On the convergence of the variational quantum eigensolver and quantum optimal control

Technical Summary: Convergence of the Variational Quantum Eigensolver and Quantum Optimal Control

Problem Statement
Variational Quantum Algorithms (VQAs), particularly the Variational Quantum Eigensolver (VQE), face a fundamental challenge: determining when the classical optimization routine converges to a globally optimal solution (the ground state of a Hamiltonian) rather than getting trapped in suboptimal local minima. While numerical evidence suggests overparameterization can improve convergence, rigorous criteria characterizing when a parameterized quantum circuit guarantees convergence to the global optimum remain elusive. The non-convex nature of the optimization landscape and the complex interplay between the circuit ansatz and the problem instance often lead to "bad" local optima, especially when the number of parameters is small relative to the problem dimension.

Methodology
The authors develop a convergence theory for VQEs by leveraging the terminology and mathematical framework of quantum control landscapes. The core approach involves analyzing the geometry of the parameterized unitary transformation $U(\theta)$ used as the VQE ansatz.

1. Differential Geometric Perspective: The cost function $J(\theta) = \langle \psi_0 | U^\dagger(\theta) H U(\theta) | \psi_0 \rangle$ is viewed as a composition of a parameterization map and a cost function defined on the unitary group. The gradient of the cost function with respect to parameters $\theta$ is expressed via the Hilbert-Schmidt inner product between the Riemannian gradient on the unitary group and the tangent vectors $\Omega_j(\theta) = U^\dagger(\theta) \partial_{\theta_j} U(\theta)$.
2. Local Surjectivity: The authors introduce the concept of local surjectivity (Definition 1). A parameterization is locally surjective if the set of Lie algebra elements $\{\Omega_j(\theta)\}$ spans the full tangent space of the special unitary group $SU(d)$ (or the relevant Lie algebra $\mathfrak{g}$ for subgroups) at every point $\theta$. This condition ensures that the optimization landscape has no "missing directions" where the gradient could vanish due to a lack of control authority, a phenomenon analogous to "gimbal lock" in robotics.
3. Critical Point Analysis: Under the assumption of local surjectivity and bounded Lie algebra elements, the authors prove that the critical points of the cost function correspond exclusively to global optima (ground states) or strict saddle points. They utilize results from non-linear stability theory to argue that gradient descent algorithms avoid strict saddle points almost surely (with respect to the Lebesgue measure) provided the gradient is Lipschitz continuous.
4. Ansatz Construction: The paper analyzes common ansatz families (SU(d)-gate and product-of-exponentials) and demonstrates that they inherently possess singular points where local surjectivity fails, regardless of overparameterization. To address this, the authors construct new parameterizations:
   • Composite Ansatz: A product of a surjective map and a locally surjective map restricted to a small domain, ensuring global reachability and local surjectivity.
   • Cayley Transform: A parameterization using the Cayley transform which is shown to be locally surjective with $d^2$ parameters.

Key Contributions and Results

• Theorem 1 (Convergence on $SU(d)$): The authors establish that if a VQE ansatz $U(\theta)$ satisfies local surjectivity and the Lie algebra elements are uniformly bounded, then for a sufficiently small step size and almost all initial parameters, the gradient descent algorithm either diverges to infinity or converges to a ground state of the Hamiltonian. Crucially, the algorithm almost never converges to a suboptimal local minimum.
• Theorem 2 (Convergence on Subgroups): The convergence guarantees are extended to optimization over closed Lie subgroups $G \subset SU(d)$. If the ansatz is locally $\mathfrak{g}$-surjective and the Hamiltonian (or initial state projector) lies within the appropriate algebraic structure, the algorithm converges to the global optimum on that subgroup.
• Theorem 3 (Reachability via Representation Theory): Using the theory of quasi-minuscule representations, the authors provide sufficient conditions under which a ground state is guaranteed to be reachable from a specific set of initial states (weight vectors) using a subgroup $G$.
• Application to Free Fermions: The theory is applied to free fermionic Hamiltonians, showing that a VQE using a locally $so(2n)$-surjective parameterization converges to the ground state for almost all initial points, provided the initial state is a weight vector of the representation.
• Analysis of Common Ansätze: The paper proves that standard ansätze like the SU(d)-gate and product-of-exponentials (even when overparameterized) always contain singular points where local surjectivity breaks down. Specifically, for the SU(d)-gate ansatz, singular points cannot be removed by overparameterization.
• Construction of Robust Ansätze: The authors propose the Composite Ansatz (Eq. 15) and the Cayley Transform (Eq. 24) as explicit constructions that satisfy local surjectivity, thereby theoretically eliminating the risk of getting stuck in suboptimal local minima due to singular controls.

Significance and Claims
The paper claims to provide a "first step" towards establishing rigorous solution guarantees for VQAs. Its primary significance lies in shifting the focus from empirical observations of overparameterization to a geometric condition—local surjectivity—that guarantees the absence of suboptimal local minima in the optimization landscape.

The authors emphasize that while satisfying local surjectivity prevents the existence of local minima, it does not guarantee that the gradient descent algorithm will terminate (i.e., it may escape to infinity). They acknowledge that ensuring both termination and convergence to the global optimum simultaneously is a non-trivial problem, potentially linked to the halting problem for certain VQE instances. Consequently, the paper does not claim to solve the convergence problem entirely but rather provides a sufficient criterion (local surjectivity) under which the quality of the solution (avoiding suboptimal local minima) is guaranteed, assuming the algorithm converges. The work also highlights the limitations of common circuit architectures and offers concrete, mathematically rigorous alternatives for constructing parameterized unitaries that avoid these pitfalls.