Quantum verification of NP problems with single photons and linear optics

TL;DR

This paper demonstrates a quantum verification protocol for NP problems, specifically SAT, using single photons and linear optics, showcasing potential for scalable optical quantum computing and quantum advantage.

Contribution

It presents the first implementation of a quantum SAT verification machine with single photons and linear optics, enabling efficient verification of satisfiable and unsatisfiable instances.

Findings

Successfully verified SAT instances with optical setup

Achieved a clear completeness-soundness gap despite imperfections

Protocol is scalable and requires only unentangled photons and linear operations

Abstract

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