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I am very new to Quantum Computing thus please excuse the layman question.

I am aware that just like classical gates Quantum Computation also has a set of universal gates. Moreover, a universal set of gates can be created that "operate on" at most 2 Quantum bits. And each such Gate can be/is represented by a set of matrices over complex numbers (https://en.wikipedia.org/wiki/Quantum_logic_gate, https://en.wikipedia.org/wiki/List_of_quantum_logic_gates).

Moreover, any set of N Qbits can always be represented by a vector of $2^N$ amplitudes such that each the square of each amplitude represents the probability that the Qbits will collapse to the corresponding state vector when measured.

Now here in lies my doubt: 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:

Query 1: 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?

Query 2: For an N Qbit system/circuit there are $2^N$ amplitudes, so in any such circuit what does a random 2 Qbit Gate do mathematically? Does it still take 2 complex numbers and multiply them with the matrix representing the Gate or something else?

Most of the places (like wiki) they show a simple 2 Qbits in classical states and what a matrix does to them. Or simply use the "operate on" word without explaining what all that means in a $N$ Qbit generic system and an example. So if someone can please explain with an example what mathematically happens in any generic circuit when we use a word "operate on" it would be immensely helpful.

Please assume a complete lay person without background in QC (only conceptual ideas as described above).

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First, do you know the concept of the Bloch sphere? If you don't you should take a look at it. Let me try to answer everything you asked.

Abstracting from quantum computing, a qubit is a vector with dimension 2 and it can be represented in the superficies of the sphere of radius 1 (Bloch sphere). When you operate over this qubit, mathematically you are multiplying a matrix with a column vector, but what you are doing is changing the position of your qubit in the Bloch sphere. Let me give you an example: by default your qubit is in state |0> and so it is in the north pole of the Bloch sphere, if you apply an X gate (for example) you are moving your state for the south.

In the case, that you have N qubits and you operate with a 2 qubit gate, in a mathematical point of view, you are acting with a matrix that is $I_{N-2}\otimes G$ over a vector of shape N, with G the two-qubit gate. Please note that is not a rule the qubits being the last two.

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  • $\begingroup$ Thank you for. I did read a bit about Bloch Sphere. As I very limitedly understand its a sphere of unit vector - the vector representing the amplitude (complex number) as you described. But the rest of it: "In the case, that you have N qubits.." I didn't understand a single bit! Is there a worked out example for laymen: i.e. say a 4 qbit system, initially described by a state vector of non trivial amplitudes. Then a universal quantum gate is applied to two qbits in it, and a worked out step by step result for the new state vector as a result. $\endgroup$ Oct 18, 2023 at 15:07
  • $\begingroup$ I have read many abstract statements/explanations (and you too tried to help explain), but couldn't find a single worked out non trivial example in sight! And I can't understand them unless I see a few [important "non-trivial"] examples of: Initial non-trivial state/state-vector of say 4 qbits, the application of gate/matrix, and the resultant state (how it is obtained). I am still missing those.. and without i can't understand these explanations in vaccum $\endgroup$ Oct 18, 2023 at 15:12
  • $\begingroup$ Maybe, I didn't understand your doubt properly at first. After your comment, I believe that I can add some comments. Are you comfortable with algebra? I believe that it is easier for you to think of the qubit as vectors and not amplitudes, i.e. think as two coordinates in the space, with the restriction that all of the possible points are in the superficie of the Bloch Sphere. Operating means a multiplication between a state and an operator. Suppose that v is a column vector (the state) with shape 4 and M is an operator or gate 4x4, if you operate on state v with M, the result will be Mv. $\endgroup$ Oct 20, 2023 at 13:59

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