In this example, a physical qubit prepared in state |0〉 is outperformed by logical qubit L a over the entire range, although our measurement uncertainty limits the significance of this to above p ~ 0.07%. The clear separation between the two logical qubits is persistent until they converge above 20% introduced error and approach the curve for the theoretical case without intrinsic errors. We further compare our experimental results to an exact simulation with optimized error parameters (see Materials and Methods). Instead of trying to reproduce a stochastic error channel, which can be tedious for low error rates ( 25), we sample the various error configurations and then multiply them by their respective statistical importance to obtain a logical error probability (see Materials and Methods). First, we deliberately introduce single- and two-qubit Pauli errors and study how errors on L a and L b scale with increasing physical qubit errors. To further investigate the robustness of the code, we add two kinds of error to the system. The yield is higher for preparation of logical states without syndrome measurements because there are fewer gates to introduce error and there is only a single selection step. We must expect to discard around half of the runs when measuring both stabilizers. The circuits succeed in discarding nearly all errors, but we pay a price because the yield is in the 65 to 75% range. These results show the power of fault-tolerant preparation and stabilizer measurement. The gauge qubit L b circuit failures occur at approximately the error rate of one two-qubit gate, whereas L a errors are suppressed substantially below that level to <1%. Note that after a circuit with seven CNOT gates, each of which introduces 3 to 4% infidelity, we obtain the correct answer |00〉 L with 97 to 98% probability. The circuit elements that dominate the intrinsic errors in our system are the two-qubit gates. Table 1 (see Materials and Methods) summarizes results for different logical states prepared with the circuits shown in Fig. The opposite is true when applying the stabilizers to |++〉 instead. S z introduces Z-type errors, which do not affect |00〉 L. S x introduces X-type errors in the system, which can be seen from a higher L b error. The errors on L a are 0.2(1)%, similar to the error floor given by the |11〉 L population, which is slightly higher than after mere state preparation due to the additional gates introduced by the stabilizers. The results of the logical states are similar, with |00〉 L populations of 97.8(2) and 97.1(3)%, respectively, and the errors occur predominantly in the non–fault-tolerantly prepared gauge qubit L b. Populations in the odd-numbered states reflect events where an error is detected by a stabilizer. 2 (E and F), for nondemolition syndrome extraction. With |00〉 L thus prepared, we apply in turn the two stabilizers S z and S x, shown in Fig. Note that similar weight 4 stabilizers have recently been implemented in superconducting qubits ( 23). In addition to the error checks provided by stabilizer measurements, only even-parity outcomes are accepted when the data qubits are measured at the end of the circuit. Both stabilizer measurements serve to determine the overall yield, that is, the fraction of runs for which no error was indicated. Because we prepare eigenstates of logical Pauli operators, only logical Pauli operations that change the ideal state result in errors. With only one ancilla qubit available, we measure the two stabilizers in separate experiments. Measuring the ancilla yields either |0〉, indicating no error, or |1〉, meaning an error has occurred and the run is to be discarded. Applying these stabilizers conditional on the state of an ancilla qubit extracts the parity of the data qubits along X or Z (see Fig. The difference is that the stabilizers have weight 4 because we simultaneously extract information about the gauge qubit L b.
#Quantum error code#
As in a Bacon-Shor code block ( 20, 21), the code space together with the logical operators and stabilizers form a subsystem that allows local syndrome extraction similar to that of Napp and Preskill ( 22), as depicted in Fig.
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