Non-Hermitian many-body systems can be spectrally unstable, so small perturbations may induce large eigenvalue shifts. The pseudospectrum quantifies this instability and provides a perturbation-robust diagnostic. For inverse-polynomially small $\epsilon$, we show that deciding whether a point $z\in \mathbb{C}$ is $\epsilon$-close to the spectrum is PSPACE-hard for 5-local operators, whereas deciding whether $z$ lies in the $\epsilon$-pseudospectrum is quantum-Merlin-Arthur-complete (QMA-complete) for 4-local operators. This identifies pseudospectrum membership as a natural computational target. We then present a concrete end-to-end quantum framework for deciding pseudospectrum membership, which combines a singular-value estimation step with a dissipative state preparation algorithm. Our quantum singular-value Gaussian-filtered search (QSIGS) combines quantum singular value transformation (QSVT) with classical postprocessing to achieve Heisenberg-limited query scaling for singular-value estimation. To prepare suitable input states, we introduce an algorithmic Lindbladian protocol for approximate ground right singular vectors and prove its effectiveness for the Hatano-Nelson model. Finally, we demonstrate the full pipeline on a trapped-ion quantum computer and distinguish points inside and outside the target pseudospectrum near the exceptional point of a minimal non-Hermitian qubit model.

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• Open quantum systems
• Quantum algorithms & computation