§ Quantum computation without complex numbers
I recently learnt that the Toeffili and Hadamard gates are universal for
quantum computation. The description of these gates involve no complex numbers.
So, we can write any quantum circuit in a "complex number free" form. The caveat
is that we may very well have input qubits that require complex numbers.
Even so, a large number (all?) of the basic algorithms shown in Nielsen and
Chaung can be encoded in an entirely complex-number free fashion.
I don't really understand the ramifications of this, since I had the intuition
that the power of quantum computation comes from the ability to express
complex phases along with superposition (tensoring). However, I now have
to remove the power from going from R to C in many cases. This is definitely
something to ponder.