Approximate exchange-only entangling gates for the three-spin-$1/2$ decoherence-free subsystem

The three-spin-$1/2$ decoherence-free subsystem defines a logical qubit protected from collective noise and supports exchange-only universal gates. Such logical qubits are well-suited for implementation with electrically-defined quantum dots. Exact exchange-only entangling logical gates exist but are challenging to construct and understand. We use a decoupling strategy to obtain straightforward approximate entangling gates. A benefit of the strategy is that if the physical spins are aligned, then it can implement evolution under entangling Hamiltonians. Hamiltonians expressible as linear combinations of logical Pauli products not involving $σ_y$ can be implemented directly. Self-inverse gates that are constructible from these Hamiltonians, such as the CNOT, can be implemented without the assumption on the physical spins. We compare the control complexity of implementing CNOT to previous methods and find that the complexity for fault-tolerant fidelities is competitive.