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How to calculate the sum of $\sum\limits_{j}{\langle A|{{B}_{j}}\rangle |{{C}_{j}}\rangle }$ with quantum circuits by qiskit, where $\sum\limits_j \langle A| B_j\rangle |C_j \rangle$, $A,B_j,$ and $C_j$ are three quantum states of the same dimension?

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    $\begingroup$ Are $A, B_j$ and $C_j$ Statevectors in Qiskit? How is the question related to Qiskit? Please provide a more elaborate formulation. $\endgroup$
    – Peter Ivashkov
    Jun 8 at 16:23
  • $\begingroup$ Yes, these are three state vectors of the same dimension, but they are not equal $\endgroup$
    – R-X Zhao
    Jun 9 at 12:58

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Here is one way to do it:

  1. You can compute the inner product $\langle A |B_j\rangle$ with inner() method of Statevector. Just type A.inner(B).
  2. You can multiply a Statevector with a scalar. So you can write $\langle A |B_j\rangle |C\rangle$ like A.inner(B)*C, since the inner product of two states is a scalar.
  3. You add can Statevector as well, so summing over j should be straightforward.
  4. Notice though, that a valid Statevector must have a norm 1. That is, S.inner(S) must be equal 1. Adding or multiplying with a scalar doesn't, in general, preserve the norm. You can check if your Statevector is normalized with is_valid() method.
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  • $\begingroup$ Sorry this is not what I want, I want to implement each step of the above formula on a quantum computer, instead of the classical computer sharing too many problems for the quantum computer $\endgroup$
    – R-X Zhao
    Jun 10 at 6:55

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