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 with the advances in high-dimensional quantum information manipulation and large scale linear-optical systems. Our results open an essentially new route toward quantum advantages and extend the computational capability of optical quantum computing.
Highlights
Quantum computing has been found to unprecedentedly speed-up classically intractable computational tasks[1,2,3,4,5,6,7]
nondeterministic polynomial-time (NP) can be abstracted as a proof system which models computation as exchange of messages between the prover and the verifier
Two properties are required in a QMA protocol: (i) Completeness: if the instance is satisfiable, there exist a proof such that Arthur accepts with at least some high probability c; (ii) Soundness: if the instance is not satisfiable, for any proof Arthur accepts with at most some probability s
Summary
Quantum computing has been found to unprecedentedly speed-up classically intractable computational tasks[1,2,3,4,5,6,7]. Error-corrected quantum computers is still challenging, the community seeks practical uses of noisy intermediate-scale quantum (NISQ) technologies in computational problems of interest and importance[5]. NP can be abstracted as a proof system which models computation as exchange of messages between the prover and the verifier. Verifying the correctness of a proof is a foundational computational model underpinning both the complexity theory and applications such as delegated
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