0
$\begingroup$

I recently have gained an interest in quantum computers and quantum computation, and I was wondering, what is the minimum number set of quantum gates that allow for arbitrary quantum computation?

$\endgroup$
0

1 Answer 1

1
$\begingroup$

marked CW

This might depend on what you mean by "universal" and also what you mean by "minimum" and further by "number".

For example, a common gate set that is quantum computationally universal is the CCNOT (Toffoli) + Hadamard set - but note that, with such a gate set, your amplitudes are always real (and never imaginary or complex). Alternatively you could consider CSWAP (Fredkin) + Hadamard, which, like CCNOT+Hadamard, is (I think) computationally universal and also only has real amplitudes, but the CSWAP gate never changes the parity (one's count) in any basis state it acts upon. So, such sets are computationally universal but do have restrictions on the Hilbert space upon which they act.

Of note, though, is that both the CCNOT and the CSWAP gates are three-input/three-output gates, while most physical implementations would only ever work with two-input gates. So, another common gate set is CNOT+Pauli+P, where the CNOT gate is a two-qubit entangling gate, the Pauli gates are our friendly X, Y, and Z gates, and P is an arbitrary phase gate. So you can do all you need without three-input gates (and, you can explore complex amplitudes with Y and P gates).

But with the P gate it's assumed that you can apply an arbitrary phase - so the number of such gates might be argued to be infinite. Accordingly and alternatively one other common gate set is the Clifford sets plus the T gate - which is a phase shift by $\pi/4$. Thus Clifford+T is, in some ways, the "minimum" gate set.


I'm currently reading Schrödinger's Killer App by Johnathan Dowling. It's (much, much) better than Kaku's more recent book; I'm reading Dowling's work for the nice insights and history of the field from the mid-90's onward. Dowling makes much of the universality of what he calls the RAT, CAT, and ENT gate sets- RAT being a gate that "rotates at twenty-two point five degrees" or a T gate; CAT being a Hadamard gate to prepare Schrödinger's cat, and ENT being an arbitrary entangling gate such as CNOT.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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