Quantum Zero Knowledge Proofs

Table of Contents

1. Introduction
2. What Is Zero Knowledge?
3. Classical Zero Knowledge: An Overview
4. Quantum Zero Knowledge: Definition and Goals
5. Models of Quantum Interactive Proofs
6. Honest-Verifier vs General Zero Knowledge
7. Types of Quantum Zero Knowledge
8. QSZK: Quantum Statistical Zero Knowledge
9. QZK: Quantum Computational Zero Knowledge
10. The Role of Quantum Entanglement
11. The Challenge of Simulation
12. Quantum Rewinding and Limitations
13. QSZK-Complete Problems
14. Quantum State Distinguishability
15. Quantum Circuit Distinguishability
16. Comparison with Classical ZK Classes
17. Quantum Zero Knowledge in Cryptographic Protocols
18. Secure Delegation and Quantum ZK
19. Quantum Coin Flipping and Bit Commitment
20. Device-Independent Zero Knowledge
21. Post-Quantum Security Considerations
22. Experimental Implementations
23. Open Problems in Quantum Zero Knowledge
24. Practical Applications and Outlook
25. Conclusion

1. Introduction

Zero knowledge proofs allow a prover to convince a verifier that a statement is true, without revealing any additional information. In the quantum realm, quantum zero knowledge extends this concept to interactive quantum protocols, leveraging quantum states, entanglement, and quantum verifiers.

2. What Is Zero Knowledge?

A zero knowledge proof satisfies three properties:

• Completeness: Honest prover can convince the verifier
• Soundness: Dishonest prover cannot cheat a verifier
• Zero Knowledge: Verifier learns nothing except the validity of the statement

3. Classical Zero Knowledge: An Overview

Introduced by Goldwasser, Micali, and Rackoff (1985), classical zero knowledge has been used in:

• Secure authentication
• Blockchain privacy (e.g., zk-SNARKs)
• Cryptographic proofs

4. Quantum Zero Knowledge: Definition and Goals

Quantum zero knowledge adapts this to quantum settings where:

• The prover and verifier may exchange quantum messages
• The proof may involve quantum states
• The simulator must reproduce the verifier’s view quantumly

5. Models of Quantum Interactive Proofs

Quantum ZK protocols use interactive quantum communication:

• Based on QIP (Quantum Interactive Proofs)
• Prover and verifier exchange qubit messages over multiple rounds

6. Honest-Verifier vs General Zero Knowledge

• Honest-Verifier ZK (HVZK): Assumes verifier follows protocol
• General ZK: Must resist malicious quantum verifiers
• HVZK is easier to construct, but general ZK is stronger

7. Types of Quantum Zero Knowledge

8. QSZK: Quantum Statistical Zero Knowledge

Defined by Watrous:

• Zero knowledge against any (possibly unbounded) quantum verifier
• Simulation must be indistinguishable statistically
• Known to be closed under complement

9. QZK: Quantum Computational Zero Knowledge

• Requires only computational indistinguishability
• More practical for cryptographic settings
• Assumes limits on verifier’s computational power

10. The Role of Quantum Entanglement

Entanglement:

• Complicates simulation
• Can be used to cheat or amplify prover’s power
• Must be carefully controlled in ZK protocols

11. The Challenge of Simulation

Classical ZK relies on rewinding the verifier for simulation.

In quantum:

• Rewinding is problematic due to no-cloning and measurement collapse
• Requires novel quantum rewinding techniques (Watrous, Unruh)

12. Quantum Rewinding and Limitations

Quantum rewinding uses:

• Unitary rewinding
• Semi-classical transcripts
• Measurement-aware branching

But simulation complexity grows significantly compared to classical.

13. QSZK-Complete Problems

Analogous to NP-complete:

• Quantum State Distinguishability
• Quantum Circuit Distinguishability
  These are QSZK-complete problems

14. Quantum State Distinguishability

Given two mixed states \( \rho_0 \), \( \rho_1 \), determine whether:
\[
| \rho_0 – \rho_1 |_{tr} > a \quad \text{or} \quad < b \] Where \( a > b \) and trace distance is used

15. Quantum Circuit Distinguishability

Similar to classical circuit problems:

• Distinguish whether two quantum circuits produce close or far-apart output states
• Proven to be QSZK-complete

16. Comparison with Classical ZK Classes

• QSZK ⊆ PSPACE
• Classical SZK ⊆ AM ∩ coAM
• No known inclusion relationships between QSZK and SZK

17. Quantum Zero Knowledge in Cryptographic Protocols

Applications:

• Quantum authentication
• Zero-knowledge arguments for quantum states
• Quantum money verification
• Blind quantum computation

18. Secure Delegation and Quantum ZK

In delegated quantum computing:

• ZK ensures privacy of user’s computation
• Verifier cannot learn input/output beyond what is revealed

19. Quantum Coin Flipping and Bit Commitment

Quantum ZK plays a role in:

• Coin flipping with fairness guarantees
• Bit commitment protocols (some now proven impossible under certain assumptions)

20. Device-Independent Zero Knowledge

• Uses Bell inequality violations to ensure security
• No need to trust internal mechanics of devices
• Links ZK to device-independent cryptography

21. Post-Quantum Security Considerations

Classical ZK schemes may break under quantum attacks. Quantum ZK aims to:

• Ensure security against quantum adversaries
• Replace RSA/Discrete Log with lattice-based or quantum-native assumptions

22. Experimental Implementations

Although theoretical, there are:

• Experimental quantum ZK implementations (e.g., IBM Qiskit)
• Protocols tested for zero knowledge coin tossing and ZK-based quantum ID

23. Open Problems in Quantum Zero Knowledge

• Is QSZK = SZK?
• Can we build quantum ZK proofs for NP?
• Is quantum rewinding always necessary?

24. Practical Applications and Outlook

• Secure quantum cloud computing
• Quantum-secure authentication systems
• Privacy-preserving quantum protocols
• Basis for quantum blockchain systems?

25. Conclusion

Quantum zero knowledge extends classical notions of provability and secrecy into the quantum world. It offers powerful tools for privacy, cryptography, and secure delegation. Though still evolving, QZK stands as a foundational building block for the future of secure quantum computing and quantum internet protocols.