## 4 New Challenges for Quantum Photonics  
MiniaturizationWith the progress of chip complexity, researchers have had to face the unavoidable problem of miniaturization, which represents a significant challenge for quantum technologies. [PERSON] et al. [44] implemented a CNOT gate with a footprint of \(4.8\times 4.45\) um\({}^{2}\) that corresponded to about \(\sim\)3\(\lambda\times 3\lambda\). The chip was realized with standard fabrication processes, and it was fully compatible with SOI waveguides. They used silicon-superlattice waveguides, and by choosing different core widths of the waveguides (the symmetry-breaking strategy) they could arrange very dense uncoupled waveguides. To prove the functionality of the chip, the authors measured the truth table of the CNOT gate, demonstrated the generation of the four Bell states, and performed quantum-process tomography with a fidelity result of \((92.5\pm 0.8)\%\). Optical elementsWith the increase in integrated quantum photonics devices, researchers have been developing more efficient optical elements. For example, in 2021, [PERSON] et al. [62] interfaced CMOS silicon and germanium-on-silicon nanophotonics with integrated electronics, to realize a balanced homodyne detector with a bandwidth of 1.7 GHz and shot noise limited up to 9 MHz. All the linear optics needed to realize the homodyne detector, useful for measuring squeezed states, were integrated on the chip [see Figure 3a]. To test the performance of the device, a continuous spectrum of squeezed light was measured from 100 MHz to 9 GHz, and homodyne tomography was performed, to prove the ability of the chip to reconstruct quantum states. In 2023, [PERSON] et al. [63] realized an on-chip electro-optic optical isolator with an achieved optical isolation of 48 dB. The optical isolator was tunable in wavelength from 1510 nm to 1630 nm, keeping an isolation larger than 37 dB. This element will help the integration of laser sources in photonics. EncodingAnother approach to optimizing and increasing quantum resources is to exploit more available degrees of freedom. In this regard, in 2022, [PERSON] et al. [64] realized the first multimode implementation of a 2-qubit quantum gate. They built a very compact CNOT gate with waveguides able to support two orthogonal transverse modes, in which the qubits were encoded. To prove the functionality of the device, two separate qubits were entangled, with an average fidelity of \((89\pm 2)\%\). The idea, to enlarge the dimension of the suitable Hilbert space by means of further degrees of freedom, was also exploited by [PERSON] et al. in Ref. [65], who realized a SWAP gate by encoding the information in the two degrees of freedom: polarization and spatial momentum. The SWAP gate was engineered, concatenating three CNOT gates. An average fidelity of \((97.3\pm 0.3)\%\) of the output spatial-momentum states validated the single-qubit conversion from polarization qubit to spatial-momentum qubit. A process fidelity of (\(94.9\pm 1.0\))% was achieved for the on-chip two-qubit SWAP operation, with a process purity of (\(93.3\pm 1.0\))%. Swapping between path and polarization degrees of freedom was also used for the first quantum photonic interconnection [66]. High-fidelity entanglement manipulation and distribution between two separate photonic chips was realized. Using on each chip a two-dimensional grating coupler, entangled states encoded in the path degree of freedom and generated on the chip were delivered to another chip by interconverting between polarization and path degrees of freedom. _Protocols--_The possibilities offered by the new devices have opened the way to reconsidering schemes and protocols able to claim quantum supremacy. In this regard, graphs are discrete mathematical tools able to model quantum-mechanical devices and systems.