Fault-Tolerant Logical Gates in the IBM Quantum Experience

Quantum computers will require encoding of quantum information to protect them from noise. Fault-tolerant quantum computing architectures illustrate how this might be done but have not yet shown a conclusive practical advantage. Here we demonstrate that a small but useful error detecting code improves the fidelity of the fault-tolerant gates implemented in the code space as compared to the fidelity of physically equivalent gates implemented on physical qubits. By running a randomized benchmarking protocol in the logical code space of the [4,2,2] code, we observe an order of magnitude improvement in the infidelity of the gates, with the two-qubit infidelity dropping from 5.8(2)% to 0.60(3)%. Our results are consistent with fault-tolerance theory and conclusively demonstrate the benefit of carrying out computation in a code space that can detect errors. Although the fault-tolerant gates offer an impressive improvement in fidelity, the computation as a whole is not below the fault-tolerance threshold because of noise associated with state preparation and measurement on this device.

Figure 1

(a) Fitting charts for the four different types of runs used to extract the $b$ and $c$ parameters in Eq. (2). The data were fit as described in the Supplemental Material [26]. Postselection on invalid code states was used to throw away runs where an error was detected. The bars show the percentage of detected (and discarded) runs for the [4,2,2] code runs. The number of initial errors are greater in the phase run because of the need to perform an initial non-fault-tolerant gate (see text). (b) Simulation showing survival probabilities for logical “real-Clifford” RB using the [4,2,2] code and the equivalent physical two-qubit RB. (Here, we have only shown the “real” runs, not the “phased” runs.) The same high fidelity four-qubit noise map (fidelity $\approx 0.99$) is used for the runs, although additional noise is applied to the cnots of the run on physical qubits. The noise map contains single-qubit errors as well as $ZZ$ errors between neighboring qubits. Although the combined noise map is high fidelity, the correlated errors are sufficiently strong that the fidelity decreases when using encoded gates, meaning the encoded computation performs worse than the bare physical computation in the correlated noise scenario.