Universal Extensions of Restricted Classes of Quantum Operations

Area of Science:

• Quantum Information Science
• Quantum Computing
• Quantum Optics

Background:

• Arbitrary unitary operations are crucial for many quantum theory applications.
• Real-world quantum systems often have restricted sets of accessible unitary operations.
• Achieving physical universality requires understanding which additional gates are needed.

Purpose of the Study:

• To determine the necessary additional unitary gates for achieving physical universality in restricted quantum systems.
• To investigate this problem for three specific paradigmatic cases: particle-number preserving bosonic linear optics, particle-number preserving fermionic linear optics, and general fermionic linear optics.

Main Methods:

• Utilized tools from group theory and control theory.
• Classified the sets of gates generated by adding a single additional gate to existing restricted sets.
• Analyzed scenarios for arbitrary numbers of particles and dimensions of the single-particle Hilbert space.

Main Results:

• Provided a complete classification of gate sets generated by adding one gate in the studied scenarios.
• Identified the conditions under which physical universality can be achieved for these restricted quantum systems.
• Solved the universality problem for particle-number preserving bosonic and fermionic linear optics, and general fermionic linear optics.

Conclusions:

• The study completely solves the problem of achieving arbitrary unitary transformations for specific restricted quantum gate sets.
• Findings are applicable to systems with arbitrary particle numbers and single-particle Hilbert space dimensions.
• This work provides a fundamental understanding of gate set universality in quantum computing.