Fault-tolerant quantum circuits on connectivity-constrained hardware with swap gates

In near-term quantum computing devices, connectivity between qubits remain limited by architectural constraints. A computational circuit with given connectivity requirements necessary for multiqubit gates has to be embedded within physical hardware with fixed connectivity. Long-distance gates have to be done by first routing the relevant qubits together. The simplest routing strategy involves the use of gates to swap the information carried by two unconnected qubits to connected ones. Ideal gates just permute the qubits; real gates, however, have the added possibilities of causing simultaneous errors on the qubits involved and spreading errors across the circuit. A general swap scheme thus changes the error-propagation properties of a circuit, including those necessary for fault-tolerant functioning of a circuit. Here, we present a simple strategy to design the swap scheme needed to embed an arbitrary circuit onto a physical hardware with constrained connectivity, in a manner that preserves the fault-tolerant properties of the abstract circuit. Our approach is generally applicable, and works on any circuit, including ones that carry out logical operations and error correction. The embedded circuit will, of course, be noisier, compared to a native implementation of the abstract circuit, but we show in the examples of embedding surface codes on heavy-hexagonal and hexagonal lattices that the deterioration is not severe. This then offers a straightforward solution to implementing circuits with fault-tolerance properties on current hardware.