Double-bracket quantum algorithms for high-fidelity ground state preparation

TL;DR

This work introduces and assesses double-bracket quantum algorithms (DBQAs) as a practical route to high-fidelity ground-state preparation on near-term quantum hardware. By combining a short warm-start circuit with DBQA refinements—grounded in Brockett’s double-bracket flows and implemented via product-form exponentials—the approach yields exponential convergence toward diagonalization with controllable gate costs. Numerical studies on XXZ/XXZ-like Hamiltonians show substantial energy reductions and fidelity gains, while hardware experiments on IBM devices and emulations for Quantinuum platforms illustrate tangible near-term advantages and platform-dependent benefits. The results suggest DBQAs can serve as a versatile, hardware-aware unitary synthesis method, potentially bridging the gap between variational methods and fault-tolerant techniques, and may be extended as warm-starts for broader quantum algorithms. Overall, the work demonstrates that warm-started DBQAs can significantly improve ground-state approximations with realistic gate counts, offering a promising path for early fault-tolerant quantum computing and hybrid quantum-classical workflows.

Abstract

Ground state preparation is a central application for quantum computers but remains challenging in practice. In this work, we quantitatively investigate the performance and gate counts of double-bracket quantum algorithms (DBQAs) for ground state preparation. We propose a practical strategy in which DBQAs refine initial state preparation circuits, and we compile them for Heisenberg chains using controlled-Z and single-qubit gates. Warm-started DBQAs consistently improve both the energy and ground-state fidelity relative to the initial states provided by variational ansätze, indicating that DBQAs offer an effective unitary synthesis method. To demonstrate compatibility with near-term hardware, we executed a proof-of-concept example on IBM devices. With error mitigation, we observed a statistically significant improvement over the corresponding warm-start circuit. Furthermore, numerical emulations for the same system size indicate that executing DBQAs on Quantinuum's hardware could achieve similar cost-function gains without requiring error mitigation. These findings suggest that DBQAs are a promising approach for enhancing ground-state approximations on near-term quantum devices.

Figures (5)

• Figure 1: We propose a two-stage ground state preparation protocol: first, apply a relatively short-depth warm-start circuit; second, apply a DBQA circuit to further the ground state preparation fidelity.
• Figure 2: Visualization of the impact of DB-DOI on cost function for a single VQE random seed, see Tab. \ref{['tab:xxz_results']} for statistical analysis. (left) Training of VQE (blue lines) for 3, 4, and 5 layers of a Hamming-weight-preserving ansatz (hues of blue) achieved ground state energy residue $\Delta E\approx 1\%$ within $500$ training epochs. For more epochs, the initially rapid decrease in the cost function saturates and shows marginal improvement afterward. We initialize DBQA with VQE for selected epochs $\in \{100, 200, 500, 1000, 2000\}$ and optimize DBQA parameters with CMA-ES cma. In the bottom panel, we show the relative difference value between the achieved energy $\tilde{E}_0$ and the true ground state energy $E_0$. (right) Token cost estimates of DB-DOI by counting the total number of CZ gates required for the complete protocol, which includes the training (using the parameter-shift rule) of the VQE until a target epoch and the optimization of DBQA. The depth of the individual circuits are instead reported in the legend.
• Figure 3: Hamming-weight-preserving architecture for $L=4$ qubits and $S = 2$. A single circuit layer consists of Reconfigurable Beam Splitter (RBS) gates (see Eq. \ref{['eq:rbs']}) connecting nearest- and next-nearest neighbors. We account for periodic boundary conditions by adding RBS gates connecting the last and first qubits. After a chosen number of layers, we add measurements in the computational basis.
• Figure 4: Performance of DB-DOI applied to the XXZ model with $\Delta = 1$ for $L=20$ qubits. The main panel shows the energy as a function of the number of CZ gates per circuit per qubit, with points corresponding to $k=0,1,2,3$ DB-DOI steps (different colors indicate different HVA ansatz depths $p$ for $k=0$). DB-DOI is warm-started from HVA states of varying depth (left inset), where the colored circles mark the DB-DOI initial energies in the main panel. In the left inset, we show the relative energy to the ground-state energy $\lambda_0$ to visualize the system size scaling of HVA for $L=12, 16 20$ qubits. To explore HVA for varying system sizes $L=12,16,20$, we off-set the changing target energy by plotting the HVA energy relative to the exact ground state ($\Delta E_{\rm HVA} = E_{\rm HVA}-E_0$) as a function layer number $p$ which suggests that for $p=1$ HVA is limited by expressivity while for $p\ge 3$ it is possible to find improved ground state approximations but training becomes a limitation. The right inset compares DB-DOI and DB-QITE for the same initialization, showing that DB-DOI achieves similar energy reductions as DB-QITE with a few hundred CZ gates per qubit. Note that the number of CZ gates plotted refers to the circuit depth and does not include the training's cost.
• Figure 5: One example of DB-DOI for $\operatorname{XXZ}$ using a hardware efficient ansatz and obtained by fixing the simulation random seed; the image is intended to provide qualitative information about the impact of DB-DOI. A more robust study of the performance is presented in Tab. \ref{['tab:xxz_results']}.(left) Training of VQE (blue lines) for $7$, $8$, and $9$ layers (hues of blue) achieved ground state energy residue of about 1% within $500$ training epochs. We initialize DBQA with VQE for selected epochs $\in [1000, \, 2000, \, 3000, \, 4000, \, 5000]$ where we apply a DBQA optimized in its parameters with CMA-ES cma. (right) Token cost estimates of DB-DOI by counting the total number of two-qubit gates required to execute the complete protocol: training the VQE until a target epoch and then optimizing and applying the DBQA.