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Where might a quantum computer actually earn its keep?

Chemistry and materials are the oldest reason to build one, and the place where the honest answer today is “not shown yet”. Here is what would have to be true, and who is checking.

You'll be able to say which proposed uses of a quantum computer have a real case and which are still hype.

Plain

The case in one idea

Electrons in a molecule are entangled with one another. Writing that down exactly takes one number for every arrangement of the electrons among the available orbitals, and the count of arrangements multiplies with every orbital you add. With 30 spin-orbitals, each either occupied or empty, there are 2³⁰, about a billion, arrangements; the standard model of FeMoco, 54 electrons in 54 orbitals, has around 10³⁰. No classical memory holds that. A register of 108 qubits holds it natively, because the machine is a quantum system. That is the one place where a quantum computer’s advantage is not a trick played on a puzzle: the problem is the machine’s own physics.

It is not automatic, though. Classical chemistry is not brute force. Methods such as density functional theory, coupled cluster and DMRG exploit structure, and for most molecules they are very good. A quantum machine only wins where the electrons are so strongly correlated that those shortcuts break, and whether such problems exist among the ones anybody needs solved is the open question.

Working

Nitrogenase, and a target that moved

Nitrogenase is the enzyme that turns nitrogen from air into ammonia at room temperature; its active site is a metal cluster called FeMoco. In 2017 Reiher, Wiebe, Svore, Wecker and Troyer proposed it as a quantum target and published resource estimates (PNAS 114, 7555). Later work cut the cost sharply: Lee and colleagues put a FeMoco simulation at about four million physical qubits and under four days, assuming a 1 µs cycle and physical gate errors no worse than 0.1% (PRX Quantum 2, 030305).

Then the classical side answered. In 2019 Li and colleagues in Garnet Chan’s group showed that the 54-orbital model used for those estimates “is not representative” of the real cofactor’s ground state, and proposed a replacement model as the benchmark (arXiv 1809.10307). In January 2026 the same group posted a preprint that estimates that model’s ground-state energy to chemical accuracy with coupled-cluster and DMRG calculations (arXiv 2601.04621). It is a preprint, and an estimate, not an exact solution. But the flagship problem is no longer safely out of classical reach.

The broader audit is in Nature Communications 14, 1952 (2023): the authors examined whether exponential quantum advantage exists for ground-state chemistry and concluded that evidence for it across chemical space has yet to be found. The question turns on whether the features that make a good starting state easy to prepare also let classical heuristics succeed, and their numerical studies found no evidence that they do not.

The near-term route hits a measurement wall

Machines that exist today are too noisy for the fault-tolerant algorithms above, so the near-term proposal is a hybrid loop, the variational quantum eigensolver (how it works, in The Machinery). Gonthier and colleagues counted what it would cost to get the combustion energies of small organic molecules to chemical accuracy and concluded that current techniques “will not be sufficient to achieve practical quantum computational advantage” for their molecule set (Phys. Rev. Research 4, 033154). The bottleneck is not the qubits but the number of repeated measurements.

Finance: real speedup, heavy overhead

Pricing a derivative is a Monte Carlo average, and amplitude estimation gives a real quadratic speedup on those. The catch is the overhead. The most complete resource estimate, by Chakrabarti and colleagues, needs about 8,000 logical qubits and a T-depth of 54 million, and to match one second of classical pricing it needs a logical clock rate of 50 MHz (Quantum 5, 463). Fifty million logical clock cycles per second is the bar the hardware has to clear; nobody has shown it yet.

Hybrid quantum–classical: the 2026 bet

How the circuit is designed

The machines of 2026 are too noisy to run long, exact algorithms, so the workable design is a short parameterised circuit inside a classical loop. Designing one is a series of trade-offs, in this order:

1. Write the problem as a cost. A molecule or material becomes a Hamiltonian H; a scheduling or portfolio problem becomes a cost function over bits. Lower cost means a better answer.
2. Choose the ansatz, the circuit shape. A hardware-efficient ansatz uses only the gates and qubit connections the chip does natively, so it is shallow but knows nothing about the problem. A problem-inspired one (chemistry-style or QAOA-style) builds the structure of the problem in, so it needs fewer parameters but more gates.
3. Spend the depth budget. Every two-qubit gate adds noise, so the circuit has to stay inside what the chip can run. IBM’s Nighthawk, 120 qubits, is aimed at circuits of about 5,000 two-qubit gates, with 7,500 targeted by the end of 2026 (The Quantum Insider).
4. Measure, then let the classical side steer. The circuit is run thousands of times, the average cost is computed, and a classical optimiser nudges the circuit’s angles θ and tries again until the cost stops falling.

VQE and QAOA, side by side

The full circuits for both are in The Machinery, topic 16.

How it is expected to help

Why “expected” is the right word

Variational circuits have a known trainability problem: on larger circuits the cost landscape can go almost flat, so the optimiser has nothing to follow (“barren plateaus”, McClean et al., arXiv 1803.11173), and hardware noise can cause the same thing (Wang et al., arXiv 2007.14384). The measurement count also balloons, as the VQE section above says. And every claimed advantage has to survive classical methods that keep improving, which is the nitrogenase story again. So the realistic 2026 outcome is one or a few carefully checked, narrow wins, not drug discovery or a trading desk changing overnight.

Formal

The arithmetic behind the 108 qubits

In second quantisation the electronic Hamiltonian is

where p, q, r, s run over spin-orbitals. The Jordan–Wigner mapping gives each spin-orbital one qubit, so N spatial orbitals need 2N qubits. For the FeMoco model, 54 electrons in 54 spatial orbitals, that is 108 qubits and a state space of 2¹⁰⁸ ≈ 3.2 × 10³². Fixing 27 spin-up and 27 spin-down electrons leaves C(54,27)² ≈ 3.7 × 10³⁰ amplitudes; stored as one 16-byte complex number each, that is about 6 × 10³¹ bytes. (A minimal-basis H₂ molecule, by contrast, is 2 spatial orbitals, 4 qubits, 16 amplitudes.)

Chemical accuracy is 1 kcal/mol, which is 1.59 milli-hartree (1 hartree = 627.5 kcal/mol). Quantum phase estimation reads the energy out to a precision ε with a cost that grows as 1/ε, and for the qubitised version the runtime scales as λ/ε, where λ is the 1-norm of the Hamiltonian’s coefficients. That is why the tensor-hypercontraction factorisation behind the four-million-qubit estimate matters: it shrinks λ.

The catch is the starting state. Phase estimation returns the eigenvalue of whichever eigenstate the input overlaps with. If your guess has overlap γ with the true ground state, each attempt succeeds with probability γ², so you repeat about 1/γ² times (about 1/γ with amplitude amplification). If γ falls exponentially as molecules grow, so does the cost, and the exponential advantage disappears. Whether good overlap is available exactly where the classical heuristics fail is the question for which the 2023 audit found no evidence.

Every figure in this section is computed in a script; the quoted resource estimates are the papers’, linked above.

Go deeper

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