I have $n$ states in superposition $|A\rangle=\dfrac{1}{2^{l/2}}\sum_{i=0}^{2^l-1}\sum_{j=0}^{2^l-1}|0\rangle^{\otimes q}|i\rangle^{\otimes l}|j\rangle^{\otimes l}$. Now I have to apply the transform in batches of $4$ i.e for example Let $l=2$, then $$A= \dfrac{1}{2^{2/2}}\left( |0\rangle^{\otimes q}|00\rangle|00\rangle+ |0\rangle^{\otimes q}|00\rangle|01\rangle+|0\rangle^{\otimes q}|00\rangle|10\rangle+|0\rangle^{\otimes q}|00\rangle|11\rangle+|0\rangle^{\otimes q}|01\rangle|00\rangle+|0\rangle^{\otimes q}|01\rangle|01\rangle+|0\rangle^{\otimes q}|01\rangle|10\rangle+|0\rangle^{\otimes q}|01\rangle|11\rangle+|0\rangle^{\otimes q}|10\rangle|00\rangle+|0\rangle^{\otimes q}|10\rangle|01\rangle+|0\rangle^{\otimes q}|10\rangle|10\rangle+|0\rangle^{\otimes q}|10\rangle|11\rangle+|0\rangle^{\otimes q}|11\rangle|00\rangle+|0\rangle^{\otimes q}|11\rangle|01\rangle+|0\rangle^{\otimes q}|11\rangle|10\rangle+|0\rangle^{\otimes q}|11\rangle|11\rangle\right)$$Now as there are $16$ states in superposition here, I need to apply a unitary transform to $4$ states at at a time i.e the first $4$, then the second $4$ , then the third $4$ , then the last $4$. The transform that I apply every time on the four states is on the first register $0\rangle^{\otimes q}$ of each state. The unitary transform between these states is $P$. How would the tensorial notation of this state look like is my question? How do I give the right notation that the operator acts on $4$ states and then the next four is my doubt.

  • $\begingroup$ can somebody suggest ? $\endgroup$
    – Upstart
    Jun 1 '19 at 15:51
  • $\begingroup$ The normalisation factor is off. It is a superposition of 16 states and each is 1/2. $\endgroup$ Jun 2 '19 at 9:53
  • $\begingroup$ Terms of a superposition cannot be separated like that. Generally we talk about applying transforms on a set of qubits. What is the context? $\endgroup$ Jun 2 '19 at 9:54

You'd need to apply $P$ controlled on the second register. I assume you want a different operation for each of the 4 sets? In terms of a tensor product you'd need operators like

$$P_1\otimes |00\rangle\langle00|\otimes I$$ $$P_2\otimes |01\rangle\langle01|\otimes I$$ $$P_3\otimes |10\rangle\langle10|\otimes I$$ $$P_4\otimes |11\rangle\langle11|\otimes I$$


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