroic individual calculations to modeling system sizes that map naturally onto real industrial R&D needs.

Bulk Solid-State Physics

For solid-state lattice structures like NiO and PdO, calculating exact ground-state energies requires double- or triple-ζ polarized basis sets to accurately model bulk material properties. While small-unit-cell simulations (8–16 atoms) need only a few thousand logical qubits, larger supercells (up to 72 atoms) require approximately 100,000 logical qubits. Under the cited compilation, at a 0.01% physical error rate, that is about 65 million physical qubits. The physical-qubit figure is highly code- and error-rate-dependent; other compilations put the same workload nearer 200 million physical qubits under different assumptions.

Lattice Gauge Theory

The crown jewel of quantum physics simulation. Rhodes, Kreshchuk, and Pathak (2024) reported up to 25 orders of magnitude reduction in spacetime volume for non-Abelian SU(2)/SU(3) lattice gauge theories via qubitization, measured against a particular earlier Trotterized construction at selected parameters rather than as a generic result. Quantum simulation avoids QCD’s notorious sign problem by construction for coherent real-time evolution, since Hamiltonian simulation dispenses with importance sampling, though state preparation, continuum extrapolation, gauge truncation, and observable estimation all remain unresolved costs. Full 3+1D QCD simulation remains well beyond current estimates.

Why Finance and Optimization Face the Quadratic Speedup Barrier

I need to address the elephant in the room. Every quantum computing pitch deck includes slides about portfolio optimization, derivative pricing, vehicle routing, and machine learning. The evidence for quantum advantage in these domains is dramatically weaker than for chemistry and physics. The reason is mostly arithmetic: at the logical clock rates on any published roadmap, a square-root improvement takes impractically long runtimes to recover a physical-to-logical overhead of 10³–10⁴.

A quantum advantage shows up in three shapes of problem: problems whose answer is a quantum state or a spectral quantity, problems with a hidden global algebraic structure, and generic search, where the advantage is quadratic and proved to stop there. Combinatorial optimization, the thing most people picture when they hear “hard problem,” falls almost entirely into the third. Finance and logistics problems fall into the same category.

Babbush, McClean, Newman, Gidney, Boixo, and Neven in their 2021 paper “Focus Beyond Quadratic Speedups for Error-Corrected Quantum Advantage” made the arithmetic explicit. Surface-code logical clock rates (~10 kHz today, potentially ~1 MHz long term), combined with physical-to-logical overhead of 10³–10⁴, make quadratic quantum speedups uncompetitive against massively parallelized classical heuristics absent unusually favorable constants or major architectural improvements. Grover-type search, quantum amplitude estimation for Monte Carlo, and quantum-accelerated simulated annealing all provide quadratic speedups. (I’ve written separately about why Grover’s algorithm won’t kill AES; the same argument applies across all these domains.)

Sanders, Berry, Gidney, Babbush, and colleagues quantified this concretely: quantum-accelerated simulated annealing would require approximately one day and one million physical qubits to match what classical simulated annealing solves in four CPU-minutes on spin-glass instances.

Derivative Pricing

Goldman Sachs and IBM’s resource estimate for pricing a 3-asset autocallable basket option came to ~8,000 logical qubits, T-depth 5.4×10⁷, total ~10¹⁰ T-gates, requiring roughly 5.4 seconds at a 10 MHz logical clock or under one second at ~54 MHz, either way about three orders of magnitude above the 10 kHz logical rates those papers cite as feasible. Stamatopoulos and Zeng (2024) used quantum signal processing to cut this to ~4,700 logical qubits and ~10⁹ T-gates but still need ~45 MHz T-gate throughput. The global OTC derivatives market is $846 trillion in notional outstanding (BIS, mid-2025), against a gross market value of roughly $21.8 trillion in the same release, so the addressable value is large but nowhere near the notional headline. Either way the resource gap is 3–4 orders of magnitude from realistic hardware.

Portfolio Optimization

The Goldman Sachs-AWS end-to-end resource analysis of quantum interior-point methods for portfolio optimization (Dalzell et al., PRX Quantum 4:040325, 2023) found per-iteration T-counts of 10²²–10²⁴ on 100-stock problems because tomography dominates. The authors explicitly conclude QIPM is not practically advantageous for portfolio optimization without fundamental algorithmic improvements.

Machine Learning’s Great Dequantization

Ewin Tang’s 2018 breakthrough demonstrated that the Kerenidis-Prakash exponential speedup for recommendation systems evaporates under matching classical sampling access. Subsequent work extended this to quantum SVM, PCA, low-rank regression, and discriminant analysis. Three categories were not dequantized: algorithms on quantum inputs (learning from quantum experiments), Hamiltonian-simulation-based linear systems without QRAM, and, since April 2026, space-bounded processing of streamed classical data. Dequantization is an argument about query complexity, and standard sample-and-query results do not, on their own, rule out an exponential separation in working memory under a streaming access model.

Zhao and colleagues make the consequence explicit. An algorithm with only a polynomial query speedup, and therefore dequantizable in the usual sense, can still carry an exponential space advantage. The authors suggest the space advantage may extend to problems previously thought to be dequantized, and name recommendation systems, Ewin Tang’s original target. Nobody has done that work yet, so I would not move recommendation systems on this ladder until someone does. Variational quantum machine learning faces separate evidence against scalability: barren plateaus, training landscape traps, and NP-hardness of optimization. Some proposed methods for avoiding barren plateaus confine training to locally classically approximable regions, which weakens their case for quantum advantage, though trainability alone does not generically imply efficient classical simulability.

For time-to-solution, finance and optimization are the weakest cases for fault-tolerant quantum advantage. Generic ML has no quantum advantage that anyone has shown how to execute. When I first wrote this section, the path forward required either (a) discovering a super-polynomial speedup for a specific problem class, or (b) engineering logical clock rates three orders of magnitude beyond current projections.

Something close to path (a) has now happened, on a different axis. Theoretical learning separations under contrived oracle and data-access models have existed for years and moved nobody’s roadmap. This one is on streamed classical data, on recognizable tasks, with an unconditional lower bound and memory figures on real datasets. Its separation is in memory rather than time, which is why a decade of dequantization results, all framed in query complexity, did not rule it out. But executing that advantage at commercial data scale leans on path (b), because the Õ(N) gate floor turns into wall-clock time at whatever logical rate the hardware runs, unless the commuting structure parallelizes well. Both paths now run largely through (b), and the streaming-ML advantage and the finance applications depend heavily on the same quantity, the logical clock rate.

The Comprehensive Utility Table

The table below consolidates selected published quantum resource estimates by application domain and logical-qubit tier, at varying levels of completeness and maturity. Some rows are compiled end-to-end fault-tolerant estimates; others are algorithmic circuit costs, width-only extrapolations, or asymptotic results, and the notes say which. I’ve excluded cryptanalysis, since I’ve covered that extensively elsewhere, to focus on the positive utility map.

Early Fault-Tolerant (25–300 Logical Qubits)

Mid-Scale Fault-Tolerant (300–1,000 Logical Qubits)

Large-Scale Fault-Tolerant (1,000–10,000 Logical Qubits)

Extreme Scale (10,000+ Logical Qubits)

The Zhao and Niroula rows are the only ones in the table where the binding constraint is memory rather than logical width, and their qubit counts should be read alongside their gate counts rather than on their own. Both are preprints. Zhao et al. was revised to v2 on 1 October 2026 and still has no journal. The Niroula result was also still a preprint when Quantinuum’s Helios paper cited it in Nature in June 2026.

Three Patterns in the Ladder

First, physical-system simulation dominates. Of the applications with strong evidence for quantum advantage, the overwhelming majority involve simulating quantum mechanical systems: molecules, materials, lattice models, gauge theories. Quantum computers are literally made of quantum stuff, so this makes sense. They simulate quantum systems natively. Richard Feynman’s original 1982 insight remains the most accurate prediction in the field: quantum computers will simulate quantum systems.

Second, the center of gravity has shifted. The marquee application of 2017 was FeMoco, the full nitrogenase cofactor simulation requiring thousands of logical qubits. The most exciting near-term targets today are condensed-matter lattice models (Hubbard, cuprate, pnictide) at 100–500 logical qubits. These require 10⁶–10⁸ Toffoli gates rather than 10⁹–10¹³, putting them within reach of announced 2029 hardware. The first fault-tolerant scientific result, almost certainly a condensed-matter Hamiltonian or a photosensitizer calculation rather than FeMoco, is plausible before 2030.

Third, the utility map is narrower than advertised, for more reasons than the quadratic speedup barrier. Finance and logistics face that barrier. Vendor marketing treats them as equivalent application domains to chemistry and physics on evidence that is far weaker. Chemistry and physics applications have plausible super-polynomial advantages on specific strongly correlated systems and strong polynomial ones on many more. Lee et al. are the reason I no longer call them provable. The classical difficulty they meet comes from the mathematics of the problem, which more hardware does not change.

I now put machine learning on classical data in a third position, one that did not exist when I built this map. Its separation is stronger than anything in chemistry, because it is unconditional. Its engineering is unknown: the Õ(N) gate floor is proven, nobody has compiled the algorithms, and no crossover point has been published.

Error Correction Overhead and the Ladder’s Timelines

Every rung of this ladder depends on cheaper error correction, and the overhead is already falling.

The surface code, the workhorse of quantum error correction since the early 2000s, encodes one logical qubit per ~2d²-1 physical qubits, counting syndrome-measurement ancillas, where d is the code distance. At distance 25, that’s roughly 1,250 physical qubits per logical qubit. For 1,000 logical qubits at distance 25, you need 1.25 million physical qubits before magic state factories and routing.

qLDPC codes are changing the math. IBM’s bivariate bicycle “gross” code [[144,12,12]] encodes 12 logical qubits; the cited memory simulation used 144 data qubits and another 144 for checks, 24 physical qubits per logical qubit in total. The comparable surface-code memory used nearly 3,000 physical qubits for the same 12 logical memories, a roughly tenfold reduction. It is a simulated memory comparison, and it does not yet establish that universal computation, routing, lattice surgery, and factories keep the same ratio.

Magic state cultivation, developed by Gidney, Shutty, and Jones (2024), reduces the projected qubit-round cost of T-gate production by roughly an order of magnitude at selected operating points. It grows and checks T states inside a small color code and grafts them into the surface code, instead of distilling them in large factories, and Google demonstrated it on a superconducting processor in December 2025. Algorithmic fault tolerance (QuEra/Harvard/Yale, 2025) cuts simulated runtime overhead 10–100× by running one error-check round per logical layer rather than dozens.

These are architectural shifts that change which rungs of the utility ladder are reachable with which hardware. A 100,000-physical-qubit machine with surface codes gives you perhaps 80–100 logical qubits. Architecture studies of advanced qLDPC codes project 1,000–4,000 from the same machine, though those projections depend on connectivity, code family, decoder, physical error rate, and workload. That range is the difference between glycine ground-state calculations and FeMoco.

The compression kept coming through 2026. Mitten codes, built from non-abelian group algebras by a Caltech and Oratomic team in July, reach a 20% encoding rate at check weight 9, with a family running from [[150,30,10]] to [[975,195,≤24]]. Google’s boundary-connected planar modules, posted in October by fourteen researchers across Quantum AI, DeepMind, and MIT, needed ten times fewer physical qubits per logical qubit than the paper’s own modularized surface-code baseline in circuit-level simulation. The Mitten and Google results both come from simulation, and each counts data-qubit blocks on both sides, with no factories or routing attached. Each also depends on how its authors set up the comparison. I take them apart in The Error Correction Revolution. For this ladder, the estimated overhead keeps falling, and no machine has yet shown the lower overhead end to end while running an algorithm.

When Will Each Rung Be Reached?

Based on current vendor roadmaps and the pace of error-correction improvements:

2026–2028: 10–50 logical qubits at modest fidelity. Validation of QPE techniques, small lattice-model demonstrations. Scientific publications, not industrial workflows.

2028–2030: 100–300 logical qubits (IBM Starling at 200 LQ, QuEra targeting 256 LQ with Libra in 2028, Quantinuum Apollo at hundreds of LQ). First classically intractable lattice-model computations. Photosensitizer calculations. Early-FT chemistry with embedded active spaces. This is the “Wright Brothers” moment for fault-tolerant quantum utility.

In 2026 SoftBank and Quantinuum published a joint use-case study, which I went through in detail when it appeared, that indexes chemistry feasibility by spin orbitals rather than logical qubits. On Helios today, under error correction at code distance 3 to 5, they put the ceiling at roughly 10 spin orbitals at logical error rates near 10⁻³, and the same for Sol in 2027. Apollo reaches about 100 in 2029, with excited-state QPE at logical error rates near 10⁻⁷, and only Lumos in the 2030s reaches 100-plus at chemical accuracy. It is a vendor roadmap, framed by its authors as an assumption-driven envelope, not a forecast, and the dates should be read on those terms. FeMoco needs roughly 152 spin orbitals in one common model of the problem. The document therefore puts FeMoco-class chemistry later than my own 2030–2033 bracket, and Quantinuum has a commercial interest in the opposite answer. That makes it more interesting than its genre usually is, though not more reliable.

2030–2033: 500–2,000 logical qubits (IBM Blue Jay at 2,000 LQ). Battery materials RIXS. FeMoco-class calculations. Pre-Born-Oppenheimer dynamics. First results feeding into real pharmaceutical and materials R&D pipelines. In this bracket the economic case starts to close, and, as I’ve analyzed elsewhere, the part of the range from roughly 850 to 1,400 logical qubits overlaps with what cryptanalysis requires. The width is only half the question: a 414-qubit RIXS algorithm needing 2×10¹⁰ Toffolis does not become practical merely because a roadmap reaches 500 logical qubits, and IBM’s own gate budgets for this era fall below several of the workloads in the table above.

2033–2040: 5,000–100,000 logical qubits. Bulk solid-state physics. Full catalytic cycle modeling. ICF target design. Lattice gauge theory approaching QCD.

These timelines assume continued progress in error correction, algorithmic compression, and physical qubit quality. They are not predictions, just scenarios. The IonQ projection of up to 80,000 logical qubits by 2030 is the most aggressive publicly stated roadmap by a wide margin and depends on successful integration of Oxford Ionics’ chip-integrated ion traps. IonQ has backed it with two pieces of work: a simulated fault-tolerant blueprint (“Walking Cat Architecture”) mapping 110 logical qubits onto 2,514 physical ones using qLDPC codes, and a photonic interconnect linking two commercial trapped-ion systems, which validates one modular-networking component rather than the forecast itself. Whether these translate to the projected qubit counts by 2030 remains to be seen. IBM itself dates Blue Jay to “2033 or maybe beyond.”

What Organizations Should Do Now

The natural question for CISOs, CTOs, and R&D leaders reading this is: what should I do about it?

For organizations in chemistry, materials science, energy, and pharma: the time to build quantum computing teams and identify target calculations is now. The algorithmic widths of several early chemistry and condensed-matter targets overlap vendor roadmaps for 2028–2030, although their non-Clifford budgets and complete physical-resource requirements often do not. Use the intervening period to validate classical bottlenecks, construct active spaces, benchmark state preparation, and define experimentally meaningful observables, rather than assuming that reaching a nominal logical-qubit count will deliver utility. Organizations that postpone problem selection and workflow development until hardware is available risk entering the field several development cycles behind early participants.

For organizations in finance and logistics: maintain awareness but manage expectations. The evidence for near-term quantum advantage in these domains is weak. The quadratic speedup barrier is an engineering-economic obstacle under current architectures rather than a mathematical impossibility theorem, and it is not moving fast. Focus quantum computing investment on understanding your organization’s actual computational bottlenecks and whether any of them involve simulating quantum mechanical systems (e.g., materials science for manufacturing, molecular modeling for biotech). The near-term utility is in those simulation problems.

For everyone: the error-correction revolution means that physical qubit counts will translate into more logical qubits than anyone projected five years ago. The qLDPC compression is real and accelerating. Machines will reach these utility thresholds sooner than the raw physical-qubit roadmaps suggest.

Cryptanalysis May Arrive Before the Industrial Applications

The logical qubit requirements for breaking RSA-2048 (~1,400 active logical qubits in the current construction) and 256-bit elliptic curve cryptography (835 in the narrowest circuit yet published) are at or below the requirements for most of the grand challenge chemistry and materials applications on this ladder. FeMoco simulation requires 1,500 to 2,142 depending on the compilation. Cytochrome P450 requires roughly 1,000 to 4,900 depending on the decomposition. Full-scale materials modeling requires tens of thousands.

That 835 figure is three months old as I write and the circuit behind it has already been revised twice. A team from Tsinghua, Peking, and Sun Yat-Sen universities and the Chinese Academy of Sciences posted it in July 2026, taking roughly a third off their own April circuit, which had itself cut the count to 1,333. Measured against the correct prior figures, Babbush et al. at 1,175 and Chevignard’s corrected space-optimized P-256 at 1,193, it is a 29 to 30% reduction, and about a third of the 2017 Roetteler baseline of 2,330.

It is also the clearest example on this page of a width figure that has to be read with its gate count. In its current version, revised in September, the 835-qubit circuit spends roughly 2×10⁹ Toffoli gates, 24 to 34 times the 5.8×10⁷ to 8.1×10⁷ of the Google and Schrottenloher circuits. When it was posted in July I put the penalty at about twenty times, against their space-optimized variants; the factor moves with the version and with the circuit you compare against. The paper includes no depth analysis, so whether the trade nets out in wall-clock time is unknown. IonQ’s September 2026 estimate walks the same trade in the opposite direction on the same curve: 1,457 logical qubits, about 39 million Toffoli gates against Schrottenloher’s 58 million, and a complete compilation down to 19,397 physical ions running 25.7 days per attempt on the company’s Walking Cat architecture. The narrow circuit needs 57% of the wide one’s logical qubits, and nobody has published a runtime for it. Anyone quoting “835 qubits” without the gate penalty leaves out what that width costs.

Every figure in those two paragraphs, and every figure in the tables above, counts qubits that work. A team led from the University of Sydney measured what happens when they don’t. On ibm_miami, the first of IBM’s 120-qubit Nighthawk processors, June 2026 calibration put the median two-qubit gate error at 2.83×10⁻³ while the worst couplers ran at 1.69×10⁻¹, about sixty times higher. The team could not place a standard distance-5 rotated surface code anywhere on the chip without using a component they had already flagged as defective. Excluding the defective components recovered up to a factor of 2.8 in logical error rate. Craig Gidney’s sub-million-qubit RSA-2048 estimate, like the 20-million-qubit figure he published with Martin Ekerå in 2019, assumes a square grid in which every qubit and every coupler works and all of them work equally well. I know of no estimate on this page that prices dead components, and ibm_miami is a production processor, not a laboratory outlier.

Cryptanalysis therefore enters the same fault-tolerant resource band as grand-challenge chemistry, and may precede much of it. Recent P-256 estimates fall below many prominent chemistry workloads in both width and non-Clifford count, while current RSA-2048 and FeMoco estimates overlap. Which arrives first depends on architecture, connectivity, factory throughput, error budget, implementation quality, and what counts as a successful chemistry result.

That removes any basis for waiting for spectacular industrial chemistry as a warning signal. There is a widespread misconception that CRQC represents the hardest quantum computing milestone, and that the field will pass through a visible sequence of scientific achievements that provides years of warning. On the estimates in this ladder, cryptographic capability may arrive first for some primitives, contemporaneously for others, and without a long sequence of unmistakable commercial milestones.

In July 2026 Physical Review X published a Perspective by Hsin-Yuan Huang, Soonwon Choi, Jarrod McClean, and John Preskill proposing five properties, which they call keystones, that an ideal quantum advantage should satisfy: predictability, typicality, robustness, verifiability, and usefulness. Shor’s algorithm is their worked example for four of the five: predictability, typicality, robustness and verifiability. The framework is conceptual and does not rank arrival dates, but it supplies an independently constructed bar, and the cryptanalytic advantage is the example its authors use to illustrate clearing it. Nothing on the chemistry rungs of this ladder has been put through the same exercise.

Nicole Holzmann’s August 2026 perspective, accepted by Philosophical Transactions of the Royal Society A, explains why chemistry has further to travel once the quantum computation finishes. She groups what lies between a quantum computation and an industrial decision into six categories of consideration, each introducing its own approximations. Her first proposed milestone for the field is one complete pathway from a quantum-computed quantity to a decision-relevant outcome, and her second is a quantitative account of how the errors introduced along it propagate. Cryptanalysis has the shortest such path on this ladder. Shor’s algorithm has classical pre- and post-processing and probabilistic repetition, so its raw output is not literally the key, but a candidate answer is checked by arithmetic and nobody needs a model to interpret it. Holzmann also argues that the value of a decryption capability diminishes as standards move to post-quantum schemes. Migration does shrink the supply of newly vulnerable traffic. Harvested ciphertext is another matter: it stays under the algorithm it was sent with when the standard changes.

Organizations waiting for quantum chemistry breakthroughs as their signal to begin post-quantum cryptography migration are reading the ladder upside down. The harvest-now-decrypt-later threat is active today, and the migration itself takes years. The deadlines are already set; the ladder confirms why they are urgent.

The View from the Top of the Ladder

Fault-tolerant quantum computers will be narrowly useful first. Physical-system simulation (lattice models, strongly correlated chemistry, spectroscopy, materials dynamics) needs 10²–10³ logical qubits and from about 10⁶ Toffoli gates for the lattice models to 10¹⁰ and more for spectroscopy. That is within one order of magnitude of announced 2029 roadmaps on width, and often several orders beyond them on gate count.

Full-scale industrial chemistry (FeMoco, P450, catalytic cycles) needs 10³–10⁴ logical qubits, one rung higher, and is a 2030–2035 prospect under optimistic assumptions. And the evidence from Lee et al. (2023) weakens the case for assuming a generic exponential advantage there, making the case for quantum investment more nuanced than the early hype implied.

Finance and generic optimization face an arithmetic problem that better hardware alone is unlikely to fix. The gap between published resource estimates and hardware roadmaps is not narrowing fast enough to close before 2035 on quadratic-speedup applications. Streamed-data machine learning introduces a genuinely different possibility, an unconditional exponential advantage in memory, but no commercially relevant wall-clock crossover has yet been established.

Since 2022 the error-correction overhead has changed far more than the algorithmic map. qLDPC codes, magic state cultivation, and algorithmic fault tolerance are the levers that will determine whether the first meaningful fault-tolerant quantum computation is a 2028 Hubbard model, a 2031 FeMoco calculation, or something further out. Which of those it turns out to be depends on the complete fault-tolerant spacetime volume a machine can execute before the best classical method finishes, more than on its logical-qubit count.

From the author