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I have some questions related to superoperators:

  1. What is the differences between quantum operators and superoperators? For instance quantum gates are also unitary operators but can we say quantum operators are superoperators?

  2. Can we write a quantum operator as superoperator?

  3. For instance, I want to write toffoli gate circuit as super operator:

The circuit is here:

enter image description here

And as I asked here before here , I wrote toffoli gate matrix by using one and two qubit gates. But how can I write this circuit as a superoperator? I initially thought that I already wrote the circuit since it is a linear map from H_a to H_b. But I am not sure..

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In this context, a "superoperator" is generally a linear map between linear operators, that is, an element of $\mathrm{Lin}(\mathrm{Lin}(\mathcal X),\mathrm{Lin}(\mathcal Y))$, if $\mathcal X,\mathcal Y$ are input and output Hilbert spaces.

Whether a gate is represented as a superoperator depends on how you are modeling states. If you describe a (pure) quantum state as a vector (modulo multiplication by a complex scalar) $|\psi\rangle\in\mathbb C^N$, then quantum gates are matrices acting on such vectors: $U|\psi\rangle$ for some $U\in\mathbf U(N)\subset \mathbf{GL}(n,\mathbb C)\subset \mathrm{Lin}(\mathbb C^N)$.

If states are instead described as density matrices $\rho$, that is, Hermitian positive-semidefinite operators with unit trace, then $\rho\in\mathrm{Herm}(\mathcal X)\subset\mathrm{Lin}(\mathcal X)$, and "quantum operations" then ought to be "superoperators", that is, linear maps between such spaces, objects of the form $\Phi\in\mathrm{Lin}(\mathrm{Lin}(\mathcal X),\mathrm{Lin}(\mathcal Y))$ satisfying certain properties. In particular, a quantum gate (that is, a unitary operation) corresponds to a superoperator of the form $\rho\mapsto U\rho U^\dagger$ for some $U\in\mathbf U(N)$.

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