nLab controlled Z-gate

Idea

In quantum computing and quantum information theory, the controlled Z gates are controlled quantum gate versions of the Z gate.

Where the Z gate acts on a single qbit, being the unitary operator

$$\begin{array}{ccc} \mathbb{C} &\overset{Z}{\longrightarrow}& \mathbb{C} \\ \vert b \rangle &\mapsto& (-1)^b \vert b \rangle \mathrlap{\,,} \end{array}$$

the $(N-1)$-controlled Z gates for $N \in \mathbb{N}_{\geq 1}$ are the unitary operators on the tensor product of $N$ qbits given by

(1)$$\begin{array}{ccc} \mathbb{C}^N &\overset{C^{N-1} Z}{\longrightarrow}& \mathbb{C}^N \\ \vert b_1,\cdots, b_N \rangle &\mapsto& (-1)^{b_1 \cdots b_N} \vert b_1, \cdots, b_N \rangle \mathrlap{\,,} \end{array}$$

where $b_i \in \{0,1\}$.

(cf. Beverland, Campbell, Howard & Kliuchnikov 2020).

Since the exponential expression in (1) is

$$(-1)^{b_1 \cdots b_N} = \begin{cases} -1 & \text{ if }\; \forall_i \colon b_i\!=\!1 \\ +1 & \text{ otherwise } \end{cases}$$

we equivalently have:

$$C^{N-1}Z \,=\, id - 2{\vert 1, \cdots, 1\rangle}{\langle 1, \cdots, 1 \vert} \mathrlap{\,.}$$

(cf. Gühne et al. 2014 (1)).

Examples

The CCZ quantum gate is the doubly-controlled form of the Pauli $Z$-gate, which plays a role as a non-Clifford gate alternative to the T-gate in the discussion of Takagi, Yoder & Chuang 2017.

Related concepts

• T-gate

References

• O. Gühne, M. Cuquet, F.E.S. Steinhoff, T. Moroder, M. Rossi, D. Bruß, B. Kraus, C. Macchiavello; eq (1) in: Entanglement and nonclassical properties of hypergraph states, J. Phys. A: Math. Theor. 47 (2014) 335303 [doi:10.1088/1751-8113/47/33/335303, arXiv:1404.6492]

• Ryuji Takagi, Theodore J. Yoder, Isaac L. Chuang: Error rates and resource overheads of encoded three-qubit gates, Phys. Rev. A 96 (2017) 042302 [arXiv:10.1103/PhysRevA.96.042302, arXiv:1707.00012]

• Michael Beverland, Earl Campbell, Mark Howard, Vadym Kliuchnikov; §4.1 of: Lower bounds on the non-Clifford resources for quantum computations, Quantum Sci. Technol. 5 (2020) 035009 [doi:10.1088/2058-9565/ab8963, arXiv:1904.01124]