jacq.io

Will quantum bring down the cost of training frontier ai models?

It’s always good – and humbling – to think about real‑world applications of quantum. Especially when the challenge is posed by my man. He never lets me get away with a yes, no, or maybe. He is a mathematician, after all. He wants proofs, not vibes.

His question was simple and brutal:

Can quantum computing bring down the cost of training frontier AI models?

And because I am still only an aspiring PhD candidate, I did what any honest physicist‑in‑training does: I went back to first principles, wrote down the optimisation equations, and followed the maths to its conclusion.

Here is my verdict — technical, defensible, and grounded in the current state of quantum hardware:

Quantum computing is unlikely to materially reduce the end‑to‑end cost of training frontier AI models over the next decade.

The more realistic possibility is niche, hybrid acceleration: quantum devices may eventually help with specific subroutines — optimisation, sampling, certain linear‑algebra primitives, model compression, or quantum‑native data tasks — but current and near‑term systems do not look capable of replacing GPU/TPU/ASIC clusters for dense transformer training.

Why? Because the optimisation landscape of frontier AI is brutally classical.

Training a frontier‑scale transformer is, at its core, a massive optimisation problem:

$$\underset{\mathit{\theta }}{\min }L(\mathit{\theta })={\mathbb{E}}_{x\sim \mathcal{D}}[\mathrm{\ell }(f_{\mathit{\theta }}(x),y)]$$

where:

• $\mathit{\theta }$ are billions of parameters,
• $L$ is the loss functional,
• $\mathrm{\ell }$ is the token‑level loss,
• and $\mathcal{D}$ is a dataset measured in petabytes.

Quantum computers do not (yet) offer meaningful speedups for this kind of dense, gradient‑based optimisation.

What they can accelerate — in theory — are subroutines like:

• quantum‑enhanced sampling
• quantum‑accelerated linear algebra
• quantum‑inspired tensor‑network compression
• variational quantum optimisation for specific bottlenecks

For example, a quantum variational subroutine might optimise a hard component of a model:

$$C(\mathit{\theta })=⟨\psi (\mathit{\theta })\mid H\mid \psi (\mathit{\theta })⟩$$

or a quantum optimal‑control routine might minimise a functional over control fields:

$$J[\{u_j(t)\}]=1-\mathrm{Tr}\text{\hspace{0.17em}⁣}({\rho }_T\rho (T))+\lambda \sum _j{\int }_0^T\text{\hspace{0.17em}⁣}\mid u_j(t){\mid }^{2}dt\mathrm{.}$$

These are beautiful equations — but they do not replace the dense matrix multiplications and massive parallelism that dominate transformer training.

And this matters because of the cost structure.

In the current economics of frontier AI, the expensive parts are:

• hardware (47–67%)
• R&D staff (29–49%)
• energy (2–6%)

(Epoch AI’s estimates.)

Electricity is a small share of direct cost – but power capacity is becoming a major infrastructure constraint. Gemini Ultra is estimated at 35 MW, and naive extrapolation implies gigawatt‑scale AI supercomputers by 2029.

The IEA projects data‑centre electricity demand to more than double to around 945 TWh by 2030, with AI‑optimised data centres’ electricity demand more than quadrupling.

Quantum does not meaningfully change this trajectory in the next decade.

So what does quantum realistically offer?

A future where quantum devices:

• accelerate specific optimisation kernels
• compress models using quantum‑inspired tensor methods
• improve sampling for generative models
• enable quantum‑native data pipelines
• support hybrid quantum‑classical training loops

This is not nothing. It is intellectually exciting. It is mathematically rich. And it is exactly the kind of frontier where my man and I end up debating late into the night — him demanding proofs, me sketching Hamiltonians on scrap paper.

But it is not a revolution in AI training cost. Not yet.

My conclusion – and the line I will tell him over a romantic candlelit dinner:

Quantum will not make frontier AI cheap. But it may make frontier AI different.

And sometimes, that is the more interesting answer.

Photo: me at the start of this challenge, trying to figure out Quantum Bayesianism (QBism), my starting point.

Download this post in pdf: https://jacq.io/wp-content/uploads/2026/08/Will-quantum-make-training-AI-frontier-models-cheaper.pdf