A New Angle on Quantum Subspace Diagonalization for Quantum Chemistry

TL;DR

An improved thresholding scheme for noisy generalized eigenvalue problems that arise in quantum subspace diagonalization that leverages eigenvector-preserving transformations (rotations) of the generalized eigenvalue problem prior to thresholding is introduced.

Abstract

Quantum subspace diagonalization and quantum Krylov algorithms offer a feasible, pre- or early-fault tolerant alternative to quantum phase estimation for using quantum computers to estimate the low-lying spectra of quantum systems. However, despite promising proof-of-principle results, such methods suffer from high sensitivity to noise (including intrinsic sources such as sampling noise), making their utility for realistic industry-relevant problems an open question. To improve the potential applicability of such methods, we introduce a new variant of thresholding for noisy generalized eigenvalue problems that arise in quantum subspace diagonalization that has the potential to better control sensitivity to noise. Our approach leverages eigenvector-preserving transformations (rotations) of the generalized eigenvalue problem prior to thresholding. We study this effect in practical settings by applying this rotation thresholding scheme to an iterative quantum Krylov algorithm for several chemical systems, including the industry-relevant Fe(III)-NTA chelate complex. We develop a particular heuristic to select the rotation angle from noisy data and find for certain systems and noise regimes that the samples required to reach a target error for ground state estimation can be reduced by a factor of up to 100. Furthermore, with oracle access to the optimal transformation, more dramatic improvements are possible and we observe reductions in sample requirements by up to $10^4$, motivating the continued development of methods that can realize these improvements in practice. While we develop our approach in the context of quantum subspace diagonalization, the improved thresholding scheme we develop could be advantageous in any context where one must solve noisy, ill-conditioned generalized eigenvalue problems.

Figures (12)

• Figure 1: A visual overview of naive versus rotation‑based thresholding for quantum subspace diagonalization. After measuring Hamiltonian ($H$) and overlap ($S$) matrix elements on a quantum device, one obtains a noisy and ill‑conditioned generalized eigenvalue problem. To solve it classically typically requires thresholding to project out the low‑lying eigenspace of $S$, where the thresholding parameter that determines what subspace we project away from is determined by the amount of noise. Our rotation approach mixes $H$ and $S$ while preserving their generalized eigenvectors, allowing a less aggressive projection for the same thresholding parameter. Thus, we obtain reduced errors and sampling requirements with minimal classical overhead.
• Figure 2: For the dimension three example discussed in the main text, panel (a) depicts the eigenvalue structure of the true Hamiltonian and indicates what the estimate $\hat{\lambda}_0$ of the true ground state $\lambda_0$ would be for naive and rotated thresholding (at the optimum rotation angle) without noise. (b) shows the eigenvalues of the rotated overlap matrix $S_\theta$ as a function of rotation angle with a parameter choice $\delta=0.1$, $\Delta=2.0$, $\xi=1.1$, and $s=0.9$. Observe that with no rotation ($\theta=0$) the subspace kept for thresholding at $\tau=1.05$ is only dimension one, whereas for the optimal rotation angle ($\theta\approx 1.4$) all states are kept and the ground state energy is recovered perfectly. (c) depicts the impact of Gaussian noise (chosen with standard deviation of $0.1$) on the estimated ground state energy, indicating that rotated thresholding will pick the perturbed second eigenstate as the best estimate of the ground state. The fact that noise lowers all eigenvalues is an artifact of the simplicity of this model and not a general feature. The fact that one state is still projected away by rotated thresholding when we add noise is explained by panel (d) showing the $(\alpha,\beta)$ plane where rotations act. Perturbation theory for generalized eigenvalue problems Epperly.2022.SIAMParlett.1998Mathias.2004.M.C.C.M.Cheng.1999.Lin.Algebra.App. tells us, roughly speaking, that the closer the eigenvector-normalized coordinates of an eigenvalue are to the origin in this space the more sensitive that eigenvalue is to perturbation. Thus, in this problem the ground state is extremely sensitive to perturbations and is eliminated by thresholding. (e) shows the result of this Gaussian noise on the error estimates as a function. Observe that the best estimate now occurs for $\theta\approx 0.5$ where two states are kept, balancing the impacts of thresholding error and error due to noise.
• Figure 3: Ratio of the absolute error with respect to the CASSCF solution made by the rotation thresholding approach with rotation angle $\theta_{\text{optimal}}$ ($\epsilon_r$) and the same error made by naively thresholding ($\epsilon_n$), calculated for in length increasing all-trans polyene chains. Noise level indicates the variance of the artificially applied Gaussian noise, and $\tau$ is the thresholding parameter. Numbers in the heatmap represent the amount of additional states kept by the rotation approach, that were discarded when applying naive thresholding. Convergence scaling factor $\gamma=1.0$.
• Figure 4: Absolute error ($\epsilon_\text{absolute}$) in log-scale with respect to the CASSCF solution made by the rotation thresholding approach with rotation angle $\theta_{\text{optimal}}$ (blue bars) and the same error made by naively thresholding (red bars) for in length increasing all-trans polyene chains and three different noise levels, at all thresholding parameters $\tau \geq$ noise level. Dashed line represents chemical accuracy. Convergence scaling factor $\gamma=1.0$.
• Figure 5: Molecular structure of the low-spin state of the Fe(III)-NTA chelate complex. Hydrogen atoms are shown in light gray, carbon atoms in dark gray, oxygen atoms in red, nitrogen atoms in blue and Fe(III) is shown in orange.