Why is the CZ-gate given Clifford index 4368?

[Wikstahl]

Hi!

Apologies for opening an issue, but I couldn’t find any contact details in the repository.

I’m currently using the Two-Qubit Clifford Interleaved Randomized Benchmarking feature with a CZ-gate as the interleaved gate. While looking through the code, I saw that the CZ-gate corresponds to Clifford index 4368:

Relevant Code in two_qubit_clifford_group.py

From the implementation of CNOT_like_gates(cls, idx), index 4368 falls into the class of CNOT-like Cliffords, and the corresponding decomposition is computed as follows:

@classmethod
def CNOT_like_gates(cls, idx):
    """
    Returns the gates for Cliffords of the cnot like class
        (q0)  --C1--•--S1--      --C1--•--S1------
                    |        ->        |
        (q1)  --C1--⊕--S1--      --C1--•--S1^Y90--
    """
    assert(idx < cls.GRP_SIZE_CNOT)
    idx_0 = idx % 24
    idx_1 = (idx // 24) % 24
    idx_2 = (idx // 576) % 3
    idx_3 = (idx // 1728)

    C1_q0 = [(g, 'q0') for g in gate_decomposition[idx_0]]
    C1_q1 = [(g, 'q1') for g in gate_decomposition[idx_1]]
    CZ = [('CZ', ['q0', 'q1'])]

    idx_2s = SingleQubitClifford._get_clifford_id(S1[idx_2])
    S1_q0 = [(g, 'q0') for g in gate_decomposition[idx_2s]]
    idx_3s = SingleQubitClifford._get_clifford_id(np.dot(C1[idx_3], Y90))
    S1_yq1 = [(g, 'q1') for g in gate_decomposition[idx_3s]]

    gates = C1_q0 + C1_q1 + CZ + S1_q0 + S1_yq1
    return gates

For idx = 4368, we obtain:

• idx_0 = 0
• idx_1 = 14
• idx_2 = 1
• idx_3 = 2

Looking these up in clifford_group_single_qubit, we get the corresponding single-qubit decompositions:

• C1_q0 -> [I, I, I]
• C1_q1 -> [I, H, S2]
• S1_q0 -> [I, I, S]
• S1_q1 -> [I, I, S2] (with an additional Y90 before)

Given the decomposition above, I don’t see how this results in a CZ-gate. My expectation was that the Clifford corresponding to index 4368 should directly implement a CZ operation, but the presence of Hadamard (H) and (S, S2) suggests additional single-qubit rotations are involved.

Could anyone please clarify the reasoning behind why this particular Clifford index is associated with a CZ-gate? Is there an implicit transformation I'm missing?

Many thanks!

[MiniSean]

The CNOT-like Clifford group consists of controlled-phase gate + Hadamards + Single qubit clifford rotations. The method returns
gates = C1_q0 + C1_q1 + CZ + S1_q0 + S1_yq1 return gates - which includes the controlled-phase gate. Note that one of the Hadamards is absorbed in one of the single qubit Clifford gates.

[Wikstahl]

The CNOT-like Clifford group consists of controlled-phase gate + Hadamards + Single qubit clifford rotations. The method returns gates = C1_q0 + C1_q1 + CZ + S1_q0 + S1_yq1 return gates - which includes the controlled-phase gate. Note that one of the Hadamards is absorbed in one of the single qubit Clifford gates.

While it’s clear to me that the Hadamard can be absorbed into a single-qubit Clifford, this still doesn't explain how the presence of single-qubit gates from the S and S2 groups will result in a CZ gate.