Quantum verification of NP problems with single photons and linear optics

Quantum computing is seeking to realize hardware-optimized algorithms for application-related computational tasks. NP (nondeterministic-polynomial-time) a complexity class containing many important but intractable problems like the satisfiability of potentially conflict constraints (SAT). According well-founded exponential time hypothesis, verifying an SAT instance size $n$ requires generally complete solution in $O(n)$-bit proof. In contrast, quantum verification algorithms, which encode into bits rather than classical bit strings, can perform task with quadratically reduced information about $\tilde{O}(\sqrt{n})$ qubits. Here we machine single photons and linear optics. By using tunable optical setups, efficiently verify satisfiable unsatisfiable instances achieve clear completeness-soundness gap even presence experimental imperfections. The protocol only unentangled photons, operations on multiple modes at most two-photon joint measurements. These features make suitable photonic realization scalable large problem sizes. Our results open essentially new route towards advantages extend capability computing.