Non-Destructive Zero-Knowledge Proofs on Quantum States, and Multi-Party Generation of Authorized Hidden GHZ States

We propose the first generalization of the famous Non-Interactive Zero-Knowledge (NIZK) proofs to quantum languages (NIZKoQS) and we provide a protocol to prove advanced properties on a received quantum state non-destructively and non-interactively (a single message being sent from the prover to the verifier). In our second orthogonal contribution, we improve the costly Remote State Preparation protocols [CCKW18,CCKW19,GV19] that can classically fake a quantum channel (this is at the heart of our NIZKoQS protocol) by showing how to create a multi-qubits state from a single superposition. Finally, we generalize these results to a multi-party setting and prove that multiple parties can anonymously distribute a GHZ state in such a way that only participants knowing a secret credential can share this state, which could have applications to quantum anonymous transmission, quantum secret sharing, quantum onion routing and more.

💡 Research Summary

This paper introduces the first framework for non‑interactive, non‑destructive zero‑knowledge proofs on quantum states (NIZKoQS) and demonstrates how to use it together with an improved remote state preparation (RSP) technique to enable efficient multi‑party generation of authorized hidden GHZ states.
The authors first observe that verifying non‑trivial properties of a received quantum state without destroying it is impossible with direct measurement, and that existing solutions rely on interactive quantum secure multi‑party computation (QSMPC), which requires multiple rounds of communication. To overcome this, they adapt classical‑client RSP protocols—originally based on Mahadev’s measurement‑based verification—and propose a “single‑superposition” method that prepares an n‑qubit state (a GHZ‑type superposition hidden between |0⟩ and |1⟩) using only one quantum superposition. This reduces the preparation cost from O(n·M·N) (with M≫N) to O((M+n)·N) ≈ O(n·N), achieving a quadratic improvement.
Building on this efficient state preparation, the paper defines NIZKoQS: a prover sends a single classical message (a NIZK proof) that encodes the classical instructions used to generate the quantum state. The verifier, using only classical computation, can check that the received quantum state belongs to a prescribed “quantum language” (e.g., the set of BB84 states or hidden GHZ states) while leaving the state untouched for later quantum processing. The security of the NIZK proof relies on the hardness of the Learning With Errors (LWE) problem, and the proof is zero‑knowledge in the sense that the verifier learns nothing beyond the validity of the statement.
The authors then extend the construction to a multi‑party setting. The system consists of a quantum server (Bob), a trusted classical coordinator (Cupid), and n quantum applicants a₁,…,aₙ. Cupid, using a secret credential (password, signature, etc.), selects a subset S of applicants that will be “supported”. Through a series of protocols—BLIND, BLIND sup, and the main AUTH‑BLIND dist can—each supported applicant receives a share of a hidden GHZ state, while unsupported applicants receive nothing. The protocol guarantees: (i) Blindness – even a colluding server and a coalition of applicants cannot learn which participants are supported; (ii) Robustness – malicious or noisy applicants cannot corrupt the GHZ state shared by the supported set; (iii) Non‑destructiveness – the GHZ state remains intact after verification and can be used for downstream quantum tasks.
Security proofs are provided in the standard simulation‑based paradigm. The LWE‑based encryption of Pauli keys ensures that the classical proof does not leak any information about the quantum state or the credential set. The authors also present an “impossible” protocol to illustrate that achieving perfect zero‑knowledge together with perfect blindness is information‑theoretically unattainable.
Beyond the core contributions, the paper discusses several applications: quantum secret sharing where only authorized parties (identified by a certification authority) obtain the secret; quantum onion routing where intermediate servers blindly generate Bell pairs hidden inside larger states to forward quantum messages without revealing routing information; and anonymous transmission protocols that benefit from the hidden GHZ distribution.
Finally, the paper outlines open problems such as optimizing LWE parameters for realistic quantum hardware, extending the framework to multiple quantum servers for fault tolerance, and quantifying the trade‑off between completeness and soundness in NIZKoQS.
In summary, this work provides a novel, efficient, and versatile toolkit for verifying quantum states without interaction, dramatically reduces the cost of multi‑qubit remote state preparation, and enables anonymous, credential‑based distribution of entanglement—opening new avenues for secure quantum networking and cryptographic protocols.