Mirror Randomized Benchmarking (MRB)

Motivations

Mirror Randomized Benchmarking protocol was introduced in 2022 by T. Proctor et al. in [1] and extended a previous paper of the same authors on mirror circuits [2]. The main property of this protocol is that it can easily scale to many qubits. It can be used to benchmark Clifford and universal gate sets. This protocol avoids heavy compilation processing that limited the previous approaches, such as the multi-qubit Clifford RB method.

Protocol

The initial state \(\rho_0\) is prepared with a random layer of single-qubit Clifford gates (24 possible gates). This layer is inverted at the end of the circuit. An \(l\)-depth circuit contains \(l/2\) layers of \(\mu\)-distributed random gates \(G_1, G_2, ... G_{l/2}\) and \(l/2\) layers of the reversed gates \(G_{l/2}^{\dagger} ..., G_{2}^{\dagger}, G_{1}^{\dagger}\). Each of the \(l/2\) \(G_i\) gates is generated from a customized distribution \(\mu\) specified by the user. The layers of random gates are interleaved with random Pauli gates used to twirl the noise (yellow boxes). This step is called randomized compiling and avoids that errors coherently add or cancel between \(G_i\) and \(G_i^{\dagger}\). In the ideal case, each mirror circuit produces a unique output bitstring. The figure of merit used for measuring the experiment’s success is the Hamming distance between the sampled and the ideal output bitstring. This metric permits the extraction of the effective polarization of the quantum circuit used to estimate the entanglement fidelity of the circuit. The measured fidelity and the distribution \(\mu\) are strongly dependent and should be reported together. Different \(\mu\) distributions can be used to estimate the error rate of various subsets of layers.
Interestingly, this protocol is robust to gate-dependent errors. MRB protocol permits quantifying errors caused by two-qubit gates, including crosstalk errors.

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 layer of random gates generated by the distribution \(\mu\) should quickly randomize the error.
• The average error of all gates is depolarizing.

Limitations

• As explained in the theoretical part of [1], if the errors of \(\mu\)-distributed gates and their inverse are correlated, MRB slightly underestimates the error rate of the gate set. This effect is shown in [3].
• This benchmarking protocol is restricted to the Clifford group.

Implementation

A tutorial for the MRB protocol is available in the QCMet software repository.
An implementation of the MRB protocol is available in the pyGSTi library.

References

1. [1]T. Proctor, S. Seritan, K. Rudinger, E. Nielsen, R. Blume-Kohout, and K. Young, “Scalable Randomized Benchmarking of Quantum Computers Using Mirror Circuits,” Physical Review Letters, vol. 129, no. 15, Oct. 2022, doi: 10.1103/physrevlett.129.150502. [Online]. Available at: http://dx.doi.org/10.1103/PhysRevLett.129.150502
2. [2]T. Proctor, K. Rudinger, K. Young, E. Nielsen, and R. Blume-Kohout, “Measuring the capabilities of quantum computers,” Nature Physics, vol. 18, no. 1, pp. 75–79, Dec. 2021, doi: 10.1038/s41567-021-01409-7. [Online]. Available at: http://dx.doi.org/10.1038/s41567-021-01409-7
3. [3]A. Hashim et al., “Practical Introduction to Benchmarking and Characterization of Quantum Computers,” PRX Quantum, vol. 6, no. 3. APS, p. 030202, 2025.