# Does having infinitely many quantum gates helpful in solving problems in any ways

A general classical gate accepting $$m$$ inputs and producing $$m$$ outputs has total $$2^{2^m}$$ number gates (which may or may not have names.) Thus number of gates functions are finite.

But QC allows to have infinitely many quantum gates (unitary operations).

• Does this fact of having infinite number of gates for n-qubits serves any advantage over classical computers?
• If yes, does this implies that to solve a particular problem, there should exist a combination of gates (a circuit)?

A classical bit is a binary piece of information - I.e it can be 0 or it can be 1 (we can choose also up/down, true/false - it doesn’t matter). Exactly 2 states for a classical bit.

A qubit can be in a quantum state out of infinite amount of possible states (though in the end it collapses to a single value out of possible 2, just like the classical bit). A quantum state of a single qubit is well represented using the a sphere called the Bloch sphere - any point on the surface of the sphere corresponds to a possible quantum statevector of a single qubit. Think of quantum gates as a movement of the statevector on the surface of the sphere. Since there are infinitely many points on the surface of the sphere there are infinitely many “movements” or “rotations” we can perform to the statevector - any such rotation is equivalent to a quantum gate. Practically, the physical ability to perform “little” rotations is limited, so there aren’t infinitely many possible gates - there are many, but it is a finite amount.

Having explained that - to my knowledge - that’s not the ingredient giving quantum computation advantage over classical computation. The power of quantum computation arises from sophisticated manipulations of 3 important quantum properties - superposition, entanglement and interference.

If yes, does this implies that to solve a particular problem, there should exist a combination of gates (a circuit)?

Not everything is computable. However, if a problem is computable on a classical computer then it is computable (theoretically) on a quantum computer, and there is some sequence of quantum gates that implements the desired computation.

• I think it is important to address the false assumption that quantum computers have an infinite amount of quantum gates available. With the current technology, we can only approximate the real line as a finite set of numbers Aug 16, 2022 at 21:51