We published a new blog post on our quantum research page (QuanTAS), about the symbol that every quantum linear solver quotes its cost in — the condition number \(\kappa\) — and what it is actually measuring.

Solving \(Ax=b\) sits underneath quantum algorithms for differential equations, optimization, and machine learning. The advertised prize is an exponential speedup in the dimension. The fine print is always \(\kappa\): runtimes are quoted in it, lower bounds are proved in it, and preconditioning exists to reduce it. But \(\kappa\) is a worst case over right-hand sides, and the worst case is usually nothing like the instance in front of you. Fix a matrix with \(\kappa=10^5\) and change only \(b\): conjugate gradients takes anywhere from 1 iteration to 3188, a factor of three thousand, while the condition number never moves.

That much is old news classically and, recently, on the quantum side too — several groups now have solvers whose cost is governed by an effective condition number rather than \(\kappa\). The post concedes all of that up front. What seems not to have been said is what happens when you put the two discounts side by side: they do not discount the same thing. The quantum effective condition number forgives you for having no weight on small singular values. Krylov methods forgive you for having weight on few eigenvalues, wherever they sit. These sets come apart, and they come apart in the direction that hurts. Put \(b\) on four eigen-directions at the small end of the spectrum and CG finishes in 3 iterations while the effective condition number stays at its worst-case 100,000 — classically trivial, quantum worst-case. Instance-easiness erodes the axis where quantum was already behind, and leaves the dimension axis, where the real speedup lives, untouched.

We also report a negative result about our own method. Our Catalyst framework (arXiv:2406.13879, AAAI 2026) wraps a linear solver in a proximal-point loop, replacing \(A\) by the better-conditioned \(I+\eta A\) at the cost of more outer steps. Swept against a classical inner solver, the trade-off has no interior optimum: total work falls with \(\eta\) and flattens onto plain CG without ever crossing it. The optimum sits on the boundary, and the boundary is “do not precondition.” This is not a contradiction of the paper — it is the paper’s own stated ceiling, observed — and it sharpens the claim into a rule. Proximal preconditioning pays only when the inner solver is at least linearly sensitive to \(\kappa\), which is exactly the quantum regime and exactly not CG’s. Wrapping CG in a proximal loop is wrapping a \(\sqrt{\kappa}\) method in machinery designed to rescue a \(\kappa\) one.

Every number in the post comes from classical numerical experiments, sanity-gated and run over five seeds. No quantum solver was executed, and the post says so plainly: the question it answers is what the shared cost axis measures, and that question is answered with linear algebra. The obvious next step — running the quantum solvers end-to-end on the same instance families, so the comparison is measured rather than computed — is stated as an open problem rather than a promise.

Read the full post here.