Diagnosing Barren Plateaus with Tools from Quantum Optimal Control

Journal Article · Thu Sep 29 00:00:00 EDT 2022 · Quantum

DOI:https://doi.org/10.22331/q-2022-09-29-824· OSTI ID:1891829

Larocca, Martin ^[1]; Czarnik, Piotr Jan ^[2]; Sharma, Kunal ^[3]; Muraleedharan, Gopikrishnan ^[2]; Coles, Patrick Joseph ^[2]; Cerezo de la Roca, Marco Vinicio Sebastian ^[4]

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1. Univ. of Buenos Aires (Argentina); Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division
2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division
3. Louisiana State Univ., Baton Rouge, LA (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division
4. Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Information Sciences; Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Center for Nonlinear Studies

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 $$\mathfrak{g}$$.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Univ. of Buenos Aires (Argentina)
• Sponsoring Organization: USDOE Laboratory Directed Research and Development (LDRD) Program; National Scientific and Technical Research Council (CONICET); Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT); National Science Foundation (NSF)
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 1891829
• Report Number(s): LA-UR-21-24973
• Journal Information: Quantum, Journal Name: Quantum Vol. 6; ISSN 2521-327X
• Publisher: Quantum Science Open Community
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
NISQ
algorithms
computing
quantum
trainability