Hybrid VQE-CVQE algorithm using diabatic state preparation

TL;DR

This work introduces a hybrid VQE-CVQE algorithm that leverages diabatic state preparation to generate a guiding state for quantum simulations. By measuring the guiding state, it builds a variational subspace through a measurement-driven procedure and diagonalizes the projected Hamiltonian classically to obtain the ground-state energy, while treating the diabatic evolution parameters as variational controls. Demonstrations on a toy spinless electronic model and a 50-level system on IBM Brisbane show energies within chemical accuracy and reveal three hardware-relevant regimes based on the number of time steps N_tau and step size Delta_tau. The framework provides a flexible, hardware-aware path to accurate ground-state energies across NISQ to FTQ regimes, with potential for near-term improvements via CVQE and future adiabatic-state preparation on fault-tolerant devices.

Abstract

We propose a hybrid variational quantum algorithm that has variational parameters used by both the quantum circuit and the subsequent classical optimization. Similar to the Variational Quantum Eigensolver (VQE), this algorithm applies a parameterized unitary operator to the qubit register. We generate this operator using diabatic state preparation. The quantum measurement results then inform the classical optimization procedure used by the Cascaded Variational Quantum Eigensolver (CVQE). We demonstrate the algorithm on a system of interacting electrons and show how it can be used on long-term error-corrected as well as short-term intermediate-scale quantum computers. Our simulations performed on IBM Brisbane produced energies well within chemical accuracy.

Figures (6)

• Figure 1: The basic process for our method. Diabatic state preparation is performed on the quantum computer to generate a guiding state. A collection of state states are measured. These state states form a subspace and this subspace can be optimized classically.
• Figure 2: Energy optimization for a $Q = 8$ orbital system with $N_e = 4$ electrons. (a) energy as a function of the number of time steps with a duration of $\Delta\tau = 1/15~ \tau_0$ where $\tau_0 = 1/t$ is the characteristic time scale. (b) energy as a function of the time step duration at $N_{\tau} = 100$. (c) the total subspace after application of the Hamiltonian as a function of the number of time steps with a duration of $\Delta\tau = 1/15~ \tau_0$. The parameters are set to $\Delta\mu = 0.75~t$, $V=t$. While $t$ is a model parameter that sets the energy scale, a reasonable value to set our model to the molecular scale is $t = 1/15~\text{Ha}$ . The dashed black lines represents the energy expectation value of the guiding state. The thick blue lines represents subspace extraction using the 14 most probable states, the medium orange line represents using the 8 most probable states, and the thin green line represents using the 2 most probable states. The dotted line in (c) marks the total size of the particle conserving state space.
• Figure 3: The difference between the energy calculated on the quantum computer and the exact ground-state energy. The data is represented by the circles. The lines are drawn to guide the eye. For this data, $Q=50$, $N_e=25$, $V=0$, $N_{\tau} = 1$, and $\Delta \tau = 1/15~\tau_0$ where $\tau_0 = 1/t$. The inset is a magnification of the low-error data.
• Figure 4: Error in energy as a function of (a) the number of time steps with $\Delta \tau = 1/15~\tau_0$ where $\tau_0 = 1/t$ and (b) the duration of the time step with $N_{\tau} = 1$. In both plots, $Q=50$, $N_e=25$, $\Delta\mu = 0.2 t$, and $V=0$.
• Figure 5: Eigenstates for an example $H_0$. The spheres represent electrons. The lines represent energy levels. On the left, the lowest energy levels are filled with electrons. In the second from left, one excitation has occurred. The two right most graphs show two excitations.