Diagnosing Barren Plateaus with Tools from Quantum Optimal Control (2105.14377v3)

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 degree of controllability of the system, and hence can be diagnosed through the dynamical Lie algebra $\mathfrak{g}$ 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. Our work establishes a link between the existence of barren plateaus and the scaling of the dimension of $\mathfrak{g}$.

• The paper proposes using tools from quantum optimal control, specifically Dynamical Lie Algebra analysis, to diagnose and predict barren plateaus in Variational Quantum Algorithms.
• Analyzing the Dynamical Lie Algebra of an ansatz's generators can forecast its expressibility and susceptibility to barren plateaus, linking system controllability to gradient vanishing.
• The research highlights the impact of initial states and subspaces on barren plateaus and suggests a pathway for designing trainability-aware ansatzes to mitigate this challenge.

Diagnosing Barren Plateaus in Variational Quantum Algorithms Using Quantum Optimal Control

The paper "Diagnosing barren plateaus with tools from quantum optimal control" investigates the barren plateau phenomenon in Variational Quantum Algorithms (VQAs) and proposes a method to diagnose such occurrences through techniques borrowed from Quantum Optimal Control (QOC). The research focuses on diagnosing whether specific problem-inspired ansatzes, like the Quantum Alternating Operator Ansatz (QAOA) and the Hamiltonian Variational Ansatz (HVA), are susceptible to the barren plateau phenomenon, which hinders the scalability of VQAs by causing gradients to vanish exponentially with problem size.

Variational Quantum Algorithms have garnered attention due to their potential to outperform classical algorithms on near-term quantum devices. These algorithms rely on parameterized cost functions evaluated on quantum circuits, trained with classical optimizers. A prevalent challenge is the occurrence of barren plateaus—flat regions in the optimization landscape where the gradients are effectively zero, making training impractically slow.

Main Contributions and Results

1. Barren Plateau Diagnostics via Quantum Optimal Control: The paper proposes employing QOC tools to understand and predict the presence of barren plateaus in problem-inspired ansatzes. The authors explore the connection between the controllability of a quantum system, characterized by its dynamical Lie algebra, and the emergence of barren plateaus.
2. Dynamical Lie Algebra (DLA) Analysis: The paper involves analyzing the DLA of the ansatz's Hamiltonian generators to forecast the expressibility of the VQA and its susceptibility to barren plateaus. Systems with full-rank DLAs (controllable systems) are more likely to exhibit barren plateaus. The depth of the circuit required for the ansatz to approximate a 2-design is calculated, which is a state where barren plateaus are likely.
3. Impact of Input States and Subspaces: The research highlights the role of initial states and subspace controllability. The presence of barren plateaus could depend on the subspace of the initial quantum state, with larger subspaces leading to greater chances of encountering barren plateaus. This connection suggests careful selection or design of initial states to mitigate this effect.
4. Numerical Verification and Practical Implications: Numerical simulations support the theoretical findings, demonstrating the importance of DLA dimensionality and controllability in determining the trainability of various VQA ansatzes. The work provides a pathway for designing trainability-aware ansatzes, potentially avoiding barren plateaus and thus improving VQA efficiency.
5. Applications to Problems Like MAXCUT: The paper applies its framework to specific quantum algorithms and problems, such as solving MAXCUT using QAOA on Erdős–Rényi graphs, verifying the insights derived from DLA analysis in practical scenarios.

Theoretical Implications and Future Directions

The paper establishes a significant link between QOC theory and VQA design, suggesting that insights from quantum control can inform the development of more effective quantum computing methods. The proposed approach could guide the selection of ansatzes with better trainability, thus mitigating the barren plateau challenge.

The authors also conjecture that the DLA dimension's scaling can predict the gradient scaling in VQAs. This conjecture implies that observing the scaling behavior of the DLA can diagnose potential training issues in VQAs and help in designing ansatzes that exhibit favorable training landscapes.

Overall, by merging insights from quantum control and variational quantum algorithms, the research provides a framework that not only enhances understanding of the barren plateau phenomenon but also aids in the design of ansatzes that are robust against such optimization challenges. Further exploration into QOC-inspired ansatz construction promises advancements in the development of efficient, scalable quantum algorithms.