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Two qubit Clifford group

The two-qubit Clifford group is generated from single-qubit unitaries and the controlled-NOT (CNOT) gate. When implementing the protocol, it's crucial to use a Clifford decomposition that minimizes the number of two-qubit gates, as these generally have lower fidelities than single-qubit gates.

As in the single qubit case, the exponential decay should be fit to

\[\begin{equation} F(m)= Ap^m + B \label{eq:survival_rate} \end{equation}\]

where \(m\) is the number of Cliffords in the sequence, and \(A\) and \(B\) are parameters related to state preparation and measurement, and \(1-p\) is the depolarization rate.

The two-qubit Clifford group can be divided into four classes 1:

  1. The Single-qubit Clifford class that consists of 576 elements (\(24^2\)) and represents all single-qubit Clifford operations:
1
2
3
(q0) --C1--

(q1) --C1--
  1. The CNOT-class that has 5184 elements (\(24^2\times 3^2\)) and contains all combinations of the following sequence:
1
2
3
(q0)  --C1--•--S1--
            | 
(q1)  --C1--⊕--S1--
  1. The ISWAP-class The third class also has 5184 elements and contains all combinations of the following sequence:
1
2
3
(q0)  --C1--*--S1--
            |    
(q1)  --C1--*--S1--
  1. SWAP-class. The fourth class is the SWAP-class and consists of all 576 (\(24^2\)) combinations of the following sequence:
1
2
3
(q0)  --C1--x--
            |  
(q1)  --C1--x--

Therefore, there are in total 11 520 elements in the two-qubit Clifford group.

Gate Decomposition

The decomposition of the two-qubit Clifford group above is the optimal in terms of the number of CNOTs. Since an iSWAP requires two CNOTs and a SWAP requires three, the average number of CNOTs per Clifford operation is 1.5. The same holds if we use CZ instead of CNOT as the building block, since the CNOT can be decomposed as a CZ with two Hadamards. The number of single-qubit gates depends on how the single-qubit Cliffords are implemented.

Below follows the decomposition of the two-qubit gates in terms of the CZ-gate and single-qubit -gates:

  1. CNOT to CZ 
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3
(q0)  --C1--•--S1--      --C1--•--S1------
            |        ->        |
(q1)  --C1--⊕--S1--      --C1--•--S1^Y90--
  1. iSWAP to CZ 
1
2
3
(q0)  --C1--*--S1--     --C1--•---Y90--•--S1^Y90--
            |       ->        |        |
(q1)  --C1--*--S1--     --C1--•--mY90--•--S1^X90--
  1. SWAP to CZ 
1
2
3
(q0)  --C1--x--     --C1--•-mY90--•--Y90--•-------
            |   ->        |       |       |
(q1)  --C1--x--     --C1--•--Y90--•-mY90--•--Y90--

Interleaved Clifford Randomized Benchmarking

Interleaved Clifford Randomized Benchmarking allows estimation of the error associated with an individual Clifford gate. The core idea is to perform two benchmarking experiments: one following the standard Clifford Randomized Benchmarking (RB) method and one with the target Clifford gate interleaved 2.

If the standard RB sequence is:

\[ C_1C_2\ldots C_m C_\mathrm{inverse}, \]

where each \(C\) is a Clifford gate, then the interleaved RB sequence introduces the Clifford gate \(C_\mathrm{target}\) spaced in between the Clifford gates:

\[ C_1C_\mathrm{target}C_2C_\mathrm{target}\ldots C_mC_\mathrm{target}C'_\mathrm{inverse} \]

where \(C_\mathrm{target}\) is the gate being characterized and must also belong to the Clifford group, and \(C'_\mathrm{inverse}\) is the final inverting gate that must be updated to invert the full gate sequence including \(C_\mathrm{target}\).

The error of \(C_\mathrm{target}\) is thus determined by comparing the decay rates Eq. \(\eqref{eq:survival_rate}\) of the standard Clifford RB and the interleaved Clifford RB experiments.

If the fitted decay parameters from the standard RB and the interleaved RB are denoted by \(p\) and \(p_\mathrm{interleaved}\) respectively. Then, the interleaved RB gate error for \(C_\mathrm{target}\) is found using

\[ r_{C_\mathrm{target}}=\frac{(d-1)(1-\frac{p_\mathrm{interleaved}}{p})}{d} \]

where \(d\) is the dimension of the system (e.g., \(d=2^n\) for an \(n\)-qubit system).

  1. A. D. Córcoles, J. M. Gambetta, J. M. Chow, J. A. Smolin, M. Ware, J. D. Strand, B. L. T. Plourde & M. Steffen. Process verification of two-qubit quantum gates by randomized benchmarking. Physical Review A, 87, 030301(R) (2013). ↩

  2. Lall, D., Agarwal, A., Zhang, W., Lindoy, L., Lindström, T., Webster, S., Hall, S., Chancellor, N., Wallden, P., Garcia-Patrón, R., Kashefi, E., Kendon, V., Pritchard, J., Rossi, A., Datta, A., Kapourniotis, T., Georgopoulos, K., & Rungger, I. (2025). A Review and Collection of Metrics and Benchmarks for Quantum Computers: definitions, methodologies and software. arXiv:2502.06717 ↩