This paper presents two new quantum algorithms for solving linear systems that avoid the traditional dependence on the spectral condition number. The condition number (denoted κ) is a common worst-case measure of how hard a matrix inversion is. The authors produce the normalized quantum solution |x⟩ of Ax = |b⟩ to accuracy ε with runtimes that do not scale directly with κ. Instead the costs depend on more refined, instance-specific quantities that can be much smaller than κ for typical inputs.

The work assumes the standard quantum input model: the matrix A is accessed through a block-encoding oracle (a common way to feed a matrix into a quantum computer) and the right-hand side |b⟩ can be prepared by a unitary. The first algorithm is truncation-based. It replaces the full matrix inverse by a truncated version that ignores very small singular values. This leads to an “effective condition number” κ_eff which governs the cost. The authors prove a family of upper bounds on κ_eff. Concretely, for different integer choices t the bound trades off sensitivity to small singular values against the inverse error term (1/ε)^{1/t}. For t = 1 one gets the simple bound κ_eff ≤ ||A^{-1†}|x⟩|| / (||x|| ε), where ||·|| denotes the usual vector norm and A^{-1†} is a matrix-related vector that appears in the analysis. The truncation solver uses an optimal number of queries to prepare |b⟩ and roughly O(κ_eff · polylog(κ_eff/ε)) queries to the block-encoding of A.

The second algorithm is filtering-based and is especially simple. When the norm of the solution ||x|| is known up to a constant, its leading-order query cost to both the matrix oracle and the state-preparation oracle is about 6 · (||A^{-1†}|x⟩|| / ||x||) · (1/ε) · ln(1/ε). The paper also gives a comparably simple estimator for the solution norm with the same asymptotic cost up to logarithmic factors, so the method can be used even when ||x|| is not known in advance. Compared with a prior beyond-κ algorithm by Li, which had cost scaling like 1/ε^2 in important regimes, these new solvers lower the dependence on the inverse accuracy and improve the constant factors.