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Given the following fusion gate (type-2) which is projecting 2 qubits to an even state

$$F_{ZZ}=(\langle00|+\langle|11|)$$

I would like to find the operator for the bigger space. For example, if I have 4 qubits, and I fusion 2 and 3:

$$U = I^1 \otimes F_{ZZ}^{2,3} \otimes I^4$$

Clearly, U is collapsing the state from 4 qubits to 2 qubits.

Question: How can I build $U$ if the 2 fusion qubits are not neighbors? For example, qubits $1$ and $3$ or qubits $2$ and $4$ or $1$ and $4$. I want a very practical approach, where I can use $U$ to multiply any state $Ux$ without changing the internal order of $x$.

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1 Answer 1

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You can exploit the distributive law. For example, the operator projecting onto the even subspace of qubits $1$ and $3$ can be written as

$$ \langle 0|_1\otimes I_2\otimes\langle 0|_3\otimes I_4+ \langle 1|_1\otimes I_2\otimes\langle 1|_3\otimes I_4\tag1 $$

or more condensed

$$ \sum_{k\in\{0,1\}}\langle k|_1\otimes I_2\otimes\langle k|_3\otimes I_4.\tag2 $$

You can reduce clutter by making identity operators implicit

$$ \langle 00|_{1,3}+\langle 11|_{1,3}.\tag3 $$

Similarly for other pairs of qubits.


The slight awkwardness of the notation is a symptom of the limitations of the traditional one-dimensional notation for linear algebra. See tensor networks and ZX-calculus for alternatives that use two-dimensional diagrams.

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