Strategies for Overcoming Gradient Troughs in the ADAPT-VQE Algorithm

TL;DR

The paper tackles gradient troughs in ADAPT-VQE, a barrier to efficient convergence for strongly correlated systems. It introduces detection methods that compare gradients across ansatz positions to distinguish troughs from true convergence and proposes four operator-position protocols (OO/OP, OO/RP, RO/OP, RO/RP) that exploit non-commutativity to insert new operators at non-end positions, thereby escaping troughs. The protocols, especially the optimized-operator variants, improve convergence and can significantly reduce measurement costs by boosting gradient magnitudes during troughs, though they may incur additional overhead in some cases. In simulations on a 12-qubit H$_6$ model, the approach demonstrates robust improvement in convergence while maintaining low circuit depth, with potential applicability to larger systems, albeit under idealized conditions. Future work includes extending analyses to noisy devices, developing deeper theoretical understanding of troughs, and integrating with operator-pruning strategies.

Abstract

The adaptive derivative-assembled problem-tailored variational quantum eigensolver (ADAPT-VQE) provides a promising approach for simulating highly correlated quantum systems on quantum devices, as it strikes a balance between hardware efficiency, trainability, and accuracy. Although ADAPT-VQE avoids many of the shortcomings of other VQEs, it is sometimes hindered by a phenomenon known as gradient troughs. This refers to a non-monotonic convergence of the gradients, which may become very small even though the minimum energy has not been reached. This results in difficulties finding the right operators to add to the ansatz, due to the limited number of shots and statistical uncertainties, leading to stagnation in the circuit structure optimization. In this paper, we propose ways to detect and mitigate this phenomenon. Leveraging the non-commutative algebra of the ansatz, we develop heuristics for determining where to insert new operators into the circuit. We find that gradient troughs are more likely to arise when the same locations are used repeatedly for new operator insertions. Our novel protocols, which add new operators in different ansatz positions, allow us to escape gradient troughs and thereby lower the measurement cost of the algorithm. This approach achieves an effective balance between cost and efficiency, leading to faster convergence without compromising the low circuit depth and gate count of ADAPT-VQE.

Figures (16)

• Figure 1: Noise-free simulation of ADAPT-VQE for a linear H$_6$ molecule with $4\text{ \AA}$ interatomic distance until iteration 50. Between iterations $20$ and $29$, a significant gradient trough is evident, accompanied by a substantial decrease in gradient norm and a convergence plateau in energy, as is typical for them. The error is defined with respect to the FCI energy.
• Figure 2: Gradient norm against operator position for a regular execution of ADAPT-VQE (i.e., appending operators). Panel (a) shows an iteration within a gradient trough, while the right panel considers an iteration at convergence. Within the gradient trough, a clear decrease in the gradient norm from the prepending to the appending position can be observed. For the case at convergence, there is no such consistent decrease visible: Instead, we can observe that the gradient norms are mostly uniform across positions.
• Figure 3: Gradient norm across ansatz positions for ADAPT-VQE iterations 1--40. The gradient trough is visible across iterations 20--29 as a decrease in gradient norm as the position moves towards the end of the ansatz (with appending being the extreme case).
• Figure 4: Workflow of the enhanced ADAPT-VQE algorithms, where red indicates our changes to the protocols and blue is the original ADAPT-VQE algorithm as stated in Sec. \ref{['subsec:ADAPT-VQE_Background']}. Unlike the original ADAPT-VQE, our enhanced algorithms check for gradient troughs. If a gradient trough occurs, the protocol measures gradients at positions that are most likely to yield higher gradients than appending. In the next step, the operator and position that yield the largest magnitude gradient are identified, and the operator is inserted accordingly into the ansatz. All three algorithms are based on this principle. For an exact description see Sec. \ref{['sec:Results_new_protocols']}. The function $E(\theta_\mu, p_s, A_\mu)$ refers to the energy expectation value of the system as defined in Eq. \ref{['eq:ADAPT-energy_expectation_value_mu']}, where in the given case $\theta_\mu$ is the variational parameter for the operator $A_\mu$, $p_s$ refers to the position of the newly added operator $A_\mu$ in the ansatz and $A_\mu$ with $\mu \in [1,N_\text{Pool}]$ is defined to be an operator within a given pool of size $N_\text{Pool}$.
• Figure 6: (c) RO/OP Protocol (random operator, optimized position) used from iteration 23 to iteration 53.