Hybrid VQE-CVQE algorithm using diabatic state preparation

Technical Summary: Hybrid VQE-CVQE Algorithm Using Diabatic State Preparation

Problem Statement
Current quantum computing algorithms face a dichotomy between those designed for Noisy Intermediate-Scale Quantum (NISQ) devices and those requiring Fault-Tolerant Quantum (FTQ) computers. Algorithms like Quantum Phase Estimation and Adiabatic State Preparation require precise, long-depth circuits only feasible on FTQ hardware, which is not yet available. Conversely, near-term algorithms like the Variational Quantum Eigensolver (VQE) and Cascaded Variational Quantum Eigensolver (CVQE) operate on NISQ devices but often struggle to scale with increasing problem sizes due to the limitations of their variational ansatzes. The authors aim to bridge this gap by proposing a hybrid algorithm that functions effectively across different stages of quantum hardware development, from current noisy devices to future error-corrected systems.

Methodology
The proposed method is a hybrid VQE-CVQE algorithm that utilizes diabatic state preparation to generate a "guiding state." The process, depicted in Figure 1 and detailed in Table I, proceeds as follows:

1. Initialization: The quantum register is initialized to the ground state $|\Phi_0\rangle$ of a simple, known model Hamiltonian $\hat{H}_0$.
2. Diabatic Evolution: A parameterized unitary operator $\hat{U}(N_\tau, \Delta\tau)$ is applied to the register. This operator evolves the system from $\hat{H}_0$ to the target Hamiltonian $\hat{H}$ over $N_\tau$ time steps of duration $\Delta\tau$. The evolution is governed by a time-dependent Hamiltonian $\hat{H}(\tau) = (1 - \frac{\tau}{T})\hat{H}_0 + \frac{\tau}{T}\hat{H}$, where $T = N_\tau \Delta\tau$. Unlike adiabatic preparation, which requires $N_\tau \to \infty$, this method employs highly discretized evolution (small $N_\tau$, large $\Delta\tau$) to produce a guiding state $|\Psi_0\rangle$ that approximates the correct subspace.
3. Measurement and Subspace Construction: The guiding state is measured in the computational basis to generate a set of basis states $B_0$. To ensure the subspace captures the necessary physics, the target Hamiltonian $\hat{H}$ is applied to $B_0$ to generate a coupled set $B_1$. The final basis set is $B = B_0 \cup B_1$.
4. Classical Optimization: The system Hamiltonian is projected onto the subspace $V = \text{span}(B)$ to form an effective Hamiltonian $\hat{H}_B$. The ground-state energy $E_B$ is obtained by diagonalizing $\hat{H}_B$ on a classical computer.
5. Variational Loop: The parameters $N_\tau$ and $\Delta\tau$ serve as variational parameters. The algorithm scans these parameters to minimize $E_B$.

Key Contributions

• Hybrid Architecture: The paper introduces a method that combines the variational nature of VQE (optimizing circuit parameters) with the subspace expansion of CVQE (classical diagonalization of a projected Hamiltonian).
• Diabatic Guiding State: Instead of complex, deep ansatzes (like Unitary Coupled Cluster), the algorithm uses a simple, highly discretized diabatic evolution to generate a guiding state. This approach explicitly approximates adiabatic state preparation but remains tractable for short-depth circuits.
• Hardware Agnosticism: The algorithm is designed to be general, functioning on current NISQ devices, intermediate-scale computers with high coherence but no error correction, and future FTQ computers.
• Measurement Efficiency: The authors compare two methods for collecting measurement shots: one involving rotations for each Pauli string (standard VQE) and one measuring directly in the computational basis followed by Hamiltonian application. They demonstrate that the direct measurement method (used in their simulations) is significantly more efficient for $k$-local Hamiltonians, requiring fewer circuits despite potentially needing more shots to compensate for cutoff thresholds.

Results
The authors validated the algorithm through classical simulations and quantum hardware execution:

• 8-Orbital System (Classical Simulation): For a system with $Q=8$ orbitals and $N_e=4$ electrons, the algorithm achieved energies well within chemical accuracy. The results identified three distinct regimes based on the number of time steps ($N_\tau$):
  • Small Regime: The guiding state is poor, but the subspace extraction yields a good approximation for specific variational parameters.
  • Medium Regime: The guiding state remains a poor approximation of the exact ground state, yet the subspace energy $E_B$ is nearly independent of parameters, allowing CVQE to find a good solution without extensive optimization.
  • Large Regime: Adiabatic preparation is effectively achieved, yielding a high-quality guiding state.
• 50-Energy Level System (IBM Brisbane): The algorithm was executed on the IBM Brisbane quantum computer for a system with $Q=50$ and $N_e=25$. The full Fock space dimension is $\approx 1.3 \times 10^{14}$, far beyond classical diagonalization capabilities. The algorithm utilized a subspace of only 2,000–2,500 states.
  • The results showed small energy errors ($\Delta E$) across various energy spacings $\Delta\mu$, even with a gate error rate of 1.9%.
  • Contrary to noise-free simulations which suggest error increases as $\Delta\tau \to 0$, the noisy hardware results showed that $N_\tau = 1$ (a single time step) provided the optimal results. The authors attribute this to hardware noise effectively applying small non-zero rotations, making deeper circuits (higher $N_\tau$) detrimental due to accumulated gate errors.

Significance and Claims
The paper claims that this hybrid VQE-CVQE algorithm offers a robust pathway for quantum simulation across the evolution of quantum hardware:

• For Current NISQ Devices: Minimizing circuit depth is critical. The algorithm performs best with $N_\tau = 1$, where no variational parameter update is necessary, as the noise floor prevents the benefits of deeper circuits.
• For Near-to-Medium Term: As coherence improves, allowing variational parameters ($N_\tau, \Delta\tau$) to be updated could further optimize the energy.
• For Future FTQ Devices: The method naturally transitions to direct adiabatic state preparation without requiring additional optimization steps.

The authors emphasize that the algorithm's ability to produce accurate results with a small subspace dimension (relative to the full Hilbert space) and its resilience to probability distribution fluctuations make it a practical candidate for immediate application on available hardware while remaining scalable for future fault-tolerant systems.