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The three-spin-$1/2$ decoherence-free subsystem defines a logical qubitprotected from collective noise and supports exchange-only universal gates.Such logical qubits are well-suited for implementation withelectrically-defined quantum dots. Exact exchange-only entangling logical gatesexist but are challenging to construct and understand. We use a decouplingstrategy to obtain straightforward approximate entangling gates. A benefit ofthe strategy is that if the physical spins are aligned, then it can implementevolution under entangling Hamiltonians. Hamiltonians expressible as linearcombinations of logical Pauli products not involving $\sigma_y$ can beimplemented directly. Self-inverse gates that are constructible from theseHamiltonians, such as the CNOT, can be implemented without the assumption onthe physical spins. We compare the control complexity of implementing CNOT toprevious methods and find that the complexity for fault-tolerant fidelities iscompetitive.

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