Rotationally Invariant Circuits: Universality with the Exchange Interaction and Two Ancilla Qubits

Area of Science:

• Quantum Information Science
• Quantum Computing
• Quantum Many-Body Physics

Background:

• Universality of local unitary transformations is fundamental to quantum computing.
• Continuous symmetries impose constraints, preventing generic symmetric unitaries from being implemented locally.

Purpose of the Study:

• To investigate if SU(2) rotationally invariant unitaries can be realized despite symmetry constraints.
• To determine the minimum number of ancilla qubits required for universality.
• To characterize constraints on realizable unitaries in k-local circuits.

Main Methods:

• Utilizing the Heisenberg exchange interaction, a 2-local and rotationally invariant unitary.
• Analyzing the role of ancilla qubits in enabling universal quantum computation.
• Studying qubit circuits composed of k-local rotationally invariant unitaries.

Main Results:

• Any SU(2) rotationally invariant unitary can be realized with the Heisenberg exchange interaction and two ancilla qubits.
• A single ancilla qubit is insufficient to achieve universality under these conditions.
• Constraints imposed by locality on realizable unitaries were fully characterized.

Conclusions:

• Universality of quantum computation can be achieved even with continuous symmetries, using specific interactions and ancilla qubits.
• The findings provide a deeper understanding of the interplay between locality, symmetry, and universality in quantum systems.
• The characterization of constraints offers insights into designing and implementing quantum algorithms.