Cycle Benchmarking

Motivations

The Cycle Benchmarking (CB) protocol was proposed in 2019 by A. Erhard et al. in [1] as an alternative to the Interleaved Randomized Benchmarking (IRB) protocol with a more practical implementation that does not require the compilation of \(n\)-qubit Clifford gates.

Protocol

A cycle is defined as a set of gates acting on a disjoint subset of qubits, all occurring in parallel (with an analogy to a clock cycle). The circuit corresponding to the CB protocol is shown in Fig. 1. The circuit starts with a layer of single-qubit gates to prepare the state in a random Pauli basis (purple boxes). The CB protocol interleaves each cycle (gate G) with a random layer of Pauli gates (yellow boxes) that aim to twirl the noise into a stochastic Pauli channel (also known as randomized compiling). Each cycle is repeated \(l\) times. The gate of interest \(G\) should satisfy \(G^l=I\) (i.e., \(G\) applied \(l\) times should equal the identity). The last layer of single-qubit gates (purple boxes at the end of the circuit) turns back the qubits in the initial Pauli basis state.

The previous circuit permits the extraction of the error associated with the interleaved G gate combined with the Pauli gates \(r_\mathrm{cbg}\). A second experiment can be done in complement to only extract the error rate of the gate G (just as in the IRB protocol). It consists of applying the same protocol but with an Identity instead of the gate G (see Fig. 2) to obtain the error rate associated with the identity \(r_\mathrm{cbi}\).

Quantum circuit associated with the cycle benchmarking protocol for the identity

It then consists of using the same approximation as in IRB to estimate the error rate of the gate \(G\):

\[r_\mathrm{G} \approx r_\mathrm{cbi} - r_\mathrm{cbg}.\]

As in the IRB protocol, this approximation induces systematic errors, but bounds extracted with CB are generally tighter than with IRB. The reader can find a detailed comparison in [2].

Assumptions

This protocol assumes that the noise model is Markovian, meaning that the noise of a gate does not depend on the sequence of previous gates (history-independent).
The protocol presented on this page is only valid for benchmarking Clifford gates. CB can be used to benchmark non-Clifford gates, but requires substantial changes as discussed in [1].

Limitations

As explained in [2], the fidelity extracted with the CB protocol does not constitute the true process fidelity in general but rather a lower bound on the process fidelity. The tightness of this bound depends on the number of samples, which should scale at least as \(\min(20, 4^n -1)\) where \(n\) denotes the number of qubits.

Extensions to Cycle Benchmarking

There are other CB-like protocols, such as Character Cycle Benchmarking (CCB), in which one may choose a broader class of target gates G and \(l\)-values rather than restricting to Clifford cycles satisfying \(G^l=I\), and Character Average Benchmarking (CAB) [3], which reduces resource requirements by averaging the \(4^{n}\) Pauli eigenvalues into \(2^n\) terms by partially depolarizing the noise via local Clifford twirling.

Implementation

A tutorial for implementing the CB protocol is available in the QCMet software repository.

References

[1]A. Erhard et al., “Characterizing large-scale quantum computers via cycle benchmarking,” Nature communications, vol. 10, no. 1, p. 5347, 2019.
[2]A. Hashim et al., “Practical Introduction to Benchmarking and Characterization of Quantum Computers,” PRX Quantum, vol. 6, no. 3. APS, p. 030202, 2025.
[3]Y. Zhang, W. Yu, P. Zeng, G. Liu, and X. Ma, “Scalable fast benchmarking for individual quantum gates with local twirling,” Photonics Research, vol. 11, no. 1, pp. 81–99, 2022.