(Clifford + T) Gate Set

The (Clifford + T) gate set typically refers to the set of gates

$(\text{Clifford} + T) := \{H, S, \text{CNOT}, T\}.$

It is universal in the sense that any quantum circuit can be approximated to arbitrary precision by this set of gates. One distinguishes between Clifford and non-Clifford gates, where the $T$ gate is a special case of a non-Clifford gate. In many fault-tolerant quantum computing settings, $T$ gates are considered the most expensive gates to implement, and their count is often the target metric to minimize in quantum compilation routines (see this blog post on T-gate optimization).

In most quantum error correction codes like the surface code, (Clifford + T) is the target gate set.

Inputs

• Arbitrary circuit $U$

Outputs

• Approximated circuit $U_\text{(Clifford + T)}$

Example

A Pauli rotation about a multiple of $\frac{\pi}{2}$ is a Clifford operation. For example, we have

$R_Z\left(\frac{\pi}{2}\right) = e^{-i \frac{\pi}{4}} S.$

However, in general, Pauli rotations are realized by a series consisting of phase gates $S$, Hadamard gates $H$ and, in particular, $T$ gates. For example, $R_Z\left(\frac{\pi}{128}\right)$ is realized by the series

SHTHTHTHTHTHTHTSHTHTHTHTSHTHTHTHTHTSHTSH
THTHTHTHTSHTHTHTSHTSHTHTSHTSHTSHTHTHTHTS
HTHTHTHTHTSHTSHTHTSHTHTSHTSHTSHTSHTHTSHT
SHTSHTSHTHTHTSHTSHTSHTHTHTHTSHTHTSHTHTHT
SHTHTHTHTSHTHTSHTHTSHTSHTSHTHTHTHTHTHTHT
SHTHTSHTHTHTSHTSHTHTHTSHTSHTSHTHTSHTHTHT
HTSHTSHTSHSSS

with a precision of $10^{-10}$.

Depending on the definition, Toffoli gates are not considered to be in (Clifford + T), but they can be decomposed in this gate set in the following way.

import pennylane as qml

toffoli = qml.Toffoli([0, 1, 2]).decomposition()
print(qml.draw(lambda: [qml.apply(op) for op in toffoli])())

0: ───────────╭●───────────╭●────╭●──T──╭●─┤  
1: ────╭●─────│─────╭●─────│───T─╰X──T†─╰X─┤  
2: ──H─╰X──T†─╰X──T─╰X──T†─╰X──T──H────────┤  

Typical usage

In most error correction schemes, quantum circuits are assumed to be executed in the (Clifford + T) gate set, where Clifford gates are executed in a fault-tolerant way and $T$ gates are performed via magic state injection. A magic state $|H\rangle = |0 \rangle + e^{i \frac{\pi}{4}} |1\rangle$ can be used as a resource from an auxiliary wire to apply $T$ on an arbitrary state $|\psi\rangle$ in the following way.

0  : |ψ> ─╭●───────S─┤  T|ψ>
aux: |H> ─╰X──┤↗├──║─┤  
               ╚═══╝   

This is called magic state injection. Magic states can be created in a noisy way and purified using magic state distillation to arbitrary precision.

References

[1] "Universal Quantum Computation with ideal Clifford gates and noisy ancillas", Sergei Bravyi, Alexei Kitaev, arXiv:quant-ph/0403025, 2004

[2] "The Solovay-Kitaev algorithm", Christopher M. Dawson, Michael A. Nielsen arXiv:quant-ph/0505030, 2005

[3] "Optimal ancilla-free Clifford+T approximation of z-rotations", Neil J. Ross, Peter Selinger, arXiv:1403.2975, 2014