Alignment between Initial State and Mixer Improves QAOA Performance for Constrained Optimization

TL;DR

This work demonstrates that aligning the QAOA initial state with the ground state of the mixing Hamiltonian $H_M$ improves constrained optimization performance, tying QAOA behavior to adiabatic quantum computation even at low depths. Using XY mixers that preserve Hamming weight, the authors show through extensive simulations and a 32-qubit trapped-ion experiment that initial-mixer alignment yields higher approximation ratios across diverse mixer connectivities and both exact and Trotterized implementations; however, hardware noise tempers gains from higher Trotter steps. They introduce an effective Hamiltonian framework $H_{\text{eff}}(\beta)=i\log U(\beta)$ and define GS fidelity to quantify alignment under Trotterization, noting rapid convergence of GS fidelity relative to Trotter error. The results inform mixer and initial-state design for NISQ devices and have potential implications for broader constrained optimization tasks, including quantum chemistry, where preparing high-fidelity mixer-ground states is critical.

Abstract

Quantum alternating operator ansatz (QAOA) has a strong connection to the adiabatic algorithm, which it can approximate with sufficient depth. However, it is unclear to what extent the lessons from the adiabatic regime apply to QAOA as executed in practice with small to moderate depth. In this paper, we demonstrate that the intuition from the adiabatic algorithm applies to the task of choosing the QAOA initial state. Specifically, we observe that the best performance is obtained when the initial state of QAOA is set to be the ground state of the mixing Hamiltonian, as required by the adiabatic algorithm. We provide numerical evidence using the examples of constrained portfolio optimization problems with both low ($p\leq 3$) and high ($p = 100$) QAOA depth. Additionally, we successfully apply QAOA with XY mixer to portfolio optimization on a trapped-ion quantum processor using 32 qubits and discuss our findings in near-term experiments.

Figures (12)

• Figure 1: An overview of the results. We show that the QAOA performance depends on the alignment between the initial state $\ket{\psi_0}$ of QAOA and the ground state of the mixing Hamiltonian $\mathbf{H}_M$. Right: the approximation ratios (ARs) obtained by QAOA applied to constrained portfolio optimization with $N=6$ assets and Hamming weight constraint $K=3$ with the complete-$\textsc{x}\textsc{y}$ mixer and the initial state set to be the ground state of complete-$\textsc{x}\textsc{y}$ ("Aligned") and ring-$\textsc{x}\textsc{y}$ ("Misaligned") mixing Hamiltonian. The error bars represent the standard error of the mean approximation ratio estimated from $10$ problem instances of portfolio optimization.
• Figure 2: Comparisons of the exact ring- and complete-$\textsc{x}\textsc{y}$ mixers in QAOA with unrestricted optimization at $p = 1, 2, 3$ and with the OLS method at $p = 100$. We reported the mean approximation ratios over $10$ instances with $N=6$ and $K=3$. The error bars represent standard errors of the mean. The alignment enhances performance in both low and high-depth QAOA.
• Figure 3: An example demonstrating the convergence of the OLS method for QAOA with exact mixers with the instances from Fig. \ref{['fig:completering_exact']}. For both mixers, the initial states are aligned. As the QAOA depth increases, the final state of the OLS method \ref{['eq:phase_diagram_optmize']} will gradually converge to the ground state of the problem Hamiltonian $\mathbf{H}_P$. The complete-$\textsc{x}\textsc{y}$ mixer needs a larger depth to converge with the OLS schedule. The error bars represent standard errors of the mean approximation ratios.
• Figure 4: An example of the six-qubit complete-$\textsc{x}\textsc{y}$ model: The complete graph is constructed by three separate chains, denoted by different colors and line styles.
• Figure 5: Comparisons of exact $\textsc{x}\textsc{y}$-mixers in QAOA with unrestricted optimization at $p=2$ and with OLS at $p = 100$. The heatmaps display the average $\mathrm{AR}$ over the $10$ instances considered with $N=6$ and $K=3$. The mixers are constructed using one or two chains, as shown in Fig. \ref{['fig:intermediadte_mixer']}. The alignment improves performance in both low and high-depth QAOA, as the diagonal pairs outperform others in the corresponding row and column.