Minimally Universal Parity Quantum Computing

Abstract: In parity quantum computing, multi-qubit logical gates are implemented by single-qubit rotations on a suitably encoded state involving auxiliary qubits. Consequently, there is a correspondence between qubit count and the size of the native gate set. One might then wonder: what is the smallest number of auxiliary qubits that still allows for universal parity computing? Here, we demonstrate that the answer is one, if the number of logical qubits is even, and two otherwise. Furthermore, we present a sufficient condition for a given parity gate set to be universal. This leads to a variety of different universal parity gate sets corresponding to different numbers of auxiliary qubits, and more generally contributes to the understanding of which entangling gates are required to augment the set of single-qubit unitaries to perform universal quantum computing. As a consequence, we obtain (i) minimal implementations of the parity framework on e.g., a triangular lattice, (ii) hardware specific implementations of the parity flow framework on e.g., a heavy-hex lattice, and (iii) novel universal resources for measurement-based quantum computation (MBQC).