Dihedral Randomized Benchmarking

Motivation

Dihedral RB was proposed by A. Carignan-Dugas in 2015 to benchmark other gate sets than the ones implementing the Clifford group [1]. The main motivation of this protocol is to design an efficient method to benchmark other gate sets than the Clifford group to evaluate the error rate of gates required for universal quantum computing (such as the T-Gate).

Protocol

This protocol evaluates the average fidelity of a single-qubit unitary group of gates called the dihedral group. For a fixed \(j\), the dihedral group \(\mathcal{D}_j\) is composed of a parameterized rotation gate \(R_j(z_i) = e^{i \pi z_i/j Z}\) (where \(Z\) corresponds to the Pauli-Z gate) and a Pauli-X gate.
The protocol starts by fixing \(j\). In the paper, the authors mainly use \(j=4\) and \(j=8\). The experiment is done for different sequence lengths \(l\). For each sequence length, two arrays of integers are uniformly and randomly generated: \(\mathbf{z} = (z_1, z_2, ..., z_l)\) with \(z_i \in \{0, 1, 2 ..., j-1\}\) and \(\mathbf{x} = (x_1, x_2, ..., x_l)\) with \(x_i \in \{0, 1\}\). The system is prepared either in state \(\ket{0}\) or in state \(\ket{+}\). Each gate of the sequence \(g_i\) is composed of the \(R_j(z_i)\) gate and the \(X^{x_i}\) gate. The last quantum gate \(g_{end}\) is used to invert the preceding sequence of quantum gates. The final measurement is done such that an ideal evolution should give a \(+1\) outcome of the POVM. This success probability is then used fit the decay function and extract the average gate fidelity. The parameters \(b_1\) and \(b_2\) are used to ease the fitting of the decaying function (see [1] for details).

Focusing on \(\mathcal{D}_4\) and \(\mathcal{D}_8\) permits to benchmark the fidelity of the T gate. Indeed, \(\mathcal{D}_4\) is only generated from Clifford gate (phase and Pauli-X). \(\mathcal{D}_8\) can be implemented by using \(\mathcal{D}_4\) and adding the T gate. If the average fidelity is the same for \(\mathcal{D}_4\) and \(\mathcal{D}_8\), it means that the T gate has an average fidelity similar to the gates of the Clifford group.
At first, the average fidelity of \(\mathcal{D}_4\) is established using the circuit in Fig 1. The fidelity of the T gate is then evaluated by combining the dihedral benchmarking method to the interleaved benchmarking approach (implemented by the circuit shown in Fig 2.). In this protocol, each gate \(g_i\) is interleaved with a T gate (\(R_8(1)\)). The average fidelity of the T gate can then be estimated by comparing the result of the interleaved experiment to the experiment done for \(\mathcal{D}_4\). The estimated value is tight if the gates from \(D_4\) have much higher fidelity than the T gate, which is argued to be the regime of interest.

The z-axis is mainly used in the examples but the protocol can be extended to characterize other single-qubit rotations.

Limitations

This protocol is only valid for benchmarking single qubit gates. Additionally, the method used to estimate the fidelity of the T gate is only valid if the quality of the Clifford gates is significantly greater than the fidelity of the T gate.

References

1. [1]A. Carignan-Dugas, J. J. Wallman, and J. Emerson, “Characterizing universal gate sets via dihedral benchmarking,” Physical Review A, vol. 92, no. 6, p. 060302, 2015.