Mirror subspace diagonalization: a quantum Krylov algorithm with near-optimal sampling cost

TL;DR

The paper tackles the prohibitive sampling cost of quantum Krylov methods for ground-state energy estimation by introducing Mirror Subspace Diagonalization (MSD), a central-difference representation of the Hamiltonian using symmetrically shifted time evolutions. MSD reduces sampling requirements by approximately approaching the theoretical lower bound (up to a logarithmic factor) and maintains comparable Hamiltonian-simulation cost, particularly when the Hamiltonian’s spectral norm is small relative to its 1-norm, as in large basis-set molecular problems. The authors provide rigorous error analysis, time-shift and energy-level shifts to optimize sampling, and demonstrate dramatic empirical savings (up to ~10^4x) in molecular benchmarks, along with Hamiltonian-moment based energy-error mitigation to enhance performance at low Krylov depth. The work also outlines practical extensions to excited states and Green’s functions, and discusses error sources and resource considerations for near-term quantum hardware. Overall, MSD offers a viable path to chemically accurate ground-state energies on NISQ and early-FTQC devices by substantially lowering sampling costs while preserving convergence guarantees.

Abstract

Quantum Krylov algorithms have emerged as a promising approach for ground-state energy estimation in the near-term quantum computing era. A major challenge, however, lies in their inherently substantial sampling cost, primarily due to the individual measurement of each term in the Hamiltonian. While various techniques have been proposed to mitigate this issue, the sampling overhead remains a significant bottleneck, especially for practical large-scale electronic structure problems. In this work, we introduce an alternative method, dubbed mirror subspace diagonalization (MSD), which approaches the theoretical lower bound of the sampling cost for quantum Krylov algorithms. MSD leverages a finite-difference formula to express the Hamiltonian operator as a linear combination of time-evolution unitaries with symmetrically shifted timesteps, enabling efficient estimation of the Hamiltonian matrix within the Krylov subspace. In this scheme, the finite difference and statistical errors are simultaneously minimized by optimizing the timestep parameter and shifting the energy spectrum. Consequently, MSD attains the lower bound of the sampling cost of the quantum Krylov algorithms up to a logarithmic factor. Furthermore, we employ classical post-processing to infer Hamiltonian moments, which are used to mitigate the ground state energy error based on the Lanczos scheme. Through theoretical analysis of the sampling cost, we demonstrate that MSD is particularly effective when the spectral norm of the Hamiltonian is significantly smaller than its 1-norm. Such a situation arises, for example, in high-accuracy simulations of molecules using large basis sets that incorporate strong electronic correlations. Numerical results for various molecular models reveal that MSD can achieve sampling cost reductions ranging from approximately 10 to 10,000 times compared to the conventional quantum Krylov algorithm.

Figures (13)

• Figure 1: Schematic illustration of the MSD algorithm. It begins with two classical preprocessing steps: time-shift optimization and energy level shift, both designed to minimize the sampling cost. Next, Hadamard test sampling of shifted time-evolution operators is executed on a quantum processor. This sampling employs a shot allocation weighted by the central finite difference formula, which approximates the Hamiltonian operator. The matrix elements, estimated from the Hadamard test sampling data, are then used to classically solve a generalized eigenvalue problem for the Krylov subspace. Optionally, energy error mitigation can be applied using Hamiltonian moments derived from the Hadamard test sampling data.
• Figure 2: Circuit diagrams for estimating the matrix element $\bra{\phi_0}\hat{P}e^{-i\hat{H}t}\ket{\phi_0}$. Here, $\hat{V}_{\phi_0}$ denotes a quantum circuit to prepare the initial state $\ket{\phi_0}$ as $\hat{V}_{\phi_0}\ket{0}^{\otimes N_q} = \ket{\phi_0}$.
• Figure 3: Illustration of the energy level shift technique to reduce the spectral norm $\| \hat{H}_{\bm{Q}} \|$ for a $\bm{Q}$-symmetry sector. Energy levels belonging to the $\bm{Q}$-symmetry sector are depicted as black lines, while energy levels for other symmetry sectors are depicted as gray lines. By shifting the energy levels so that the center of the $\hat{H}_{\bm{Q}}$ spectrum is at the origin, the spectral norm $\| \hat{H}_{\bm{Q}} \|$ is minimized to $\Delta{E}_{\bm{Q}}/2$.
• Figure 4: Numerical comparison of the sampling cost between KQD and MSD for various molecules. The number of spatial orbitals $N_{\rm orb}$ is increased for the same molecule by enlarging the basis set in the following order: STO-3G, 6-31G, cc-pVDZ, and cc-pVTZ. Dependence of the sampling cost ratio $R[M]:=M_{\rm msd}/M_{\rm kqd}$ on (a) the number of spatial orbitals $N_{\rm orb}$ and (b) the ratio $\Delta{E}_{(N_e,S)}/\lambda$. Here, $M_{\rm kqd}$ and $M_{\rm msd}$ are theoretical sampling costs for KQD and MSD, estimated based on Eqs. \ref{['eq:kqd_sampling_cost']} and \ref{['eq:msd_sampling_cost_formal']}, respectively, using the 1-norm $\lambda$ and spectral range $\Delta{E}_{\bm{Q}}$ for benchmark molecular models shown in Table \ref{['tab:molecule_info']}. The shade regions in (a) depict the ratio $R[M]$ achievable by adopting the 1-norm reduction to KQD. The upper bound of the shading is determined by the reduced 1-norm values in Table \ref{['tab:molecule_info']}, which are obtained by performing the Hamiltonian optimization in Appendix \ref{['append:1norm_reduction']}. The parameters are chosen as $n=N_{\rm orb}$, $J=n=N_{\rm orb}$, and $\eta_{\mathbf{H}}=0.0016$.
• Figure 5: Numerical comparison of the simulation cost between KQD and MSD for various molecules. Dependence of (a) the maximal evolution time ratio $R[T_{\rm max}]$ and (b) the total evolution time ratio $R[T_{\rm total}]$ on the number of spatial orbitals $N_{\rm orb}$. Here, the total evolution time $T_{\rm total}$ is estimated based on theoretical sampling costs for KQD and MSD, determined by Eqs. \ref{['eq:kqd_sampling_cost']} and \ref{['eq:msd_sampling_cost_optimized']}, respectively, using the 1-norm $\lambda$ and spectral range $\Delta{E}_{\bm{Q}}$ for benchmark molecular models shown in Table \ref{['tab:molecule_info']}. The shade regions in (b) depict the ratio $R[T_{\rm total}]$ achievable by adopting the 1-norm reduction to KQD. The upper bound of the shading is determined by the reduced 1-norm values in Table \ref{['tab:molecule_info']}, which are obtained by performing the Hamiltonian optimization in Appendix \ref{['append:1norm_reduction']}. The parameters are chosen as $n=N_{\rm orb}$, $J=n=N_{\rm orb}$, and $\eta_{\mathbf{H}}=0.0016$.

Theorems & Definitions (7)

• Theorem 1: Sampling perturbation of KQD lee2024sampling
• Theorem 2: Optimized sampling cost of MSD
• proof
• Lemma 3: Norm behavior of Gaussian matrix series tropp2015introductionlee2024sampling
• Lemma 4: Matrix variance statistics for Toeplitz-Hermitian matrix lee2024sampling
• Lemma 5
• proof