Towards End-to-End Quantum Estimation of Non-Hermitian Pseudospectra

Abstract

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 $ε$, we show that deciding whether a point $z\in\mathbb{C}$ is $ε$-close to the spectrum is PSPACE-hard for $5$-local operators, whereas deciding whether $z$ lies in the $ε$-pseudospectrum is 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 post-processing 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.

Figures (8)

• Figure 1: An end-to-end approach for estimating non-Hermitian pseudospectra on a quantum computer.
• Figure 2: Preparing ground singular vectors of the Hatano-Nelson model using dissipative dynamics. (a) The eigenvalues of the Hatano-Nelson model under OBC and PBC ($n=20$). (b) The $0.1$-pseudospectrum of the Hatano-Nelson model under PBC, and the mixing time of the Lindbladian dynamics for preparing the ground singular vector of $H_{\rm HN}-zI$.
• Figure 3: Probing exceptional points of a non-Hermitian qubit system using IonQ quantum computer. (a) Illustration of eigenvalues and pseudospectra of the model Hamiltonian $H(g)$. The closed curves at each level shows the $0.1$-pseudospectrum at the corresponding $g$ value. (b) The $0.1$-pseudospectrum of $H(1)$, and the spectral gap of $\mathcal{L}_z$ over the complex plane. (c) The filter functions in QSIGS constructed using the measurement data from IonQ. The peak of the filter function indicates $\sigma_0(H(1)-zI)$. We fix $\epsilon=0.002$, and the results show that $z=0$ lies in the $\epsilon$-pseudospectrum while $z=0.1$ does not.
• Figure 4: An illustrative quantum circuit for generating the data pairs $(t_n, Z_n)$ in \ref{['algo:qsigs']}. The gate here is a block-encoding of $\sin^{\mathrm{SV}}(t_n A)$, which can be implemented using QSVT. The measurement outcome $Z_n$ is determined by whether the ancilla register is in the all-zero state or not.
• Figure 5: Illustration of the spectrum of the Hatano-Nelson model (with $n = 4k$) and the effects of the chosen jump operators $K_0$ and $K_1$.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• proof : Proof of \ref{['thm:pseudo-spectrum-qma']}
• Theorem 3: Theorem 24, fefferman2018complete
• proof
• Lemma 4
• proof
• proof : Proof of \ref{['thm:distance-to-spec-pspace']}
• Lemma 5: Cost of implementing $\sin^{\text{SV}}$ transformation
• proof