Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes

One of the most promising routes towards fault-tolerant quantum computation utilizes topological quantum error correcting codes, such as the Z₂ surface code. Logical qubits can be encoded in a variety of ways in the surface code, based on either boundary defects, holes, or bulk twist defects. However, proposed fault-tolerant implementations of the Clifford group in these schemes are limited and often require unnecessary overhead. For example, the Clifford phase gate in certain planar and hole encodings has been proposed to be implemented using costly state injection and distillation protocols. In this paper, we show that within any encoding scheme for the logical qubits, we can fault tolerantly implement the full Clifford group by using joint measurements involving a single appropriately encoded logical ancilla. This allows us to provide low overhead implementations of the full Clifford group in surface and color codes. It also provides the first proposed implementations of the full Clifford group in hyperbolic codes. We further use our methods to propose state-of-the art encoding schemes for small numbers of logical qubits; for example, for code distances d=3, 5, 7, we propose a scheme using 60, 160, 308 (respectively) physical data qubits, which allow for the full logical Clifford group to be implemented on two logical qubits. To our knowledge, this is the optimal proposal to date, and thus may be useful for demonstration of fault-tolerant logical gates in small near-term quantum computers.