Microscopic quantum dynamics study on the noise threshold of fault-tolerant quantum error correction

Quantum circuits implementing fault-tolerant quantum error correction (QEC) for the three-qubit bit-flip code and five-qubit code are studied. To describe the effect of noise, we apply a model based on a generalized effective Hamiltonian where the system-environment interactions are taken into account by including stochastic fluctuating terms in the system Hamiltonian. This noise model enables us to investigate the effect of noise in quantum circuits under realistic device conditions and avoid strong assumptions such as maximal parallelism and weak storage errors. Noise thresholds of the QEC codes are calculated. In addition, the effects of imprecision in projective measurements, collective bath, fault-tolerant repetition protocols, and level of parallelism in circuit constructions on the threshold values are also studied with emphasis on determining the optimal design for the fault-tolerant QEC circuit. These results provide insights into the fault-tolerant QEC process as well as useful information for designing the optimal fault-tolerant QEC circuit for particular physical implementation of quantum computer.

Figure 1

Quantum gate symbols used to denote unitary transformations implemented with single-qubit $X$ and $Z$ and two-qubit $ZZ$ interactions.

Figure 2

Constructions used to implement controlled-$Z$ (upper one) and controlled-NOT (bottom one) gates. The definitions of elementary gates are shown in Fig. 1.

Figure 3

The fault-tolerant circuit for the preparation and verification of the four-qubit cat state. Note that the final result is conditioned by the outcome of the measurement at the end of the circuit. If the measurement outcome is zero, we accept the state; otherwise, the state is discarded and the circuit is started over again.

Figure 4

The fault-tolerant circuit for a QEC step on the three-qubit bit-flip code. Note that to ensure the fault-tolerant condition, the syndrome detection has to be repeated for several times and then the result of a majority vote taken.

Figure 5

A circuit implementing the fault-tolerant syndrome detection for the three-qubit bit-flip code.

Figure 6

Crash rate constants as a function of the noise strength. We show crash rate constants for a quantum memory using the three-qubit bit-flip code (upper left) and five-qubit code (upper right) and for a logical $X$ gate on the three-qubit bit-flip code (bottom left) and on the five-qubit code (bottom right). For the five-qubit code circuits, curves for different types of noise are presented. To show the threshold result, we also present curves for the error rate of a single physical qubit (dotted line). The noise threshold values are summarized in Table .

Figure 7

${\Gamma }_t$ as a function of noise strength for the fault-tolerant QEC circuit using three-qubit bit-flip code at different level of measurement errors. The error rate of a single physical qubit is also shown (dotted line). The measurement error has little effect on the threshold value.

Figure 8

The crash rate per computational step, ${\Gamma }_n$, for the three-qubit bit-flip code as a function of the noise strength. Curves for the distinct bath system (solid line) and collective bath system (dotted line) are shown. The result for the collective bath is close to the result for localized baths. This result suggests that collective bath has minor effect on the efficiency of fault-tolerant QEC.

Figure 9

The crash rate constant per step, ${\Gamma }_n$, as a function of the noise strength for the two different repetition protocols for a quantum memory using three-qubit bit-flip code. Using protocol B reduces the crash rate constant by a factor of 2.

Figure 10

A circuit implementing the fault-tolerant syndrome detection for the three-qubit bit-flipping code. In this circuit, we assume that the quantum computer can perform quantum gates on different qubits in parallel.

Figure 11

The crash rate constant per unit time, ${\Gamma }_t$, as a function of the noise strength for quantum memories using the three-qubit bit-flip code. Curves for three syndrome detection circuits different in the level of parallelism are shown. The solid line is for the circuit shown in Fig. 5, the dashed line is the increased parallelism circuit shown in Fig. 10, and the dash-dotted line is the maximal parallelism circuit that finishes all controlled-$Z$ operations in a single pulse. We see dramatic improvement in the noise threshold values when the level of parallelism is increased. The result indicates that by increasing the level of parallelism, the threshold value can be significantly improved.