Quantum-Selected Configuration Interaction: classical diagonalization of Hamiltonians in subspaces selected by quantum computers

TL;DR

QSCI introduces a hybrid quantum-classical approach that uses quantum sampling to identify a compact subspace of important electron configurations, then performs classical diagonalization of the Hamiltonian restricted to that subspace to compute ground- and excited-state energies with a guaranteed variational upper bound. By treating the subspace as the primary quantum-defined component and performing all heavy matrix elements and diagonalization classically, QSCI achieves robustness to noise and reduces quantum circuit requirements, while enabling eigenstate tomography since the eigenvectors are explicitly represented. The authors demonstrate both ground- and excited-state benchmarks, including noiseless simulations and an 8-qubit experimental demonstration, and analyze scaling for larger systems such as Cr2 and hydrogen chains. They also discuss the role of QSCI as a post-processing refinement for VQE, its relation to selected CI methods, and future directions for integrating QSCI with existing quantum-chemical techniques to tackle more complex molecules on NISQ-era and beyond.

Abstract

We propose quantum-selected configuration interaction (QSCI), a class of hybrid quantum-classical algorithms for calculating the ground- and excited-state energies of many-electron Hamiltonians on noisy quantum devices. Suppose that an approximate ground state can be prepared on a quantum computer either by variational quantum eigensolver or by some other method. Then, by sampling the state in the computational basis, which is hard for classical computation in general, one can identify the electron configurations that are important for reproducing the ground state. The Hamiltonian in the subspace spanned by those important configurations is diagonalized on classical computers to output the ground-state energy and the corresponding eigenvector. The excited-state energies can be obtained similarly. The result is robust against statistical and physical errors because the noisy quantum devices are used only to define the subspace, and the resulting ground-state energy strictly satisfies the variational principle even in the presence of such errors. The expectation values of various other operators can also be estimated for obtained eigenstates with no additional quantum cost, since the explicit eigenvectors in the subspaces are known. We verified our proposal by numerical simulations, and demonstrated it on a quantum device for an 8-qubit molecular Hamiltonian. The proposed algorithms are potentially feasible to tackle some challenging molecules by exploiting quantum devices with several tens of qubits, assisted by high-performance classical computing resources for diagonalization.

Figures (15)

• Figure 1: Schematic description of the QSCI algorithm for finding the ground state. When selecting the configurations, one may post-select the configurations by using conserved quantities such as the electron number or spin $S_z$ to mitigate the errors.
• Figure 2: Schematic descriptions of the QSCI algorithms for finding the ground state and the first excited state: (a) single diagonalization scheme, and (b) sequential diagonalization scheme. In both panels, $|\psi_{\rm in}^{(0)}\rangle$ ($|\psi_{\rm in}^{(1)}\rangle$) is the input state for the ground (first excited) state. In the panel (b), the overlap term is constructed from the preobtained output state $|\psi_{\rm out}^{(0)}\rangle$ to define the effective Hamiltonian for the first excited state, $\hat{H}^{(1)} =\hat{H}+\beta_0|\psi_{\rm out}^{(0)}\rangle\langle \psi_{\rm out}^{(0)}|$.
• Figure 3: The result of QSCI, the proposed method, for the ground state of H2O molecule by noiseless simulation, shown with optimization history of VQE, which is used to prepare the input states of QSCI. Each of the resulting energies is shown as the difference to the CASCI result $E_{\rm exact}$ in Hartree. The dash-dotted line shows the result by the state-vector simulation of VQE. The lines specified by the parameter $R$ show the results of QSCI, $E_R - E_{\rm exact}$, for the given value of $R$, using the parametrized state at each iteration of VQE as the input state. The parameter $R$ determines the classical computational cost for QSCI, as explained in the main text.
• Figure 4: Same as Fig. \ref{['fig:noiseless-vqe']} but for the first ($T_1$) and second ($S_1$) excited states of H2O with $S_z = 0$, along with optimization histories of VQD, shown by dotted lines, for input-state preparation; the energy differences are plotted by the absolute values. For the QSCI calculation of the $T_1$ ($S_1$) state, the input state(s) corresponding to the lower energy states, i.e., $S_0$ ($S_0$ and $T_1$) state(s), are prepared by converged sets of parameters of VQD. QSCI results are shown for the sequential diagonalization and the single diagonalization with two types of configuration selection, as described in the main text. The QSCI calculations with sequential diagonalization are done with $R_i=16$ for $i=0,1,2$, while the value of $R$ is set to $R=16$ for single diagonalization.
• Figure 5: Estimated $R$ required for a given energy error $\epsilon$. Results with (a) expanding active spaces (Cr2) or (b) various numbers of atoms (hydrogen chain) are shown by markers, along with the linear fit of each plot.