Bosonic unstructured search: quantum enhancement without entanglement
We consider unstructured search using multimode bosonic quantum systems under energy constraints. The task is to locate an unknown marked mode among $N$ possibilities using arbitrary control, subject to a time-averaged mean-boson-number budget $n_{\text{b}}\gg 1$. We analyze a continuous-time phase-shift oracle of strength $B_{0}$, with cost given by the total oracle time, and a discrete oracle with fixed phase $ϕ_{0}$, with cost given by the number of queries. We give algorithms that identify the target with query complexity $O(\sqrt{N}/{n_{\text{b}}}B_{0})$ for continuous-time oracles or $O(\sqrt{N}/n_{\text{b}} |{\sin(ϕ_{0}/{2})}|)$ for discrete oracles with $n_{\text{b}} |{\sin(ϕ_{0}/{2})}|=O(1)$. In contrast, for discrete oracles with $n_{\text{b}} |{\sin(ϕ_{0}/{2})}|=Ω(1)$, we provide an algorithm using $O(\max\{1,\sqrt{N/n_{\text{b}}|{\sin(ϕ_{0}/{2})}|}\})$ queries. We also prove matching lower bounds on the query complexity, showing that these algorithms are asymptotically optimal. Furthermore, we show that, in the absence of loss, non-Gaussianity is necessary to achieve the optimal joint query scaling in $N$ and $n_{\text{b}}$, but entanglement is not. Under uniform boson loss at rate $γ$, we prove a query lower bound of $Ω( γN / n _{\text{b}} B_{0}^{2} )$, attained for $γ= Ω(B_{0} \sqrt{n _{\text{b}}/N})$ using coherent-state probes. Finally, we propose experimental implementations using multipass optical microscopy or superconducting circuits as potential routes to speeding up imaging or device calibration with a low boson-number budget.