Diagnosing barren plateaus with tools from quantum optimal control

Abstract

Variational Quantum Algorithms (VQAs) have received considerable attention due to their potential for achieving near-term quantum advantage. However, more work is needed to understand their scalability. One known scaling result for VQAs is barren plateaus, where certain circumstances lead to exponentially vanishing gradients. It is common folklore that problem-inspired ansatzes avoid barren plateaus, but in fact, very little is known about their gradient scaling. In this work we employ tools from quantum optimal control to develop a framework that can diagnose the presence or absence of barren plateaus for problem-inspired ansatzes. Such ansatzes include the Quantum Alternating Operator Ansatz (QAOA), the Hamiltonian Variational Ansatz (HVA), and others. With our framework, we prove that avoiding barren plateaus for these ansatzes is not always guaranteed. Specifically, we show that the gradient scaling of the VQA depends on the controllability of the system, and hence can be diagnosed trough the dynamical Lie algebra g obtained from the generators of the ansatz. We analyze the existence of barren plateaus in QAOA and HVA ansatzes, and we highlight the role of the input state, as different initial states can lead to the presence or absence of barren plateaus. Taken together, our results provide a framework for trainability-aware ansatz design strategies that do not come at the cost of extra quantum resources. Moreover, we prove no-go results for obtaining ground states with variational ansatzes for controllable system such as spin glasses. We finally provide evidence that barren plateaus can be linked to dimension of g.

Subtleties in the trainability of quantum machine learning models

“…Since BPs have been mainly studied in the context of Variational Quantum Algorithms (VQAs) [23,[35][36][37][38][39][40][41][42][43][44], we here briefly recall that in a VQA implementation, the goal is to minimize a linear cost function that is usually of the form…”

“…BPs for cost functions such as that in Eq. ( 13) have been shown to arise for global operators H [36], as well as due to certain properties of U (θ) such as its structure and depth [35,37,44], its expressibility [40,41], and its entangling power [38,39,50]. There are several ways in which the BP phenomenon can be characterized.…”

“…This field aims to optimize over sets of quantum circuits to discover efficient versions of quantum algorithms for accomplishing various tasks, such as finding ground states [24], solving linear systems of equations [25][26][27], quantum compiling [28,29], factoring [30], and dynamical simulation [31][32][33][34]. In this field, a great deal of effort has been put forward towards analyzing and avoiding the barren plateau phenomenon [35][36][37][38][39][40][41][42][43][44]. When a VQA exhibits a barren plateau, the cost function gradients vanish exponentially with the problem size, leading to an exponentially flat optimization landscape.…”

F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits

“…Let us briefly provide some motivation for this approach. One of the most fundamental challenges associated with hybrid quantum-classical algorithms is the barren plateau phenomenon, whereby the gradient of the cost function vanishes exponentially in the number of qubits [36,37,[80][81][82][83][84][85][86][87][88]. This phenomenon can arise due to deep unstructured ansätze [80], large entanglement [83,84], high levels of noise [88], and global cost functions [36,37].…”

Error-mitigated photonic variational quantum eigensolver using a single-photon ququart

“…It is being actively studied and new techniques are being developed to understand and reduce its effects in optimizing parameterized quantum circuits. Some possible remedies to avoid the barren plateus include by appropriately modifying the cost function, detecting the existence of barren plateus using the quantum control theory, and trading off the size of the solution space in favor of convergence by reducing the expressibility of ansatz states [31,32].…”