Multi-qubit Clifford Randomized Benchmarking (CRB)

Motivations

The primary motivation behind the development of multi-qubit Clifford Randomized Benchmarking was to generalize the single-qubit randomized benchmarking protocol to systems comprising multiple qubits. This extended protocol was introduced in 2010 by E. Magesan et al. [1].

Protocol

The protocol utilizes the multi-qubit Clifford group \(C_n\) to benchmark a quantum system with \(n\) qubits. For each sequence length \(l\), Clifford gates are uniformly and efficiently sampled from \(C_n\) [2]. Each gate of the Clifford group is efficiently decomposed in a sequence of single and two-qubit gates, with depth scaling as \(O(n^2 / \log(n))\). The inverse unitary \(R\) is efficiently computed from the sequence \(g_lg_{l-1}...g_1\) [3], and a final unitary \(P\) is randomly and uniformly chosen to produce an eigenstate of the observable \(\sigma^z\) (The unitary \(P\) was not used in the initial protocol but is considered to be a best practise). The success metric is the probability of observing the Identity up to the random Pauli gate \(P\). It is estimated for the different lengths \(l\) and used to fit the exponential decay function.

Assumptions

• This protocol assumes that the noise model is Markovian.
• The average error of all the Clifford gates should be depolarizing (assumption that is for all Clifford based benchmarks).

Limitations

• One limitation identified by the community concerns the depth scaling associated with the decomposition of each gate of the Clifford group \(C_n\). As the number of qubits increases, the implementation becomes increasingly challenging, and circuit fidelity tends to degrade rapidly in the presence of noise.

• The output fidelity of the quantum circuit also strongly depends on the compilation process used to map each Clifford gate to the gate set natively used by the quantum computer. Inefficiencies or suboptimal strategies in this step can significantly impact the benchmarking results.

References

1. [1]E. Magesan, J. M. Gambetta, and J. Emerson, “Scalable and robust randomized benchmarking of quantum processes,” Physical review letters, vol. 106, no. 18, p. 180504, 2011.
2. [2]R. Koenig and J. A. Smolin, “How to efficiently select an arbitrary Clifford group element,” Journal of Mathematical Physics, vol. 55, no. 12, Dec. 2014, doi: 10.1063/1.4903507. [Online]. Available at: http://dx.doi.org/10.1063/1.4903507
3. [3]D. Gottesman, Stabilizer codes and quantum error correction. California Institute of Technology, 1997.