I'm new in quantum computing, I have this question.

Qubits $x$ and $y$ are in $\mathbb{C}^2$ (column vector) and $A, B$ are unitary matrices ($A$ 8x8 and $B$ 4x4 matrix).

If I'm not wrong the input of $A$ is $x_1 \otimes x_2 \otimes x_3$ which is in $\mathbb{C}^8$ (column vector). Now given the output $z=A(x_1 \otimes x_2 \otimes x_3)$ how can I extract $y_3$ from $z$ to calculate $y_3 \otimes y_4$ ($x_4 = y_4$) which is the input of $B$?

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  • $\begingroup$ Suppose the circuit goes on. I don't want to measure y's. I want to how to perform calculation after A matrix-gate. $\endgroup$
    – asv
    Apr 28 '20 at 15:12
  • $\begingroup$ I want to know how to calculation are peformed. I don't want to use software I want to understand the calculation. $\endgroup$
    – asv
    Apr 28 '20 at 15:15
  • $\begingroup$ Use the partial trace: en.wikipedia.org/wiki/Partial_trace $\endgroup$
    – draks ...
    Apr 28 '20 at 15:19
  • $\begingroup$ Thank you for your link $\endgroup$
    – asv
    Apr 29 '20 at 10:37

In general, you cannot just extract the part of the state that corresponds to $y_3$ and $y_4$. Instead, you have to consider the entire state (which you will describe using a 16-element vector), and you apply to it the unitary $I\otimes B$ where $I$ is the $4\times 4$ identity matrix.

  • $\begingroup$ Thank you very much! So this is the way the calculation are made in any quantum circuit? $\endgroup$
    – asv
    Apr 28 '20 at 15:21
  • 1
    $\begingroup$ It's the general way that you have to perform the calculation if you want to simulate what a quantum circuit does via a classical calculation. That's the whole "problem" with quantum computation - essentially, you have to keep the whole exponentially large vector in memory, which we cannot do classically. $\endgroup$
    – DaftWullie
    Apr 29 '20 at 8:25

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