Quantum interactive proofs using quantum energy teleportation

We present a simple quantum interactive proof (QIP) protocol using the quantum state teleportation (QST) and quantum energy teleportation (QET) protocols. QET is a technique that allows a receiver at a distance to extract the local energy by local operations and classical communication (LOCC), using the energy injected by the supplier as collateral. QET works for any local Hamiltonian with entanglement and, for our study, it is important that getting the ground state of a generic local Hamiltonian is quantum Merlin Arthur (QMA)-hard. The key motivations behind employing QET for these purposes are clarified. Firstly, in cases where a prover possesses the correct state and executes the appropriate operations, the verifier can effectively validate the presence of negative energy with a high probability (Completeness). Failure to select the appropriate operators or an incorrect state renders the verifier incapable of observing negative energy (Soundness). Importantly, the verifier solely observes a single qubit from the prover's transmitted state, while remaining oblivious to the prover's Hamiltonian and state (Zero-knowledge). Furthermore, the analysis is extended to distributed quantum interactive proofs, where we propose multiple solutions for the verification of each player's measurement. The complexity class of our protocol in the most general case belongs to QIP(3)=PSPACE, hence it provides a secure quantum authentication scheme that can be implemented in small quantum communication devices. It is straightforward to extend our protocol to Quantum Multi-Prover Interactive Proof (QMIP) systems, where the complexity is expected to be more powerful (PSPACE$\subset$QMIP=NEXPTIME). In our case, all provers share the ground state entanglement, hence it should belong to a more powerful complexity class QMIP$^*$.