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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?

Thanks..

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3 Answers

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.

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Ok, good explanation, thanks. –  user901366 Sep 25 '12 at 17:33
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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.

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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.

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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
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