Quantum Linear Systems and the HHL Algorithm

HHL solves linear systems in log dimension in principle, but its fine print makes it a poor fit for Kronos PDE solves; we say why plainly.

The promise and the fine print

Many fusion computations reduce to solving large linear systems A x = b: discretized equilibrium equations, implicit time steps, linearized transport. The Harrow-Hassidim-Lloyd (HHL) algorithm prepares a quantum state proportional to the solution x in time polylogarithmic in the system dimension N, an exponential-looking speedup. The fine print is where it usually fails to help.

# HHL solves A x = b, output = quantum state |x> ~ sum_i x_i |i>
# runtime  ~  O( kappa^2 * s * log(N) / eps )
#   kappa = condition number of A
#   s     = sparsity (nonzeros per row)
#   eps   = target error
# NOT O(N): but the caveats below usually dominate

The four catches

• Input: b must be loadable as a quantum state efficiently (needs QRAM or special structure).
• Output: you get the state |x>, not the vector x; reading all N components destroys the speedup.
• Conditioning: runtime scales as kappa^2, and stiff fusion PDE operators are often ill-conditioned.
• You only cheaply extract summary quantities , not the full field.

Fit to Kronos, honestly

The Kronos PDE workloads, free-boundary Grad-Shafranov, MHD stability, transport, are exactly the cases where the catches bite: we usually want the full field for the digital twin, the operators are ill-conditioned, and state preparation of the source is nontrivial. Classical multigrid and Krylov solvers, plus physics-informed neural networks, already solve these well and give the whole solution.

We therefore do not plan HHL for equilibrium or transport solves. It stays on the shelf as a primitive that could matter only for a narrow problem where we need a scalar functional of a well-conditioned, structured system, which the operational solves are not. This is a case where the honest answer is 'quantum does not help here'; see the broader PDE assessment.