Fault-tolerant quantum error detection

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Science Advances  20 Oct 2017:
Vol. 3, no. 10, e1701074
DOI: 10.1126/sciadv.1701074


  • Fig. 1 Graphical representation of the logical operators and stabilizers defining the [[4, 2, 2]] code on physical qubits 1 to 4.

    The structure of the logical operators X and Z for the two encoded qubits La and Lb, and for the two stabilizers Sx and Sz, is defined in Eqs. 3 and 4.

  • Fig. 2 Circuit diagrams.

    (A to D) Circuits for the encoding of four different logical states constructed such that logical qubit La is prepared fault-tolerantly. Any logical state can be achieved by applying single logical qubit operators to states encoded as shown here. (E and F) Circuits for the two stabilizers Sx and Sz, which project Z- and X-type errors, respectively, onto an ancilla qubit a. Note that a controlled Z-gate is realized by an inverted CNOT with the ancilla in the Z-basis as the target. (G) Example of fault-tolerant construction of circuits for logical qubit La: The encoding circuit for |00〉L has a single nondetectable error channel. A bit-flip error E occurring as shown can change the state to |01〉L, which is an error on the logical gauge qubit Lb. Logical qubit La is prepared fault-tolerantly. This property holds for all circuits (A to F).

  • Fig. 3 Results achieved with logical state |00〉L.

    (A) Results from the preparation of state |00〉L. The abscissa represents the five-qubit states in decimal. We succeed in preparing (prep) and measuring the state with ~90% probability. The inset shows the result after postselection on the state being in the logical basis, that is, even parity. It is broken down by the logical state of the fault-tolerantly (FT) prepared qubit La and the non–fault-tolerantly (NFT) prepared qubit Lb. (B) Results of the stabilizer measurements after preparation of |00〉L. The yields are 77.8(6) and 65.2(7)% for Sz and Sx, respectively. The insets show that the error probability on the fault-tolerantly prepared and stabilized logical qubit La is an order of magnitude below the non–fault-tolerantly prepared and stabilized qubit Lb.

  • Fig. 4 Performance of the code under different kinds of artificial errors.

    (A) Logical error probability under artificially introduced stochastic Pauli errors. Uncertainties shown in gray with dashed outlines. We prepare state |00〉L, introduce a specific error, and apply Sz before readout. The parameter values for the curves (see Materials and Methods) corresponding to the two logical qubits are determined either experimentally (solid lines) or from simulation (dashed lines). The black curve shows the limiting theoretical case without intrinsic errors (see Materials and Methods). At low added error rates, the intrinsic errors dominate, and the fault-tolerantly constructed qubit La starts about an order of magnitude below the non–fault-tolerantly constructed qubit Lb. With increasing inserted error probability, the added Pauli errors become dominant, and the La/b curves converge and approach the theory curve without intrinsic error. The solid black line shows the error rate for a single physical qubit. La results in a lower error across the entire range relative to the physical qubit, although our measurement uncertainty means that this is no longer significant below p ~ 0.07%. The Lb error is lower than the physical qubit for added errors >4%. (B and C) Preparing |00〉L and measuring Sx/z with purposefully miscalibrated two-qubit gates, known as XX-gates. A miscalibration of α means that the Bell state produced by the gate is imbalanced: Embedded Image. The yields diminishing with miscalibration for the stabilizer measurements are shown in (B), whereas the errors on the logical qubits presented in (C) remain similar, with La errors about an order of magnitude lower than Lb errors.


  • Table 1 Probability distributions (in percentage) of measured logical states |LaLb〉 for various prepared logical states in each row, with and without stabilizers Sx or Sz applied.

    The measurement basis is shown in the last column. The logical states are |00 > L … |11 > L, measured in the Z-basis, and | + + 〉L … | − − 〉L, measured in the X-basis. The very low error probability on the first logical qubit La compared to Lb shows the action of its fault-tolerant construction. We run every circuit 5000 to 6000 times. The results without stabilizer show the number of rejected runs from the parity check on the data qubits (typically ~8%) whereas the additional discard on the results with stabilizer (typically ~20%) is due to the ancilla result. The physical errors for state preparation and measurement are 0.3(1)% for states |0〉 and | + 〉, and 1.2(1)% for states |1〉 and | − 〉.

    YieldMeasured logical state |LaLbMeasurement basis
    | + + 〉| + − 〉| − + 〉| − − 〉X
    | + + 〉L91.1(4)95.7(3)3.9(3)0.24(8)0.22(8)X
    | + + 〉LSz68.2(7)93.0(5)4.2(4)1.3(2)1.5(2)X
    | + + 〉LSx72.1(6)94.3(4)4.5(4)0.5(1)0.7(2)X
    | − 1〉L90.1(4)0.22(8)50.5(8)0.09(6)49.2(8)Z
    | − 1〉L87.0(5)0.3(1)0.3(1)50.4(8)48.9(8)X
    | − 1〉LSz79.9(6)0.15(7)50.0(8)0.10(6)49.8(8)Z
    | − 1〉LSz75.5(6)0.4(1)0.3(1)50.1(8)49.2(8)X
    | − 1〉LSx72.1(6)0.6(1)50.2(8)0.5(1)48.7(8)Z
    | − 1〉LSx76.2(5)0.4(1)0.4(1)50.0(7)49.2(7)X
    |0 + 〉L93.2(3)47.4(5)52.5(5)0.06(3)0.05(3)Z
    |0 + 〉L92.4(4)50.0(8)0.04(4)49.8(8)0.09(5)X
    |0 + 〉LSz81.6(6)48.3(8)51.4(8)0.17(8)0.17(8)Z
    |0 + 〉LSz68.5(7)47.1(9)2.4(3)47.4(9)3.1(3)X
    |0 + 〉LSx72.0(6)48.3(8)51.5(8)0.16(7)0.12(7)Z
    |0 + 〉LSx70.9(7)49.4(9)0.4(1)49.7(9)0.5(1)X

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