[Paper Review] Fundamental limitations on optimization in variational quantum algorithms

Hao-Kai Zhang, Chengkai Zhu+2 more|arXiv (Cornell University)|May 10, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper establishes a fundamental limitation on variational quantum algorithms (VQAs) by proving that, for randomly initialized parameterized quantum circuits, the variation range of the cost function under local unitary operations decays exponentially with the number of qubits. This result unifies and generalizes the barren plateau phenomenon, revealing that optimization in VQAs faces intrinsic exponential scaling challenges due to the high-dimensional Hilbert space, even in the absence of noise.

ABSTRACT

Exploring quantum applications of near-term quantum devices is a rapidly growing field of quantum information science with both theoretical and practical interests. A leading paradigm to establish such near-term quantum applications is variational quantum algorithms (VQAs). These algorithms use a classical optimizer to train a parameterized quantum circuit to accomplish certain tasks, where the circuits are usually randomly initialized. In this work, we prove that for a broad class of such random circuits, the variation range of the cost function via adjusting any local quantum gate within the circuit vanishes exponentially in the number of qubits with a high probability. This result can unify the restrictions on gradient-based and gradient-free optimizations in a natural manner and reveal extra harsh constraints on the training landscapes of VQAs. Hence a fundamental limitation on the trainability of VQAs is unraveled, indicating the essential mechanism of the optimization hardness in the Hilbert space with exponential dimension. We further showcase the validity of our results with numerical simulations of representative VQAs. We believe that these results would deepen our understanding of the scalability of VQAs and shed light on the search for near-term quantum applications with advantages.

Motivation & Objective

• To identify and formalize a fundamental limitation on the trainability of variational quantum algorithms (VQAs) beyond gradient-based analyses.
• To investigate the variation range of the cost function when adjusting a single local quantum gate in a random parameterized quantum circuit.
• To unify the constraints on both gradient-based and gradient-free optimization in VQAs under a single theoretical framework.
• To provide a rigorous scaling theorem that explains the inherent optimization hardness in high-dimensional quantum Hilbert spaces.

Proposed method

• Proves that for a random parameterized quantum circuit forming a 2-design, the expected variation range of the cost function under local unitary operations decays exponentially with the number of qubits.
• Uses unitary 1-designs and 2-designs to model random circuits and applies fidelity-based bounds to analyze cost function sensitivity.
• Applies the Bures fidelity and its monotonicity under quantum channels to derive upper bounds on the maximum and minimum achievable fidelities during local optimization.
• Employs concentration inequalities and Markov's inequality to bound the variance and tail probabilities of the variation range.
• Derives a general upper bound on the variation range of the cost function as dA/dB, where dA is the dimension of the local subsystem and dB that of the rest of the system.
• Validates the theoretical results through numerical simulations on representative VQAs, including VQE for the 1D antiferromagnetic Heisenberg model, showing agreement with the predicted exponential decay rate.

Experimental results

Research questions

• RQ1 What is the scaling behavior of the cost function variation range when optimizing a single local gate in a random VQA circuit?
• RQ2 How does the variation range of the cost function relate to the gradient vanishing problem in VQAs?
• RQ3 Can a unified theoretical framework explain both gradient-based and gradient-free optimization limitations in VQAs?
• RQ4 What is the fundamental mechanism behind the exponential scaling of optimization difficulty in high-dimensional quantum systems?
• RQ5 To what extent do random circuit structures and design properties (e.g., 2-designs) govern the trainability of VQAs?

Key findings

• The expected variation range of the cost function under local unitary operations decays exponentially with the number of qubits, with a decay rate of −0.5 in the log-scale, matching numerical simulations.
• The theoretical upper bound on the variation range is dA/dB, which for a local subsystem of m qubits and the rest of n−m qubits gives an upper bound of 1/2n−2m.
• The variance of the variation range is also bounded and decays exponentially, indicating that the cost function is highly insensitive to local gate changes in random circuits.
• The result unifies the barren plateau phenomenon and extends it beyond gradients, showing that even cost function differences vanish exponentially.
• Numerical simulations of VQE for the 1D antiferromagnetic Heisenberg model confirm the predicted exponential decay with a slope of −0.5 in the semi-log plot of variation range vs. qubit count.
• The analysis reveals that the transition to a 2-design occurs at a circuit depth of approximately 10×n, where the theoretical bounds become valid.