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Dependence of quantum error correction criteria on concentration of qubits in an ion-trap quantum computer

In order to carry out different quantum algorithms on the same quantum circuit, we have to start the next algorithm from the qubit arrangement at the end of the previous one. The arrangement of trapped ions after some algorithm execution is generally not optimal for the following algorithm in an ion-trap quantum computer. We simulated the sequences in the Grover's quantum circuits for searching an unsorted database in order to obtain sequence error rates under a random initial arrangement of qubits. When the sequence error rate for the circuit with QEC is less than that for the circuit without QEC, it is worth introducing the QEC circuit. From the condition, we obtained the quantum error correction criteria that depends on two parameters: concentrations flip of qubits and error rates. If we can know the flip error rate in an ion-trap quantum computer and determine the concentration, the criteria help us to realize a quantum circuit execution with less error rates in an ion-trap quantum computer.