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Single Qubit Clifford Randomized Benchmarking

Summary

The protocol used to benchmark single qubit Clifford gates.

  1. Initialize the qubit in the ground state.
  2. Apply a sequence of \(m\) Cliffords to the qubit.
  3. Apply an additional \((m+1)\) gate which inverts the whole sequence.
  4. Measure the resulting ground state probability.
  5. Repeat this procedure \(k\) times.
  6. Repeat for multiple lengths \(m\) to build up an exponential decay.

Clifford Based Randomized Benchmarking

Performing long sequences of random Clifford gates uniformly sampled from the Clifford group results in an exponential decay of the survival probability. The exponential decay (also called average sequence fidelity or survival rate) \(F\) should be fit to

\[ F(m)= Ap^m + B \]

where \(m\) is the number of Cliffords in the sequence, \(A\) and \(B\) are parameters related to state preparation and measurement, and \(1-p\) is the depolarization rate. Using \(p\), the average gate error per Clifford, \(r_\mathrm{Clifford}\), can be computed as

\[ r_\mathrm{Clifford} = 1-p-\frac{1-p}{d} \]

where \(d=2^N\) is the dimension of the Clifford gates for \(N\) qubits. By dividing the Clifford error by the average number of physical gates per Clifford, the error per gate, \(r_\mathrm{gate}\) can be obtained.

Single Qubit Clifford Group

To do single-qubit Clifford randomized benchmarking, one must first establish how to perform Clifford rotations on the qubit. This is done by enumerating the single-qubit Clifford group, which consists of all rotations on the Bloch sphere that map each of the six axial states (i.e., the six eigenvectors of the Pauli matrices) to one another while preserving their orthogonality.

For the single-qubit Clifford group there are 24 different Clifford operators. A convenient way to decompose these is to introduce the Pauli Group \(\mathbb{P}=\{I,X,Y,Z\}\), the Exchange Group \(\mathbb{S}=\{I,S,S^2\}\), and the Hadamard Group \(\mathbb{H}=\{I,H\}\)1. The Pauli Group, in the Pauli Transfer Matrix (PTM) representation, is given by:

\[ X = \begin{pmatrix} 1&&&\\ &1&&\\ &&-1&\\ &&&-1 \end{pmatrix},\quad Y = \begin{pmatrix} 1&&&\\ &-1&&\\ &&1&\\ &&&-1 \end{pmatrix},\quad Z = \begin{pmatrix} 1&&&\\ &-1&&\\ &&-1&\\ &&&1 \end{pmatrix} \]

which correspond to \(\pi\) rotations around the \(x\), \(y\), and \(z\) axes, respectively. The exchange-axis group is given by:

\[ S = \begin{pmatrix} 1&&&\\ &&&1\\ &1&&\\ &&1& \end{pmatrix},\quad S^2 = \begin{pmatrix} 1&&&\\ &&1&\\ &&&1\\ &1&& \end{pmatrix}, \]

which exchange \((x,y,z)\rightarrow (z,x,y) \rightarrow (y,z,x)\), and the Hadamard group:

\[ H = \begin{pmatrix} 1&&&\\ &&&1\\ &&-1&\\ &1&& \end{pmatrix}, \]

which exchanges \((x,y,z)\rightarrow (z,-y,x)\). The single-qubit Clifford group is generated by all combinations of elements in \(\mathbb{H}\), \(\mathbb{P}\), and \(\mathbb{S}\), leading to a total of \(2\times 3\times 4 = 24\) elements.

To implement single-qubit randomized benchmarking, these elements need to be decomposed into physical gates that can be implemented on a quantum computer. Many experiments use fundamental operations like \(e^{-iX\theta/2}\) or \(e^{-iY\theta/2}\), represented in the PTM as:

\[ X_\theta = \begin{pmatrix} 1&&&\\ &1&&\\ &&\cos\theta&-\sin\theta\\ &&\sin\theta&\cos\theta \end{pmatrix},\quad Y_\theta = \begin{pmatrix} 1&&&\\ &\cos\theta&&\sin\theta\\ &&1&\\ &-\sin\theta&&\cos\theta \end{pmatrix} \]

The table below lists the decomposition of all 24 Clifford elements and their physical decomposition in terms of \(X_\theta\) and \(Y_\theta\), following Epstein et. al.1. The "–" symbol signifies application in time (from left to right). The 24 Clifford elements and their physical decomposition in terms of \(X_\theta\) and \(Y_\theta\). The mean number of physical gates per Clifford element is \(1.875\).

Clifford elements Physical decomposition
\(I - I - I\) \(I\)
\(I - I - S\) \(Y_{\pi/2} - X_{\pi/2}\)
\(I - I - S^2\) \(X_{-\pi/2} - Y_{-\pi/2}\)
\(X - I - I\) \(X\)
\(X - I - S\) \(Y_{-\pi/2} - X_{-\pi/2}\)
\(X - I - S^2\) \(X_{\pi/2} - Y_{-\pi/2}\)
\(Y - I - I\) \(Y\)
\(Y - I - S\) \(Y_{-\pi/2} - X_{\pi/2}\)
\(Y - I - S^2\) \(X_{\pi/2} - Y_{\pi/2}\)
\(Z - I - I\) \(X - Y\)
\(Z - I - S\) \(Y_{\pi/2} - X_{-\pi/2}\)
\(Z - I - S^2\) \(X_{-\pi/2} - Y_{\pi/2}\)
\(I - H - I\) \(Y_{\pi/2} - X\)
\(I - H - S\) \(X_{-\pi/2}\)
\(I - H - S^2\) \(X_{\pi/2} - Y_{-\pi/2} - X_{-\pi/2}\)
\(X - H - I\) \(Y_{-\pi/2}\)
\(X - H - S\) \(X_{\pi/2}\)
\(X - H - S^2\) \(X_{\pi/2} - Y_{\pi/2} - X_{\pi/2}\)
\(Y - H - I\) \(Y_{-\pi/2} - X\)
\(Y - H - S\) \(X_{\pi/2} - Y\)
\(Y - H - S^2\) \(X_{\pi/2} - Y_{-\pi/2} - X_{\pi/2}\)
\(Z - H - I\) \(Y_{\pi/2}\)
\(Z - H - S\) \(X_{-\pi/2} - Y\)
\(Z - H - S^2\) \(X_{\pi/2} - Y_{\pi/2} - X_{-\pi/2}\)

  1. J. M. Epstein, A. W. Cross, E. Magesan, J. M. Gambetta. Investigating the limits of randomized benchmarking protocols. Phys. Rev. A 89, 062321 (2014) ↩↩