Question

Conceptually in any quantum Circuit, a 'step' in computation is defined as operating on one or two qbits using the relevant gates (as far as I understand). But mathematically I am unclear what the word operating on two Qbits mean? Forgetting about Quantum Computing and everything else:

Purely mathematically, what is the thing a Gate operates on mean? I understand an amplitude is a complex number, so does that mean any Gates takes in 2 Complex Numbers and transform them depending on the kind of Gate?

Answer 1

Quantum gates operate on the quantum state of qubits, which are represented by complex vectors in a multi-dimensional complex vector space. The action of a gate is mathematically described as a matrix multiplication that transforms the qubits' state.

Step-by-step explanation:

In quantum computing, a gate operates not on two complex numbers but rather on the quantum state of one or two qubits. Mathematically, we describe a single qubit as a vector in a two-dimensional complex vector space, where the state of the qubit can be a linear combination of its basis states, usually represented as |0> and |1>. The states of a qubit are represented by complex amplitudes (probability amplitudes) that, when squared, give the probabilities of the qubit being measured in one of the basis states. When we talk about a gate operating on two qubits, it means the gate acts on a four-dimensional complex vector space formed by the tensor product of the two qubit states. This operation is described through a matrix multiplication involving the gate's matrix and the qubits' state vectors, resulting in a transformation of the qubits' state.