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

TL;DR

The paper develops a convergence theory for the variational quantum eigensolver by importing quantum-control landscape ideas. It proves a sufficient criterion: if the parameterized unitary $U(oldsymbol{ heta})$ is locally surjective on $ ext{SU}(d)$ and the Lie-algebra directions $oldsymbol{ ext{Ω}}_j(oldsymbol{ heta})$ are uniformly bounded, then gradient-descent updates converge to the ground state for almost all initial configurations, provided the descent terminates. To make the criterion practical, it analyzes two common ansatz families, showing they often fail local surjectivity, and then introduces two constructions—composite ansätze and a Cayley-transform-based parameterization—that ensure local surjectivity (with $2(d^2-1)$ or $d^2$ parameters respectively). The work discusses limitations like gimbal-lock, regularization strategies to enforce termination, and connections to fundamental questions such as the halting problem, offering design principles for VQE ansätze with favorable convergence guarantees.

Abstract

When does a variational quantum algorithm converge to a globally optimal solution? Despite the large literature around variational approaches to quantum computing, the answer is largely unknown. We address this open question by developing a convergence theory for the variational quantum eigensolver (VQE). By leveraging the terminology of quantum control landscapes, we prove a sufficient criterion that characterizes when convergence to a ground state of a Hamiltonian can be guaranteed for almost all initial parameter settings. More specifically, we show that if (i) a parameterized unitary transformation allows for moving in all tangent-space directions (local surjectivity) in a bounded manner and (ii) the gradient descent used for the parameter update terminates, then the VQE converges to a ground state almost surely. We develop constructions that satisfy both aspects of condition (i) and analyze two commonly employed families of quantum circuit ansätze. Finally, we discuss regularization techniques for guaranteeing gradient descent to terminate, as for condition (ii), and draw connections to the halting problem.

Figures (3)

• Figure 1: Schematic representation of the key criterion (local surjectivity) that establishes convergence of VQEs to the ground state. If local surjectivity is satisfied, the variation $\frac{\partial U(\bm{\theta})}{\partial\theta_{j}}$ (green) of a parameterized unitary transformation $U(\bm{\theta})\in\mathrm{SU}(d)$ with respect to all variational parameters $\theta_{j}$ spans the tangent space $\text{T}_{U(\bm{\theta})}\mathrm{SU}(d)$ of the special unitary group $\mathrm{SU}(d)$ at all points of the optimization landscape. In this case the critical point structure of the VQE is determined by the Riemannian gradient $\operatorname{grad}J$. We show that then the Riemannian gradient only vanishes at global optima and strict saddle points that are avoided almost surely by gradient descent when gradients are Lipschitz continuous. Thus, if the gradient descent used for the parameter update converges, it converges for almost all initial configurations to a global minimum (main Theorem).
• Figure 2: Example of a singular point in the Euler angle parameterization of $\mathrm{SU}(2)$ on the Bloch sphere. We consider a that uses the X-Y-X Euler angle parameterization of $\mathrm{SU}(2)$ given by equation \ref{['eq:euler_angles']} to find the ground state $\ket{\psi(\boldsymbol{\theta}^*)}$ (red dot) of some problem Hamiltonian, e.g. $H = \mathds{1} - \ket{\psi(\boldsymbol{\theta^*})}\bra{\psi(\boldsymbol{\theta^*})}$. The first and last rotation will rotate the state around the $X-$axis on the Bloch sphere, the second rotation around the $Y$-axis. In principle one can map any point on the Bloch sphere to any other point in this way. However, when choosing $\ket{\psi_0} = \ket{-}$, i.e. the minus one eigenstate of $\sigma_x$, as the initial state, at any point $\boldsymbol{\theta} = {\left(\theta_1, \frac{\pi}{2}, \theta_3\right)}^T$ of the parameter landscape, only the second rotation will affect the state and move it to the opposite end of the Bloch sphere. Hence the derivatives $\frac{\partial}{\partial \theta_1} \ket{\psi(\boldsymbol{\theta})}$ and $\frac{\partial}{\partial \theta_3} \ket{\psi(\boldsymbol{\theta})}$ vanish and $\frac{\partial}{\partial \theta_2} \ket{\psi(\boldsymbol{\theta})}$ points along the $Z$-direction. Since the cost functional only decreases further when the state is moved along the equator, the Riemannian gradient (in this case, defined on the Bloch sphere itself) points into the $Y$-direction and is therefore perpendicular to all available derivatives. Hence, the Euclidean gradient vanishes and the optimization stops at this point --- although the ground state has not yet been reached.
• Figure 3: Example of a gradient algorithm escaping to infinity for an inappropriate choice of parameterization. The optimization is done on the complex unit circle $\mathrm{U}(1)$ (black). The real line of parameters (blue) is wrapped around the circle such that the negative half extends up to the point $-1$, but never reaches it. The positive half is wrapped the other way around until $- i$ so that it overlaps with the negative half in the third quadrant of the circle. The increasing radius of the blue spiral only serves visibility purposes. When a gradient search with the target $1$ is initialized at a point $\theta$ that is mapped to the third quadrant of the circle, the gradient (red) will always point into the positive direction, driving the parameter towards positive infinity, without the image point on the circle ever reaching $1$.

Theorems & Definitions (10)

• Definition 1: Local surjectivity
• Theorem
• Proposition 1: Composite ansätze
• Corollary
• Lemma 1: cf. Theorem 4 in pmlr-v49-lee16 and Corollary 2 in lee_first-order_2019
• Definition 2
• Lemma 2
• proof
• Lemma 3
• proof