Variational quantum simulation: a case study for understanding warm starts

Abstract

The barren plateau phenomenon, characterized by loss gradients that vanish exponentially with system size, poses a challenge to scaling variational quantum algorithms. Here we explore the potential of warm starts, whereby one initializes closer to a solution in the hope of enjoying larger loss variances. Focusing on an iterative variational method for learning shorter-depth circuits for quantum real time evolution we conduct a case study to elucidate the potential and limitations of warm starts. We start by proving that the iterative variational algorithm will exhibit substantial (at worst vanishing polynomially in system size) gradients in a small region around the initializations at each time-step. Convexity guarantees for these regions are then established, suggesting trainability for polynomial size time-steps. However, our study highlights scenarios where a good minimum shifts outside the region with trainability guarantees. Our analysis leaves open the question whether such minima jumps necessitate optimization across barren plateau landscapes or whether there exist gradient flows, i.e., fertile valleys away from the plateau with substantial gradients, that allow for training. While our main focus is on this case study of variational quantum simulation, we end by discussing how our results work in other iterative settings.

11 + 34 pages, 5 + 3 figures

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• Algorithmic Shadow Spectroscopy
• Efficient quantum-enhanced classical simulation for patches of quantum landscapes
• Tensor-Programmable Quantum Circuits for Solving Differential Equations
• Pitfalls when tackling the exponential concentration of parameterized quantum models
• Double-bracket algorithm for quantum signal processing without post-selection
• Typical Machine Learning Datasets as Low-Depth Quantum Circuits
• Generative flow-based warm start of the variational quantum eigensolver
• Efficient Online Quantum Circuit Learning with No Upfront Training
• Exploiting many-body localization for scalable variational quantum simulation
• Estimates of loss function concentration in noisy parametrized quantum circuits
• A coherent approach to quantum-classical optimization
• Learning complexity gradually in quantum machine learning models
• Imaginary Time Spectral Transforms for Excited State Preparation
• Gradient Scalability and Taylor Surrogation of Quantum Cost Landscapes
• Information Processing in Quantum Thermodynamic Systems: an Autonomous Hamiltonian Approach