Quantum interactive proofs using quantum energy teleportation

TL;DR

This work presents a quantum interactive proof scheme that harnesses quantum energy teleportation (QET) alongside quantum state teleportation (QST) to verify ground-state properties of local Hamiltonians over quantum networks. By demonstrating Completeness, Soundness, and Zero-Knowledge, the protocol encodes ground-state data into a two-party QET circuit and shows that a verifier can detect correct proofs via negative energy transfer $\langle E_B\rangle$ while remaining ignorant of the prover's Hamiltonian or ground state. A minimal QET circuit is analyzed in detail, with explicit Hamiltonians $H=H_1+H_2+V$ and parameters $(k,h)$, along with a rigorous Zero-Knowledge argument and finite-shot completeness/soundness results. The framework is extended to quantum state distinguishability and scaled to distributed and multi-prover settings (QMIP$^*$), suggesting practical, secure quantum authentication for near-term quantum networks and devices, under the broader complexity bounds $QIP(3)=PSPACE$ and potential $QMIP^*$ advantages. The approach offers a path to secure, verifiable quantum information processing that can operate with current quantum technologies, while maintaining strong theoretical foundations in quantum complexity theory.

Abstract

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$^*$.

Figures (6)

• Figure 1: Schematic picture of quantum interactive proof by using both QET and QST.
• Figure 2: The full quantum circuit for quantum interactive proofs using the minimal model of quantum energy teleportation. Circuit complexity is 13 and circuit width is 4. In the circuit, each $H$ is an Hadamard gate.
• Figure 3: [Left] Histogram of the energy computed by the 500 random unitary operators that generate initial states. The energy was computed for the Hamiltonian $(k=h=1)$ by 600 different $\theta$s sampled between 0 and $2\pi$ at equal intervals. $E_\text{min}$ corresponds to the lowest energy with the exact ground state at $k=h=1$. Only the lower 0.8% of the total samples generate negative energy value. [Right] Histogram of the fidelity between the exact ground state and the random initial states.
• Figure 4: This figure shows the minimum value for $\langle E_B \rangle$ as a function of $\delta$. When $\delta=0$, it corresponds to the correct $\theta$.
• Figure 5: Expectation values of the interaction terms $\langle V^{C}\rangle,\langle V^{I}\rangle$ and the single qubit operators $\langle H_{1}^{C} \rangle,\langle H_{1}^{I} \rangle$. See eq. \ref{['eq:expectation_val']} for the definition. $\langle V^{C}\rangle$ and $\langle H^{I}\rangle$ are always negative, whereas $\langle V^{I}\rangle$ and $\langle H^{C}\rangle$ are always positive, by which the prover can distinguish whether $Q_1$ or $Q_2$ was applied.

Theorems & Definitions (5)

• Proposition 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3