nLab T gate

Idea

In quantum information theory and quantum computing, by the T-gate and the S-gate one refers to the quantum gates acting on single qbits that in the defining measurement-basis $\big\{ {\vert 0 \rangle}, {\vert 1 \rangle} \big\}$ are given by the $2 \times 2$ complex matrices

$$T \;=\; \left[ \begin{array}{cc} 1 & 0 \\ 0 & e^{\pi \mathrm{i}/4} \end{array} \right]$$

and

$$S \;=\; T^2 \;=\; \left[ \begin{array}{cc} 1 & 0 \\ 0 & \mathrm{i} \end{array} \right] \mathrlap{\,,}$$

respectively, where “$\mathrm{i}$” denotes the imaginary unit.

In particular, the Pauli Z-gate is decomposable into these gates as

$$Z \;=\; S^2 \;=\; T^4 \,.$$

Remark

Beware of these alternative names and their subtleties:

• The T-gate is also known as the “$\pi/8$-gate” (e.g. in Nielsen & Chuang 2000 p xxx), even though the phase rotation is by $\pi/4$ – but differs by only a global phase from the $R_Z(\pi/8)$ rotation gate.

• The S-gate is also known as the “phase gate”, but that term is ambiguous.

Related gates

• Hadamard gate

• Pauli gates

• rotation gate

• CNOT gate

• Toffoli gate

References

• Michael A. Nielsen, Isaac L. Chuang, p xxx in: Quantum computation and quantum information, Cambridge University Press (2000) [doi:10.1017/CBO9780511976667, pdf, pdf]