Quantum chaos and circuit parameter optimization

Abstract

We consider quantum chaos diagnostics of the variational circuit states at random parameters and explore their connection to the circuit expressibility and optimizability. By measuring the operator spreading coefficient and the eigenvalue spectrum of the modular Hamiltonian of the reduced density matrix, we identify the emergence of universal random matrix ensembles in high-depth circuit states. The diagnostics that use the eigenvalue spectrum, e.g. operator spreading and entanglement entropy, turn out to be more accurate measures of the variational quantum algorithm optimization efficiency than those that use the level spacing distribution of the entanglement spectrum, such as r-statistics or spectral form factors.

Exploring the role of qubit connectivity in expressibility and entanglement of parameterized quantum circuits architectures

“…Distinctly, recent proposals more aligned to VQAs and parameterized quantum circuits involve the application of parameterized local gates and two qubits gates, instead of Haar sampling (38,39). These circuits are very close to the structures proposed here and replicate the behavior of a random circuit, up to some point.…”

“…In some contexts, highly entangling circuits are needed, so the entangled ground states of systems can be built (37). However, there are situations where too much entanglement will harm the performance and a well-suited amount is necessary to optimization (38,39). Also, entanglement is a quantum resource and is very important to quantum advantage, so a careful study of its generation is essential.…”

“…In the context of pseudorandom circuits, the analysis of their characteristics is aligned to the objective of generating Haar-random unitaries. In this more recent approach of characterizing parameterized quantum circuits (20,38,39), the final objective is not the generation of random states, but to understand how the randomness affects optimization. Understanding how the structure of the circuits will affect optimization is, therefore, essential.…”

“…Important examples are the parameterized quantum circuits presented in Refs. (38)(39)(40), whose Meyer-Wallach, von 4…”

“…This would lead to important conclusions about the states generated by both kinds of systems and how they can be used in the advancement of physical knowledge. One could apply quantum circuits to generate states analogous to chaotic many-body systems (110), to understand complexity and entanglement generation of unitaries in a more general approach (43,111) or even to comprehend how information scrambling can harm optimization of parameters in parameterized quantum circuits (39). All these possibilities are the center of many recent research approaches.…”