Symmetry dilemmas in quantum computing for chemistry: A comprehensive analysis.

Area of Science:

• Quantum computing
• Computational physics
• Quantum chemistry

Background:

• Symmetry adaptation, universality, and gate efficiency are key considerations in quantum algorithms.
• Balancing these requirements presents significant challenges, often leading to competing demands in algorithm design.

Purpose of the Study:

• To analyze the trade-offs between symmetry adaptation, universality, and gate efficiency in quantum algorithms.
• To provide a theoretical and numerical guide for designing effective operator pools for quantum simulations.

Main Methods:

• Theoretical proof demonstrating that a specific gate-efficient operator pool is not universal under spatial symmetry enforcement.
• Numerical simulations using an adaptive algorithm with three types of operator pools: fully symmetry-adapted and universal, fully symmetry-adapted and non-universal, and symmetry-breaking and universal.
• Simulations cover scenarios targeting global ground states, states differing in multiple symmetry properties, and states differing in a single symmetry property.

Main Results:

• Proved that the singlet spin-adapted singles and perfect-pairing doubles operator pool is not universal when spatial symmetry is enforced.
• Demonstrated the performance of different operator pools across various quantum state targeting scenarios.
• Identified conditions under which symmetry-breaking pools are safe and when specific symmetries must be preserved to prevent variational collapse.

Conclusions:

• The study offers a practical framework for designing and evaluating symmetry-adapted operator pools.
• Provides insights into balancing universality, resource efficiency, and robust state targeting in quantum simulations.
• Aims to guide the development of quantum algorithms for electronic structure and many-body physics.