A New Angle on Quantum Subspace Diagonalization for Quantum Chemistry

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.

💡 Research Summary

Quantum subspace diagonalization (QSD) and quantum Krylov diagonalization (QKD) have emerged as promising early‑fault‑tolerant strategies for estimating low‑lying eigenvalues of molecular Hamiltonians without the deep circuits required by quantum phase estimation. The practical obstacle is that the effective Hamiltonian H₀ and overlap S₀ matrices, obtained from noisy measurements on a quantum device, are both ill‑conditioned and contaminated by sampling and hardware noise. The standard remedy—“naive thresholding”—projects out the subspace spanned by the low‑lying eigenvectors of S₀ whose eigenvalues fall below a chosen threshold τ. While this regularizes the generalized eigenvalue problem H µ = µ S µ, it also discards potentially useful information and becomes highly sensitive to noise, especially when S₀ is nearly singular.

The authors propose a fundamentally different regularization: before applying any threshold, they rotate the pair (H, S) by a real‑valued angle θ using a 2×2 orthogonal matrix from SO(2). The rotation is defined as
H_θ = H cosθ − S sinθ, S_θ = S cosθ + H sinθ.
Crucially, this transformation preserves the generalized eigenvectors (up to a global phase) while linearly mixing the eigenvalues. Because thresholding breaks the equivalence between the original and rotated problems, an appropriately chosen θ can make the spectrum of S_θ more favorable: more eigenvalues lie above τ, so the projection removes fewer directions. Consequently, the same amount of measurement noise leads to a much smaller perturbation of the final Ritz values.

The paper first develops the mathematical foundation, showing that the rotation belongs to the projective general linear group PGL₂(ℝ) and that restricting to rotations avoids amplifying the noise norms. A three‑dimensional illustrative example demonstrates that, for a simple Hamiltonian diagonal in the eigenbasis of S₀, naive thresholding can completely eliminate the ground‑state component, whereas a rotation of about θ ≈ 1.4 rad keeps the full subspace and recovers the exact ground‑state energy in the noiseless limit. Adding Gaussian noise (σ = 0.1) still yields a substantial error reduction compared with the naive approach.

Algorithmically, the method integrates seamlessly into a real‑time QKD workflow: (1) prepare an initial state ψ₀, (2) generate a Krylov basis by repeated short‑time evolution ψ_j = e^{−iĤδt} ψ_{j−1}, (3) estimate the matrix elements of H and S via Hadamard tests or multi‑fidility estimation, (4) apply the rotation θ to obtain (H_θ, S_θ), (5) perform thresholding with the same τ as in the naive scheme, (6) solve the resulting generalized eigenvalue problem classically, and (7) undo the rotation to retrieve the physical eigenvalues. The only additional classical cost is the evaluation of a few trigonometric functions and a scan over candidate θ values.

Empirical validation is performed on two families of chemical systems. First, a set of all‑trans polyenes of varying length (4–12 carbon atoms) serves as a benchmark for increasing multi‑reference character. As the conjugation length grows, the overlap matrix becomes more ill‑conditioned, and the rotation‑based thresholding reduces the number of required measurement samples by factors of 30–100 to achieve chemical accuracy (≈1 kcal/mol). Second, the industrially relevant Fe(III)‑nitrilotriacetate (NTA) chelate, a transition‑metal complex with strong electron correlation, is examined. Here, the rotation scheme cuts the sample budget by roughly two orders of magnitude, and when the optimal rotation angle is supplied by an “oracle” (i.e., perfect knowledge of the noise‑free matrices), the reduction reaches up to 10⁴‑fold. These results establish a practical upper bound on the gains achievable with perfect rotations and demonstrate that even a heuristic angle selection yields substantial improvements.

Beyond quantum chemistry, the authors argue that any computational pipeline requiring the solution of noisy, ill‑conditioned generalized eigenvalue problems—such as non‑orthogonal configuration interaction, quantum Monte‑Carlo overlap matrices, or quantum machine‑learning kernels—could benefit from the same rotation‑before‑thresholding principle.

The paper concludes with several avenues for future work: (i) developing data‑driven or Bayesian strategies to infer the optimal θ directly from noisy measurements, (ii) exploring multi‑angle or adaptive rotation schemes that could further mitigate noise, (iii) combining rotation with other regularizations like Tikhonov damping, and (iv) experimental implementation on actual quantum hardware to assess robustness against realistic gate errors and readout noise.

In summary, by introducing eigenvector‑preserving rotations prior to thresholding, the authors provide a theoretically sound and empirically validated technique that dramatically lowers the measurement overhead of QSD/QKD. The method bridges a critical gap between the promise of early‑fault‑tolerant quantum algorithms and the practical noise constraints of near‑term quantum devices, opening a pathway toward chemically accurate simulations of industrially relevant molecules.