Simultaneous Randomized Benchmarking

Motivations

The simultaneous Randomized benchmarking protocol was proposed by J. Gambetta et al. in 2012. This protocol aims to quantify the rate of cross-talk errors arising during the parallel execution of quantum gates in a quantum circuit. This method evaluates the correlations in the error rates.

Protocol

This section develops an example of protocol implementation based on benchmarking the single-qubit Clifford group on a two-qubit chip. The protocol begins with the single-qubit Clifford RB protocol performed for each qubit (see Step 1 in the figure): A single-qubit Clifford RB is executed on one qubit while others are kept idle. The decay parameters \(\alpha_1\) and \(\alpha_2\) are then extracted respectively for qubit 1 and qubit 2. In a second step (see Step 2 in the figure), a single-qubit RB is run simultaneously on both qubits, resulting in three decay parameters: \(\alpha_{1 \vert 2}\), \(\alpha_{2 \vert 1}\), and \(\alpha_{12}\). The parameters \(\alpha_{1 \vert 2}\) and \(\alpha_{2 \vert 1}\) are obtained from the first and second qubit measurements. The joint decay parameter \(\alpha_{12}\) is retrieved from the measurements of qubit 1 and 2. The values \(\alpha_{1}\) and \(\alpha_{1 \vert 2}\) (resp. \(\alpha_{2}\) and \(\alpha_{2 \vert 1}\)) can be used to measure the errors on qubit 1 (resp. 2) due to simultaneous quantum gate operations on qubit 2 (resp. 1). Potential correlations in the errors are obtained with the value \(\alpha_{12} -\alpha_{1 \vert 2} \alpha_{2 \vert 1}\).
This protocol can be combined with other RB protocols. In this exemple, SRB is based on the single-qubit Clifford RB protocol. However, it could be combined with other protocols to benchmark gates that are not Clifford, such as Direct RB or Binary RB.

Assumptions

• This protocol uses the same assumptions as the protocol being combined with it.The example would use the same assumptions as the single-qubit Clifford RB protocol (for instance, it considers a Markovian noise).

• This protocol uses the same assumptions as the single-qubit Clifford RB protocol (for instance, it considers Markovian noise).

Limitations

• This protocol suffers from the same limitations as the protocol being combined with it. The example would use the same limitations as the single-qubit Clifford RB protocol.

Extension

This protocol has been extended to quantify the cross-talk error of multi-qubit gates (see for example, the work of D. McKay et al. [1]). Notice that the EPLG score also gets some inspiration from the simultaneous RB protocol.

References

1. [1]D. C. McKay, S. Sheldon, J. A. Smolin, J. M. Chow, and J. M. Gambetta, “Three-Qubit Randomized Benchmarking,” Physical Review Letters, vol. 122, no. 20, May 2019, doi: 10.1103/physrevlett.122.200502. [Online]. Available at: http://dx.doi.org/10.1103/PhysRevLett.122.200502