Quantum-selected configuration interaction with time-evolved state

TL;DR

The paper introduces Time-Evolved QSCI (TE-QSCI), a method to prepare optimization-free input states for quantum-selected configuration interaction by time-evolving a chosen initial state under the target Hamiltonian and sampling the evolved states to define a subspace for classical diagonalization. It analyzes two implementations—single-time TE-QSCI and time-average TE-QSCI—and compares Hartree-Fock and UCCSD-based initial states, showing that TE-QSCI can yield ground-state energies within a few milli-Hartrees for small molecules and hydrogen chains, with more favorable classical overhead than GS-QSCI and UCCSD-QSCI in many cases. The work provides detailed scaling insights, noting linear scaling of the required subspace size $R$ with system size and substantial quantum gate counts dominated by CNOTs, yet remaining feasible for early fault-tolerant quantum devices. Overall, TE-QSCI offers a practical, optimization-free pathway to leverage quantum hardware for quantum chemistry, guiding future improvements in time-evolution schemes and initial-state selection for larger, more complex systems.

Abstract

Quantum-selected configuration interaction (QSCI) utilizes an input quantum state on a quantum device to select important bases (electron configurations in quantum chemistry) that define a subspace in which to diagonalize a target Hamiltonian, i.e., perform selected configuration interaction, on classical computers. Previous proposals for preparing a good input state, which is crucial for the quality of QSCI, based on optimization of quantum circuits may suffer from optimization difficulty and require many runs of the quantum device. Here, we propose using a time-evolved state by the target Hamiltonian (for some initial state) as an input of QSCI. Our proposal is based on the intuition that the time evolution by the Hamiltonian creates electron excitations of various orders when applied to the initial state. We numerically investigate the accuracy of the energy obtained by the proposed method for quantum chemistry Hamiltonians describing electronic states of small molecules. Numerical results reveal that our method can yield sufficiently accurate ground-state energies for the investigated molecules. Systematic analysis when increasing the number of qubits in a hydrogen chain shows that the subspace size required for sufficiently accurate results is reasonable at system sizes that cannot be solved by naive classical diagonalization. Our proposal provides a systematic and optimization-free method to prepare the input state of QSCI and could contribute to practical applications of quantum computers in quantum chemistry calculations.

Figures (8)

• Figure 1: Average of the probability of measuring the computational basis state $\ket{\mu}$ for the time-evolved state, $P_\mu(t)=\abs{\braket{\mu}{\psi(t)}}^2$, among the $R=850$ states with largest probability at $t= 1$ for the H8 molecule. The average is taken for first- and second-order, third- and fourth-order, and fifth- and sixth-order electron excitations.
• Figure 2: Difference between the exact ground-state energy and the energy obtained by single-time QSCI with various Trotter step sizes $\Delta t$ for the approximation of the time-evolution operator. The solid black line corresponds to GS-QSCI, while the dashed black line corresponds to $10^{-3}$ Hartree. The black circles represent the results of the exact time evolution. The dimension of the subspace in TE-QSCI is $R=90, 850, 6000, 130, 160$ and $100$ for panels (a)-(f), respectively.
• Figure 3: Difference between the exact ground-state energy and the energy obtained by single-time TE-QSCI with the Hartree-Fock initial state as a function $R/R_{GS}$ at fixed times for different molecules. The striped black line corresponds to $10^{-3}$ Hartree and the black circles correspond to the results of the exact time evolution. The sampling times for panels (a)-(c) are all $t=1.4$ except for $\Delta t = 0.5$, where we set $t=1.5$. The sampling times for panels (d),(e), and (f) are $t=1, 1,$ and $1.4$, respectively.
• Figure 4: The results of single-time TE-QSCI with the Hartree-Fock initial state considering the effect of the fluctuation of the measurement result with a finite number of shots $n_{\text{shots}}$ at a fixed time $t$. The dimension of the subspace $R$ is chosen as the number of all distinct bases (electron configurations) obtained by the sampling. (a), (c), (e): Difference between the exact ground-state energy and the energy obtained by single-time QSCI with various Trotter step size $\Delta t$. We run ten simulations and the mean of the difference is plotted by the markers connected with a full line, while the standard deviation of the obtained energy is plotted by the markers connected by the striped line. The striped black line corresponds to $10^{-3}$ Hartree. (b), (d), (f): The average value of $R$ for ten simulations. The sampling time $t$ is the same as in Fig. \ref{['fig:Rdependentqsci']} except for NH3 with $\Delta t=0.5$, where we set $t=1.5$.
• Figure 5: System-size scaling of the classical and quantum resources of TE-QSCI. (a): The ratio $R/R_{GS}$, where $R_{GS}$ is the subspace dimensions required for $10^{-3}$ Hartree accuracy for GS-QSCI, while $R$ are the subspace dimensions required for TE-QSCI with the Hartree-Fock initial state (HF-TE-QSCI) at $t=1.4$, TE-QSCI with the UCCSD initial state at $t=0.4$ (UCCSD-TE-QSCI) and UCCSD-QSCI. The lines represent linear fits. (b): The estimate of the required number of shots to achieve $10^{-3}$ Hartree accuracy, corresponding to the results in (a). (c): The number of gates required to implement a single Trotter step of the time-evolution operator $e^{-i\hat{H} t}$. The lines show power-law fits.