CANOE: Classically Assisted Non-Orthogonal Eigensolver

Abstract

In the early fault-tolerant regime, where quantum resources remain limited, hybrid quantum-classical strategies offer one possible route toward quantum advantage. We introduce CANOE, the Classically Assisted Non-Orthogonal Eigensolver, as such an approach, distributing Rayleigh-Ritz basis states between quantum and classical hardware. This approach leverages the expressive power of quantum states, which are costly to reproduce classically, while augmenting them with a large pool of classically generated basis states that can be incorporated at negligible computational cost. We validate this through numerical simulations of a 76-qubit chromium atom system, quantifying how each additional quantum basis state enhances ground-state representability and how the inclusion of classical states further amplifies this improvement. Such a hybrid basis framework necessarily requires an efficient protocol on quantum hardware for evaluating overlaps between quantum and classical states in the resulting generalized eigenvalue formulation. We address this by introducing a histogram-based protocol and demonstrate through numerical simulations that it can approach chemical accuracy at moderate sampling cost. To solve the resulting generalized eigenvalue problem stably, CANOE incorporates a Schur-complement-based stabilization procedure that mitigates ill-conditioning caused by linear dependencies in the hybrid basis. Taken together, these results position CANOE as a practical framework for combining limited quantum resources with expansive classical resources for early fault-tolerant quantum simulations.

Figures (7)

• Figure 1: Overview of the CANOE protocol. The workload is distributed between classical (left) and quantum (right) processors. Hybrid ansatz: CANOE combines a large number of classical determinants with a small number of quantum-prepared states. Sampling the overlap: The projected matrices decompose into $\langle Q|Q\rangle$ (evaluated on quantum hardware using Hadamard test), $\langle C|C\rangle$ (computed on classical hardware), and $\langle Q|C\rangle$ blocks; the cross terms are estimated by sampling the quantum states and postprocessing classically via a histogram-based estimator. Subspace truncation: Ill-conditioning of the overlap matrix is mitigated entirely on classical hardware using a Schur-complement formulation with deflation or pseudo-inverse stabilization.
• Figure 2: Classical-Quantum Subspace hybridization analysis on Cr system with 38 spatial orbitals and 14 electrons in cc-pVDZ-DK basis. (a): ground-state energy error (Ha) versus basis-set size for a purely classical SHCI determinant (blue) and a purely quantum Krylov subspace (orange). The gray band indicates "chemical accuracy" ($1.5936\times10^{-3}\,\mathrm{Ha}$). (b): ground-state energy error as a function of classical ($N_c$) and quantum ($N_q$) basis sizes. The white dashed line indicates the iso-error contour corresponding to chemical accuracy. (c): marginal reduction in the number of classical basis states, $\Delta N^{\text{cont}}_{\text{c}}$, that can be eliminated when one additional quantum basis state is introduced ($k\!\to\!k+1$) while remaining on the same ground-state energy-error contour. (d): ground-state classical-amplitude landscape over classical ($N_c$) and quantum ($N_q$) basis sizes. The color map shows the total classical amplitude $\sum_{i=1}^{N_c} |c_i|^{2}$ in the normalized ground state obtained from the generalized eigensolver. The contour labeled $0.5$ marks the boundary where the classical and quantum sectors contribute equal weight. The bottom left and bottom right insets show the number of discarded quantum states along the lines $N_c=4$ and $N_q=8$, respectively, in the upper panel.
• Figure 3: Numerical benchmark of histogram-based overlap estimation. (a): Comparison between histogram-based and classical-shadow estimation for the Hamiltonian-matrix error $\Delta H$ as a function of the number of shots for H$_4$ (solid) and H$_2$ (dashed). The reference values are taken from the infinite-shot limit of the sampling simulation. For H$_2$, we use $N_c=2$(the full SHCI space, used here to test sampling noise) and $N_q=2$, while H$_4$ uses the same basis sizes as in Table II. (b): Determinant probabilities versus determinant rank for $\mathrm{H}_4$, shown at $10^6$ shots, comparing the histogram-based estimation and classical shadow against the "exact" result, where "exact" denotes the infinite-shot limit. (c): Classical--quantum Hamiltonian-matrix error $\Delta H_{cq}$ obtained using the histogram-based estimation method as a function of the number of shots for the benchmark systems. The reference values are obtained from analytically computed Hamiltonian-matrix elements, corresponding to the infinite-shot limit. (d): Ground-state energy error $\Delta E$ as a function of the number of shots for the same benchmark systems. The error is defined relative to the reference ground state energy, which is SHCI variational energy. The dashed lines indicate the gap between the SHCI variational energy and the infinite-shot-limit CANOE energy in the truncated subspace. The infinite-shot-limit ground-state energy is obtained by loading the exact truncated $(H,S)$ matrices, Hermitizing them, solving $Hc=ESc$ with LOBPCG, deflation, and pseudo-inverse over a threshold sweep, and comparing the resulting solver outputs with $E_{\mathrm{true}}$.
• Figure 4: Benchmark of eigensolver performance versus overlap-matrix noise for $\mathrm{H_4}$, $\mathrm{H_2O}$, and $\mathrm{COCl_2}$. Columns correspond to the three molecular systems. (a) and (b) show the median ground-state energy error, $|E - E_{\mathrm{true}}|$, as a function of $\|\Delta S\|_F$ for $\tau_{\mathrm{rank}} = 10^{-6}$ and $10^{-2}$, respectively, where $E$ denotes the ground-state energy estimated from the noisy matrices. In these two rows, solid curves denote the sampled-matrix benchmark and dashed curves denote the synthetic $\alpha$-noise benchmark. Blue, green, and orange denote LOBPCG, LOBPCG+pseudo-inverse, and LOBPCG+deflation, respectively. Green "x" markers indicate cases in which the pseudo-inverse solver failed to return a result, and star-shaped markers indicate cases in which all quantum directions were truncated so that only the classical subspace remained. (c) shows the corresponding infinite-shot-limit benchmarks, $E_{\mathrm{inf}} - E_{\mathrm{true}}$, together with the classical-classical only (CC-only) reference for each system shown in black. Values closer to zero indicate better agreement with the SHCI variational reference energy.
• Figure 5: Ancilla-assisted implementation of histogram-based overlap estimation. The ancilla-assisted preparation yields $\left\vert \Psi \right\rangle=\frac{1}{\sqrt{2}}(\left\vert 0 \right\rangle\left\vert \phi_j^q \right\rangle+\left\vert 1 \right\rangle\left\vert \chi \right\rangle)$. Measuring the ancilla in the $X$ or $Y$ basis and the system register in the computational basis gives the joint histograms $J_R$ and $J_I$, respectively.