# An Enhanced Photonic Quantum Finite Automaton
## 7 Conclusions  
In this work, we have presented an enhanced photonic implementation of 1 qfa for the recognition of unary language that significantly improves the performance obtained by the one originally proposed in [6]. The protocol uses the polarization degree of freedom of single photons, and exploits the possibility of detecting not only the horizontal polarization, as in [6], but also the vertical one. The resulting scheme largely outperforms the original automaton for smaller values of the mean number of sent photon \(\langle N_{c}\rangle\). In addition, we have extended the results previously found with a detailed analysis of the conditions for which such 1 qfa can work with high reliability. We have evaluated the minimum number of photons that must be sent in order to solve faithfully the inherent binary discrimination  
Figure 4: Probability of error for the different strategies as functions of the average number of input photon \(\langle N_{c}\rangle\) in a semi-log plot. Red line: \(m=5\); blue line: \(m=11\); green line \(m=23\). (**Top panels**): \(H\) (solid lines) and \(V\) (dashed lines) strategies in the absence of dark counts (**left**) and for \(\langle N_{\text{dc}}\rangle=100\) (**right**). (**Bottom panel**): joint strategy in the case \(\langle N_{\text{dc}}\rangle=100\). The dots on the lines refer to the threshold values evaluated according to (22) and (23). See the text for details.  
problem. As one would expect, the minimum \(\langle N_{c}\rangle\) is smaller for the automaton that relies on the new strategy based on the \(V\) detector.  
In our analysis, we have discussed the presence of dark counts in the detection of both strategies, and we have evaluated their effects both on the probability of error and on the minimum \(\langle N_{c}\rangle\). Eventually, we also examined a joint strategy in which we combine both the \(H\) and the \(V\) detection, which can indeed be used at no additional cost. We have therefore proved that when the number of sent photon is constrained to small values, the \(V\) detection version of the 1 qfa should be preferred.  
Our results pave the way to the effective implementation of 1 qfa using quantum optical platform, thus opening the possibility of processing strings of input symbols using feasible devices and, in turn, to introduce quantum languages and compare the complexity of classes of languages in classical and quantum cases. More generally, as the assessment of the actual power of quantum computers is one of the most significant challenges of quantum technology, implementing quantum automata provides a relevant arena to better understand the computing capabilities offered by quantum devices.  
Conceptualization, A.C., C.M., B.P., S.C., M.G.A.P. and S.O.; Data curation, A.C., S.C. and S.O.; Investigation, A.C., C.M., B.P., S.C., M.G.A.P. and S.O.; Methodology, A.C., C.M., B.P., S.C., M.G.A.P. and S.O.; Supervision, S.O.; Writing--original draft, A.C., C.M., B.P., S.C., M.G.A.P. and S.O.; Writing--review and editing, A.C., C.M., B.P., S.C., M.G.A.P. and S.O. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
[PERSON] is member of GNFM-INdAM. [PERSON] and [PERSON] are members of GNCS-INdAM. We thank [PERSON] for useful discussion.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
1 qfa measure-once one-way quantum finite automaton  
PBS polarizing beam splitter  
dc dark counts