Testing quantum fault tolerance on small systems

We extensively test a recent protocol to demonstrate quantum fault tolerance on three systems: (1) a real-time simulation of five spin qubits coupled to an environment with two-level defects, (2) a real-time simulation of transmon quantum computers, and (3) the 16-qubit processor of the IBM Q Experience. In the simulations, the dynamics of the full system is obtained by numerically solving the time-dependent Schrödinger equation. We find that the fault-tolerant scheme provides a systematic way to improve the results when the errors are dominated by the inherent control and measurement errors present in transmon systems. However, the scheme fails to satisfy the criterion for fault tolerance when decoherence effects are dominant.

Figure 1

Schematic representation of the system of five spin qubits (blue) coupled to an environment (red), described by the model Hamiltonian given in Eqs. (4, 5, 6, 7). The five qubits representing the quantum computer have a tunable all-to-all connectivity (dashed lines). The two-level systems in the environment form a ring with an always-on coupling between nearest neighbors and to the qubits of the quantum computer (solid lines). The latter is controlled by the coupling strength $\lambda$.

Figure 2

Test of the fault-tolerance criterion in system (1) for different coupling strengths (a) $\lambda =0.01$, (b) $\lambda =0.1$, (c) $\lambda =0.2$ between the qubits and the environment. Shown are the statistical distances to the ideal result for the selected bare (dashed red line) and encoded (solid green line) circuits as defined in Eqs. (2) and (3), and the postselection ratios (blue dots). All plotted quantities are dimensionless. The simulations were done for inverse temperature $\beta =1$ and environment $N_E=20$. Lines connecting the data points are guides to the eye.

Figure 3

Schematic image showing the five transmon qubits and six resonators described by the Hamiltonian given in Eqs. (8)–(10). The system represents a subset of the 16-qubit device ibmqx5 [4] with an additional resonator $r_5$ to enable the implementation of all bare and encoded circuits. Without this resonator, the encoded circuits with initial state $|00\rangle$ cannot be fault-tolerantly implemented [15].

Figure 4

Test of the fault-tolerance criterion in system (2), i.e., the real-time transmon simulation for different optimized gate sets (a) without frequency tuning, (b) with frequency tuning, and (c) with frequency tuning and measurement error $p=0.08$. Shown are the statistical distances to the ideal result for the selected bare (dashed red line) and encoded (solid green line) circuits as defined in Eqs. (2) and (3), and the postselection ratios (blue dots). All plotted quantities are dimensionless. Lines connecting the data points are guides to the eye.

Figure 5

Test of the fault-tolerance criterion in system (3), i.e., the 16-qubit device ibmqx5 using the qubits $(Q_4,Q_3,Q_2,Q_{15},Q_{14})$ on (a) April 3, 2018, (b) April 9, 2018, and (c) April 19, 2018. Shown are the statistical distances to the ideal result for the selected bare (dashed red line) and encoded (solid green line) circuits as defined in Eqs. (2) and (3), and the postselection ratios (blue dots). All plotted quantities are dimensionless. Only the circuits that could be mapped on the topology were run on the real device. Lines connecting the data points are guides to the eye.

Figure 6

Test of the fault-tolerance criterion in system (3) (see Sec. 5) for all 465 circuits using the qubits $(Q_4,Q_3,Q_2,Q_{15},Q_{14})$ of the IBM 16-qubit device ibmqx5 on April 19, 2018. Shown are the statistical distances to the ideal result for the selected bare (red plusses) and encoded (green crosses) circuits, and the postselection ratios (blue dots). All plotted quantities are dimensionless. Only the circuits that could be mapped on the topology were run on the real device. Lines connecting the data points are guides to the eye.

Figure 7

Test of the fault-tolerance criterion in system (3) (see Sec. 5) for all 465 circuits using the qubits $(Q_4,Q_3,Q_2,Q_{15},Q_{14})$ of the IBM 16-qubit device ibmqx5 on April 20, 2018. Shown are the statistical distances to the ideal result for the selected bare (red plusses) and encoded (green crosses) circuits, and the postselection ratios (blue dots). All plotted quantities are dimensionless. Only the circuits that could be mapped on the topology were run on the real device. Lines connecting the data points are guides to the eye.

Figure 8

Test of the fault-tolerance criterion in the decoherence model [system (1), see Sec. 3] using $N_E=5$, $\beta =1$, and $\lambda =0.1$ for the full set of circuits. Shown are the statistical distances to the ideal result for the selected bare (red plusses) and encoded (green crosses) circuits, and the postselection ratios (blue dots). All plotted quantities are dimensionless. Lines connecting the data points are guides to the eye.

Figure 9

Test of the fault-tolerance criterion for different environment sizes (a) $N_E=5$, (b) $N_E=20$, and (c) $N_E=27$. Shown are the statistical distances to the ideal result for the selected bare (dashed red line) and encoded (solid green line) circuits, and the postselection ratios (blue dots). All plotted quantities are dimensionless. The simulations were done for inverse temperature $\beta =1$ and coupling strength $\lambda =0.1$. Lines connecting the data points are guides to the eye.

Figure 10

Test of the fault-tolerance criterion for different inverse temperatures (a) $\beta =0$, (b) $\beta =1$, and (c) $\beta =5$. Shown are the statistical distances to the ideal result for the selected bare (dashed red line) and encoded (solid green line) circuits, and the postselection ratios (blue dots). All plotted quantities are dimensionless. All simulations were done for environment size $N_E=20$ and coupling strength $\lambda =0.1$. Lines connecting the data points are guides to the eye.