Interleaved Randomized Benchmarking (IRB)

Motivation

The Interleaved Randomized Benchmarking (IRB) protocol was proposed in 2012 by E. Magesan et al. [1] after the multi-qubits Clifford Randomized Benchmarking protocol. The IRB aims to evaluate the average error rate of individual Clifford gates, which was impossible with the multi-qubit CRB method.

Protocol

The protocol consists of two experiments. The first experiment consists of running a multi-qubit CRB to extract the decay parameter \(\alpha_\mathrm{crb}\) and the corresponding Error Per Clifford \(r_\mathrm{crb}\). The following figure recalls the circuit corresponding to the multi-qubit CRB protocol:

The second experiment aims to evaluate the error rate of the n-qubit Clifford gate of interest \(G\). The gate \(G\) is interleaved with random \(n\)-qubit Clifford gates \(g_1\) up to \(g_l\). The final gate \(g_\mathrm{end}\) aims to reverse the complete sequence of gates.

This second experiment aims to extract the decay parameter \(\alpha_\mathrm{irb}\) and the corresponding error rate \(r_\mathrm{irb}\). In practise, \(\alpha_\mathrm{irb}\) should decay faster than \(\alpha_\mathrm{crb}\) due to the additional inserted gates \(G\). The infidelity \(r_\mathrm{G}\) of the gate \(G\) is then estimated as:

This method also permits to extract bounds on the true error rate of \(G\) \(\epsilon_\mathrm{G}\) (see. equation 5 of [1]).

Assumptions

• The noise variation between the Clifford gates \(g_i\) should be small.
• This protocol relies on the same assumptions of the multi-qubit CRB protocol (i.e., noise should be depolarizing and Markovian).

Limitations

• This method does not provide a very reliable estimate of the fidelity of the gate \(G\) as unitary errors can coherently add or cancel with each others. Consequently, this protocol can produce negative error rates. It can leads to large differences between the measured error rate \(r_\mathrm{G}\) and the true error rate \(\epsilon_\mathrm{G}\) associated with the gate \(r_\mathrm{G}\).
• This protocol inherits from the same limitations of the multi-qubit CRB protocol (as depth scaling and dependence over compilation strategies).

Extensions

The IRB protocol has been extended to other gates that are not from the Clifford group [2] [3].

References

1. [1]E. Magesan et al., “Efficient measurement of quantum gate error by interleaved randomized benchmarking,” Physical review letters, vol. 109, no. 8, p. 080505, 2012.
2. [2]R. Harper and S. T. Flammia, “Estimating the fidelity of T gates using standard interleaved randomized benchmarking,” Quantum Science and Technology, vol. 2, no. 1, p. 015008, 2017.
3. [3]S. Garion et al., “Experimental implementation of non-Clifford interleaved randomized benchmarking with a controlled-S gate,” Physical Review Research, vol. 3, no. 1, Mar. 2021, doi: 10.1103/physrevresearch.3.013204. [Online]. Available at: http://dx.doi.org/10.1103/PhysRevResearch.3.013204