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.

Properties

Role in fault-tolerant quantum computing

The Clifford group of quantum gates on $n$ qbits (the normalizer of the Pauli group in $U(2^n)$, generated by the Hadamard gate $H$, the phase gate $S = diag(1,\mathrm{i})$ and the CNOT gate) is computationally weak: By the Gottesman-Knill theorem, Clifford quantum circuits acting on stabilizer states and followed by Pauli quantum measurements are efficiently simulable on a classical computer. But adjoining any one non-Clifford gate yields a universal gate set.

The prominent choice of that extra non-Clifford gate is the T-gate $T = \mathrm{diag}(1, e^{\pi \mathrm{i}/4})$: The set Clifford+$T$ generates, up to global phase, a dense subgroup of $SU(2^n)$, so that every quantum circuit is efficiently approximated by Clifford+$T$ circuits (by the Solovay-Kitaev theorem).

The catch is that:

• The prominent topological quantum error-correcting codes like the surface code (lattice quantum simulations of topological order) protect only Clifford gates. And this is a problem of principle (Eastin & Knill 2009, Bravyi & König 2013).

• Also actual topological quantum computation as long as its anyons are only of Ising type (such as expected, but not established, for the $\nu = 5/2$ FQH states) yields exactly only the Clifford gates (Bravyi 2006, Nayak et al. 2008, §IV.A).

The now standard strategy to work around this problem, trades the missing gate for a resource state: $T$-gates are enacted by gate teleportation, consuming magic states $\vert A \rangle = T \vert + \rangle$ via a CNOT, a $Z$-measurement and a classically controlled $S$-correction (which are all Clifford operations).

For more see at magic state.

Related gates

• Hadamard gate

• Pauli gates

• rotation gate

• CNOT gate

• Toffoli gate

References

General

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

As a non-Clifford gate

• Earl T. Campbell, Barbara M. Terhal, Christophe Vuillot: Roads towards fault-tolerant universal quantum computation, Nature 549 (2017) 172–179 [doi:10.1038/nature23460, arXiv:1612.07330]