Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

A lot of known quantum gates are in the Pauli group (I,X,Z,Y) or in the Clifford group (H,P,Cnot). I need examples of the quantum gates that aren't in this groups. Also, are there are matlab functions to check if a quantum gate (2x2 or 4x4) is in any of these groups? Or maybe there are matlab routines that generate quantum gates outside of this groups?


share|improve this question

3 Answers 3

up vote 3 down vote accepted

Any gate of the form diag$(1,1,1,\exp(i\phi))$ is not in $C_n$ for any $n$ unless $\phi = 2\pi k/2^n$ for some integers $k$ and $n$. This can be proven by induction using the similar result for single-qubit gates. I'm not sure if this is included in any published paper.

We don't have a good characterization of gates in $C_n$ for $n > 2$, so there is no known more general method of generating them, or even checking if a gate has this property.

share|improve this answer
Ok, good explanation, thanks. –  user901366 Sep 25 '12 at 17:33

The Pauli and Clifford groups only contain finitely many elements, so almost any unitary is not going to be in them.

Just ask matlab to make you a random unitary. For example, almost any one qubit phase gate is not in these groups.

I'm not aware of a matlab function that checks for membership in these groups. However, you could write a simple code for the small gate sizes you mentioned. Since elements $U$ of the Clifford group satisfy $U (Pauli) U^\dagger = (Pauli)'$ you could run through all Pauli operators and make sure they are mapped to each other e.g. by computing the overlap of operators using something like the matrix inner product $(M,N) = tr(M^\dagger N)$ since one has $(\sigma^a, \sigma^b) = tr(\sigma^a \sigma^b) = 2 \delta^{ab}$.

There is probably a better way, but this silly algorithm should work if you only care about 2x2 and 4x4 gates.

share|improve this answer
I will try this. I want search for quantum gates that is not in C_{n} as decribed in arxiv.org/abs/quant-ph/9908010, section II. –  user901366 Sep 9 '12 at 18:54
It seems from your comment that you're now asking something much more general and complicated than before. The classes $C_n$ described in your reference are an infinite series of groups built from the Pauli group. Only $C_1$ and $C_2$ were mentioned in your original question, are these all you're interested in? Or do you also care about $C_n$ for $n>2$? –  Physics Monkey Sep 9 '12 at 19:09
Initially I need a quantum gate that is not in $C_{2}$. But examples, or to know how to build it, of the gates out of $C_{n}$ could be very useful. Any ideia? –  user901366 Sep 9 '12 at 23:13
@user901366: perhaps you should ask your follow-up question separately, and accept one of the answers to this question. –  Niel de Beaudrap Sep 13 '12 at 9:54

Precisely because the Clifford group is generated by the operators $$ S = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \qquad H = \tfrac{1}{\sqrt 2} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \qquad \mathbf{cnot} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} $$ and tensor products with the identity, it follows that every Clifford operator is of the form $2^{-n/2} M$, where $M$ is a matrix over the Gaussian integers (i.e. complex numbers where the real and imaginary parts are both integers). Any unitary which is not of this form is therefore not a Clifford group operator.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.