No-go theorem for iterations of unknown quantum gates

We propose a no-go theorem by proving the impossibility of constructing a deterministic quantum circuit that iterates a unitary oracle by calling it only once. Different schemes are provided to bypass this result and to approximately realize the iteration. The optimal scheme is also studied. An interesting observation is that for a large number of iterations, a trivial strategy like using the identity channel has the optimal performance, and preprocessing, postprocessing, or using resources like entanglement does not help at all. Intriguingly, the number of iterations, when being large enough, does not affect the performance of the proposed schemes.

Figure 1

In a possible scenario, the gate iterator apparatus, ${\mathsf{Iter}}_{\mathit{n}}$, takes $\mathcal{U}, |\psi \rangle$, and $|0\rangle$ as inputs and gives $U^n|\psi \rangle$ as the output.

Figure 2

The Stinespring realization of the quantum circuit of ${\mathsf{Iter}}_n$.

Figure 3

Fidelity of the random guess and the estimation strategies for various orders of iteration in dimensions $d=2,3$, and 4. The advantage of the estimation strategy over the random guess is clear. It is also seen that this advantage tends to decrease with increasing the dimension $d$.

Figure 4

Fidelity of the identity and direct channel strategies. Both schemes have equal performances for $n>d$, and, in fact, they achieve the optimum fidelity in this case.

Figure 5

(a) The channel ${\mathcal{C}}_n$ with one open slot is a replacement for the Stinespring realization shown in Fig. 2; (b) the unitary gate is inserted into the slot to realize ${\mathsf{Iter}}_n$.

Figure 6

The optimum fidelity in approximating ${\mathsf{Iter}}_n$.