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We compare two different implementations of fault-tolerant entangling gateson logical qubits. In one instance, a twelve-qubit trapped-ion quantum computeris used to implement a non-transversal logical CNOT gate between two five qubitcodes. The operation is evaluated with varying degrees of fault tolerance,which are provided by including quantum error correction circuit primitivesknown as flagging and pieceable fault tolerance. In the second instance, atwenty-qubit trapped-ion quantum computer is used to implement a transversallogical CNOT gate on two [[7,1,3]] color codes. The two codes were implementedon different but similar devices, and in both instances, all of the quantumerror correction primitives, including the determination of corrections viadecoding, are implemented during runtime using a classical compute environmentthat is tightly integrated with the quantum processor. For differentcombinations of the primitives, logical state fidelity measurements are madeafter applying the gate to different input states, providing bounds on theprocess fidelity. We find the highest fidelity operations with the color code,with the fault-tolerant SPAM operation achieving fidelities of 0.99939(15) and0.99959(13) when preparing eigenstates of the logical X and Z operators, whichis higher than the average physical qubit SPAM fidelities of 0.9968(2) and0.9970(1) for the physical X and Z bases, respectively. When combined with alogical transversal CNOT gate, we find the color code to perform thesequence--state preparation, CNOT, measure out--with an average fidelitybounded by [0.9957,0.9963]. The logical fidelity bounds are higher than theanalogous physical-level fidelity bounds, which we find to be [0.9850,0.9903],reflecting multiple physical noise sources such as SPAM errors for two qubits,several single-qubit gates, a two-qubit gate and some amount of memory error.

Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code