Are SU($n$) operations enough for quantum computation?

Question

Usually we want a quantum computer that can perform all foreseeable unitary operations U($n$). A quantum processor that can naturally perform at least 2 rotation operators $R_k(\theta)=\exp(-i\theta\sigma_k/2)$, where $\sigma_k$ are the Pauli matrices; can generate any SU(2) rotation of the Bloch sphere. And the usual Pauli operations (up to a global phase) can be generated by just choosing the right angle $R_k(-\pi)=i\sigma_k$. By creating some controlled-$i\sigma_x$, I guess we could generate all SU($n$). Would a quantum processor restricted to SU($n$) operations limit the power to simulate quantum systems and other algorithms of interest?

Edit: note that controlled-SU(2) gates are still in SU(4).

Answer 1

If you want to be more precise about it, quantum (pure, ket) states are elements of complex projective spaces, $\mathbb{CP}^n$. This is the set of equivalence classes of elements of $\mathbb C^{n+1}$ modulo multiplication by complex scalars.

So "gates" should really be described as maps between such equivalence classes. This is the projective unitary group, ${\rm PU}(n)\simeq{\rm PSU}(n)$. You can think of this as the set of equivalence classes of unitaries modulo multiplication by a scalar phase. Note that when you consider transformations on elements of $\mathbb{CP}^n$, the difference between special and "regular" unitary matrices disappears, as reflected by the fact that ${\rm PU}(n)={\rm PSU}(n)$.

Of course, it is generally easier to work on regular linear spaces rather than on their projective counterparts, and simply remember to interpret the results of the calculations appropriately at the end. So in the more standard language, yes ${\rm SU}(n)$ operations are sufficient, though you don't need to only use gates in there, as all gates differing by only a complex phase factor represent the same physical operation. Case in point, the Pauli $X$ gate is a common choice, despite it not being special, as $i X\in\mathrm{SU}(2)$.

Answer 2

Two states that are the same up to a global phase are physically indistinguishable; some even go as far as to say that they are exactly the same state. Any $U_{1}$ and $U_{2} = e^{i\alpha}U_{1}$ will thus give physically indistinguishable states, meaning that there is nothing that $U_{2}$ can do that $U_{1}$ cannot.

Would a quantum processor restricted to SU(n) operations limit the power to simulate quantum systems and other algorithms of interest?

No. There's no removed physical property by 'removing' the global phase.

In theoretical analysis, it is often chosen to restrict to $SU(n)$ - but this is not necessarily the only choice, or the most obvious choice. Even more, the usual Pauli matrices $X, Y, Z$ actually have determinant $-1$, so are not even in $SU(2)$.

Finally, note that once you start talking about controlled-$U(n)$ gates vs. controlled-$SU(n)$ gates, there actually is a physical difference, because it stops being a global phase. But, any controlled-$U(n)$ can be implemented using its controlled-$SU(n)$ gate plus an $R_{z}$ rotation on the control qubit.