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Interleaved randomized benchmarking of CNOT gate

This experiment measures the average \(\text{CNOT}\) gate fidelity using interleaved randomized benchmarking. The interleaved randomized benchmarking is an extension of the standard randomized benchmarking experiment.

Description

In the standard randomized benchmarking experiment, we used Clifford gates to determine the average fidelity of the gateset:

\[ [\mathrm{G_1} - \mathrm{G_2} - \mathrm{G_3} - \mathrm{G_{recovery}}], \]

where \(\mathrm{G_1}\), \(\mathrm{G_2}\) and \(\mathrm{G_3}\) are different Clifford gates, and \(\mathrm{G_{recovery}}\) is chosen such that the qubits return to their initial state at the end of the sequence. The probability that this happens in practice is known as the survival probability, \(P_\text{surv}\), and is given by

\[ P_{\text{surv}}(x) = Ap^{x} + B , \]

where \(A\), \(p\), and \(B\) are fit parameters, and \(x\) is the gate sequence length. For a single qubit, the average gate fidelity, \(\alpha\), can be obtained from \(p\) using

\[ \alpha = 1-\frac{(1-p)}{2} . \]

Clifford gates

A Clifford gate is a quantum gate that maps each Pauli operator (X, Y, Z) to another Pauli operator under conjugation. The main Clifford gates are the Hadamard (\(\text{H}\)), Phase (\(\text{S}\)), and \(\text{CNOT}\) gates. They’re important in randomized benchmarking because they’re easy to simulate and preserve stabiliser states.

To benchmark a specific gate, such as the \(\text{CNOT}\), we use interleaved randomised benchmarking, in which the gate of interest is inserted between each Clifford gate.

As a quick reminder, the \(\text{CNOT}\) gate transforms the two-qubit state

\[ a\ket{00} + b\ket{10} + c\ket{01} + d\ket{11} \]

to

\[a\ket{00} + b\ket{10} + d\ket{01} + c\ket{11}\]

and can represented by the matrix

\[ \text{CNOT} = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ \end{bmatrix}. \]

The interleaved RB sequence for the \(\text{CNOT}\) gate would therefore look like this:

\[ [\mathrm{G_1} - \mathrm{CNOT} - \mathrm{G_2} - \mathrm{CNOT} - \mathrm{G_3} - \mathrm{CNOT} - \mathrm{G'_{recovery}}]. \]

Here, \(\mathrm{G'_\text{recovery}}\) may differ from the recovery gate used at the end of the standard RB sequence; it ensures the qubits return to their initial state after the interleaved sequence. Since the \(\text{CNOT}\) gate acts on computational basis states, it is itself a Clifford gate, and therefore the recovery gate should also come from the Clifford group.

The survival probability of this sequence is then given by

\[ P^{\text{CNOT}}_{\text{surv}}(x) = A'(p \cdot p_{\text{CNOT}})^{x} + B' . \]

By comparing the average fidelity of both sequences (standard and interleaved), it's possible to isolate the impact of the \(\text{CNOT}\) gate on the overall sequence fidelity and calculate the \(\text{CNOT}\) gate fidelity, \(\alpha_{\text{CNOT}}\), using1

\[ \alpha_{\text{CNOT}} = 1-\frac{(1-p \cdot p_{\text{CNOT}})}{2}. \]

Experiment steps

  1. \(N\) different sequences of the same length, \(x\), are defined. Each conventional RB sequence consists of gates from the set \(R_x(\frac{\pi}{2})\), \(R_x(-\frac{\pi}{2})\), \(R_y(\frac{\pi}{2})\), \(R_y(-\frac{\pi}{2})\), \(I\). The interleaved RB sequence consists of these gates as well as \(\text{CNOT}\) gates.

  2. The conventional and interleaved RB sequences are applied to the qubit, which begins in its ground state \(|0\rangle\).

  3. The resonator transmission is measured for each of the \(N\) sequences.

  4. Steps 1 to 3 are repeated for varying lengths of the gate sequence, \(x\).

Analysis steps

  1. The qubit state is determined from the \(I\)-\(Q\) data by applying the discriminator trained in the Readout discriminator training experiment.

  2. The survival probabilities for the conventional (\(P_{\text{surv}}(x)\)) and interleaved (\(P^{\text{CNOT}}_{\text{surv}}(x)\)) sequences are obtained by averaging over the \(N\) different sequences.

  3. \(P_{\text{surv}}(x)\) and \(P^{\text{CNOT}}_{\text{surv}}(x)\) are plotted against the sequence length, \(x\), and exponential curves are fit to the data.

  4. The fit parameters \(p\) and \(p_{\text{CNOT}}\) are used to calculate the average gate fidelities, \(\alpha\), for the conventional and interleaved sequences.

  5. The average fidelity of the \(\text{CNOT}\) gate is then extracted.

  1. Easwar Magesan, Jay M. Gambetta, B. R. Johnson, Colm A. Ryan, Jerry M. Chow, Seth T. Merkel, Marcus P. da Silva, George A. Keefe, Mary B. Rothwell, Thomas A. Ohki, Mark B. Ketchen, and M. Steffen. Efficient measurement of quantum gate error by interleaved randomized benchmarking. Phys. Rev. Lett., 109:080505, Aug 2012. doi:10.1103/PhysRevLett.109.080505. ↩