Optimal Complexity of Parameterized Quantum Circuits

Area of Science:

• Quantum Computing
• Quantum Information Theory

Background:

• Parameterized quantum circuits are essential for variational quantum algorithms in the Noisy Intermediate-Scale Quantum (NISQ) era.
• Their expressibility, the ability to generate diverse quantum states, is key to approximating solutions for complex problems.
• Assessing expressibility via random parameter sampling relates to fundamental concepts of quantum complexity.

Purpose of the Study:

• To compare the convergence rate of different parameterized quantum circuits towards the Haar measure (asymptotic complexity).
• To investigate the role of circuit topology in entanglement generation and quantum complexity.
• To evaluate the utility of majorization-based measures in understanding random quantum state generation.

Main Methods:

• Comparison of various parameterized quantum circuit classes against universal random circuits.
• Quantification of circuit expressibility using random parameter sampling.
• Application of majorization-based complexity measures.
• Analysis of qubit connection topology's impact on entanglement.

Main Results:

• Parameterized circuits demonstrate faster convergence to asymptotic complexity compared to universal random circuits, requiring fewer gates.
• Qubit connectivity topology significantly influences entanglement generation and the growth of quantum complexity.
• The majorization criterion provides a complementary perspective on random state generation dynamics.

Conclusions:

• Parameterized quantum circuits are efficient for achieving high expressibility and complexity in NISQ algorithms.
• Optimizing qubit topology is critical for maximizing entanglement and computational power.
• Majorization measures offer a valuable tool for analyzing quantum state ensembles and complexity.